# The Logistic Map: From Fixed Points to Chaos via Period-Doubling > One equation, next year's population equal to a growth rate times this year's times whatever room is left, taken from the setting where everything settles onto a single value all the way into chaos. The cobweb construction carries the whole argument: we watch a fixed point hold because the curve crosses the diagonal gently, watch that grip weaken as the growth rate rises, and watch an orbit spiral away from a point that still satisfies the equation exactly. From there the period doubles to two, to four, to eight, the gaps between doublings shrink by a factor near 4.669, and past the point where they pile up the orbit stops repeating altogether. The lecture ends inside the chaotic band, on a window of period three and the copy of the entire picture hidden in it. No background in dynamical systems is assumed. - Canonical watch page: [The Logistic Map: From Fixed Points to Chaos via Period-Doubling](https://academa.ai/lectures/period-doubling-logistic-map) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Mathematics - Published: 2026-08-28T22:51:38.000Z - Updated: 2026-08-28T22:51:38.000Z - Duration: PT1222S (20 minutes 22 seconds) - Chapters: 6 - Views: 0 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M14TZCA2S8TZQ5H5XEFF4NZW/0/dark/master.m3u8) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M14TZCA2S8TZQ5H5XEFF4NZW/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M14TZCA2S8TZQ5H5XEFF4NZW/0/dark/poster.jpg) ## Description Watch one population equation lose its fixed point, double through two, four and eight cycles, and slide into chaos. ## Chapters - [00:00–03:30.207 · One Innocent Equation](https://academa.ai/lectures/period-doubling-logistic-map?t=0) - [03:30.207–07:32.071 · Why It Holds, and Why It Lets Go](https://academa.ai/lectures/period-doubling-logistic-map?t=210.2067708333333) - [07:32.071–11:1.154 · The First Split](https://academa.ai/lectures/period-doubling-logistic-map?t=452.07149999999996) - [11:1.154–14:36.375 · Again, and Again, and Again](https://academa.ai/lectures/period-doubling-logistic-map?t=661.1541874999999) - [14:36.375–17:28.741 · The Whole Picture, and What Lies Past It](https://academa.ai/lectures/period-doubling-logistic-map?t=876.3748333333332) - [17:28.741–20:22 · Windows of Order](https://academa.ai/lectures/period-doubling-logistic-map?t=1048.7411874999998) ## Transcript ### [00:00 · One Innocent Equation](https://academa.ai/lectures/period-doubling-logistic-map?t=0) Here is an equation you could write on a biology exam without anybody blinking. Next year's population is this year's, multiplied by a growth rate, multiplied by however much room is left. Nothing about it looks dangerous. And yet if we turn that growth rate up slowly and watch, this little rule stops settling down and starts doing something nobody had any right to expect. So let us write it down properly. Call x sub n the population in year n, measured not in animals but as a fraction of the most the environment could support. Then next year's is r times x n, times one minus x n. Read the two factors separately. The first is plain growth: more of them this year means more of them next year. The second is crowding. While the population is small that factor sits near one and hardly bites, and as the population approaches capacity it drags the whole product down towards nothing. Two things are fixed for the rest of the lecture. The population x always lies between zero and one, nought being extinction and one being a full house. And r is a knob, which we shall turn from one up to four and no further. The equation has a name, the logistic map, and the word map is worth noticing. This is not a differential equation and there is no time flowing smoothly along. There is this year, and there is next year, and a rule joining them. Here it is drawn as a picture, with the knob set at two point six. Input along the bottom, output up the side. Feed in a population, read off the one that follows it. It is a hump. A small population gives a small answer. A population near capacity gives a small answer too, because there is nowhere to put the young. And the best year of all comes from starting somewhere near the middle. Now the only thing that ever happens in this lecture is that we apply the rule again, and again, and again. So start somewhere. Say a fifth of capacity, nought point two. One turn of the handle takes nought point two to nought point four one six. On the picture that is going straight up from nought point two until you hit the curve. There is the answer, measured up the side. And here is the move that makes the whole construction work. That answer has to become next year's input, so somehow it must get from the vertical axis back onto the horizontal one. The green line, y equals x, does exactly that. Go across to it, and you are now standing above nought point four one six. Then climb to the curve for the year after, and cross back down to the line, and carry on. Each pair of moves is one year. Let it run for ten years and watch where it goes. The staircase spirals inwards and parks itself exactly where the curve crosses the line. A little over six tenths of capacity, and it stays there for ever after. Start anywhere else and it goes to the same place. That crossing is the whole story of the settled state, because a crossing is precisely a year in which next year's population equals this year's. So there are two questions. What is it? And why did the staircase walk towards it rather than away? ### [03:30.207 · Why It Holds, and Why It Lets Go](https://academa.ai/lectures/period-doubling-logistic-map?t=210.2067708333333) Start with the settled state, since that is what we just watched happen. A population that has stopped moving is one where next year equals this year. Call that value x star. Then x star must equal r x star times one minus x star. Now x star could of course be zero, and we shall come back to that in a moment. Otherwise divide both sides by it. One equals r times one minus x star. Rearrange, and there it is. The settled population is one minus one over r. With the knob at two point six that comes to nought point six one five, which is precisely where our staircase parked. But now the question this whole lecture turns on. Why did the staircase walk towards that point? Nothing in the algebra we just did says it has to. Zero satisfies the very same equation, and orbits run away from zero as fast as they can go. The answer is the steepness of the curve where it crosses the line. So differentiate the rule. F prime of x is r times one minus two x. Substitute the fixed point, one minus one over r, and the r cancels beautifully. The slope at the crossing is simply two minus r. At two point six that is nought point six, with a minus sign in front. Here is what that number does for a living. Suppose you are a small distance away from x star. Very near a crossing the curve is almost exactly its own tangent line, so one turn of the handle takes that distance and multiplies it by the slope. Multiply a distance by minus nought point six and it shrinks, and it changes sides. Multiply again and it shrinks and changes sides again. Which is exactly the inward spiral we watched: over, under, over, under, closing in. And notice there are two flavours. A positive slope smaller than one shrinks distances without changing sides, and the staircase climbs to the crossing like an ordinary flight of stairs. A negative one gives the spiral. Either way, less than one in size means the point holds. So the condition is this, and it is the only piece of theory in the whole lecture. The fixed point holds on as long as the size of two minus r is less than one, and that is the same as saying r lies between one and three. Watch it get feeble. Turn the knob up to two point nine. The crossing has shifted a little to the right, and the slope there is now minus nought point nine. Start from a fifth of capacity again. Each year the distance from the crossing shrinks by only a tenth of itself, so the staircase takes an age. But a tenth of a tenth of a tenth is nothing in the end, and it still parks. At r exactly three, the slope at the crossing is exactly minus one. Distances are no longer shrunk at all. They are simply reflected, over and back, over and back, at the same size for ever. The fixed point has stopped pulling, and it has not yet started pushing. So go past it. Three point two. The slope at the crossing is minus one point two, and each year the distance from it is multiplied by one point two. Now start almost exactly on the fixed point, a hair above it, and watch what the construction does. It leaves. The staircase spirals outwards, away from a value that satisfies the fixed point equation perfectly well. The point has not gone anywhere. It has simply lost the ability to hold on to anything, and everything near it is now on its way out. Which leaves an obvious gap in the story. The population is not going to the fixed point. It is trapped between zero and one so it cannot run off. Where on earth is it going? ### [07:32.071 · The First Split](https://academa.ai/lectures/period-doubling-logistic-map?t=452.07149999999996) So here is the knob at three point two again, and here is an orbit starting from a fifth of capacity. I have drawn the first ten years in grey, because they are only the transient: the population finding its feet. And here is what it finds. Not a point at all. A square. Read that square as a timetable. Up to the curve and across to the line is one year. Then again is the next year. And the fourth move brings you back exactly where you started, so year three is a repeat of year one. So the population never settles. It alternates. A lean year, a crowded year, a lean year, a crowded year, without end. Nought point five one three, then nought point eight, then nought point five one three again. And notice how far apart those two values are. This is not the old fixed point with a small wobble on top. The population is swinging from half capacity to four fifths and back, every single year, and it will do that for ever. The two values are locked to each other. F of p is q, and f of q is p, so applying the rule twice brings each of them home. This is called a two cycle, and it appeared at exactly the value of r where the fixed point lost its grip. That is no coincidence, and there is a clean way to see why. If a population repeats every two years, then applying the rule twice must bring it back to itself. So let us look at the map that takes this year straight to the year after next. That is f of f of x, and here it is in red, sitting on top of the original hump. It has two humps of its own, and it meets the line y equals x in four places. Two of those crossings are old friends. Zero is fixed under one step, so it is certainly fixed under two. And so is x star, the fixed point we already had, which is still sitting there at nought point six eight seven five. The other two are new, and they are our lean year and our crowded year. Each is a perfectly good fixed point of this doubled map. And look at how the red curve meets the line at those two. It crosses gently. A shallow slope means attracting. Now here is the event at r equal to three. Below three the red curve looked like this: tucked up against the line, touching it only at x star. As r passed three it pushed straight through, and made two new crossings, one on each side. One solution became three, and the middle one was the one that failed. It is worth saying plainly what did not happen. Nothing was destroyed and no solution went missing. The fixed point is exactly where it always was, and no population will ever go to it again. So here is the pattern, and it is about to happen over and over. A point holds on while the slope there is shallower than one in size. When the slope reaches minus one, the point does not vanish. It stays, it turns repelling, and a cycle of twice the period is born around it. Which invites the obvious question. Our new two cycle has a slope of its own, measured on that red curve. What do you suppose happens when that one reaches minus one? ### [11:1.154 · Again, and Again, and Again](https://academa.ai/lectures/period-doubling-logistic-map?t=661.1541874999999) Turn the knob again. At r equal to three point four four nine the two cycle fails, in exactly the way the fixed point failed. It is the same mechanism one level down. The two cycle is a fixed point of the doubled map, so it has a slope of its own, and that slope walks down through minus one as we turn. When it crosses, each of the two values sheds a pair, and we have four. Here is the knob at three point five. Grey for the transient again, and then the settled orbit in yellow. Four points, and the population visits them in a fixed order. Nought point three eight three, then nought point eight two seven, then nought point five zero one, then nought point eight seven five, and then back to the first. So the years no longer come in two kinds. They come in four, and the pattern closes only every fourth year. A good year, a terrible year, a middling year, a very good year, and only then a repeat. Push the knob a little further, to three point five five, and every one of those four splits again. The orbit now needs eight years to close. The picture is already becoming hard to read, and that is exactly the point. So look instead at how little I had to turn the knob. From three to three point four five was a long way. From three point four five to three point five four was a fifth of that. From there to three point five six, shorter still. The doublings are arriving faster and faster. Which means we should stop looking at pictures for a moment and look at the numbers. Here are the values of r at which the period doubles, the gap from each one to the next, and the ratio of one gap to the one after it. Two to four, then four to eight, then eight to sixteen, then sixteen to thirty two. The gaps collapse: nearly a half at the start, and under a thousandth four steps later. But the ratio does not collapse with them. Look down the last column. Four point seven five, four point six six, four point six seven, and then it simply stops moving. Carry the cascade on and that ratio converges on a definite number. Four point six six nine two zero one, and on it goes. It is called the Feigenbaum constant, and it is written delta. Two things about it deserve a pause, and the first is only arithmetic. Because the gaps shrink by a fixed factor every time, they behave like a geometric series, and a geometric series with ratio bigger than one in the denominator adds up to something finite. So the doublings do not go on for ever in r. They pile up at a finite value, three point five six nine nine and a bit, and by the time you arrive there the period has doubled infinitely often. The second thing is much stranger. That number has nothing whatever to do with our particular hump. Replace r x times one minus x with a sine arch, or with almost any smooth curve that has one rounded maximum, run the same cascade, and out comes the same constant. Mitchell Feigenbaum found that in nineteen seventy five, on a pocket calculator, and it took him three years to get it published. Which leaves the place where the doublings pile up, and everything past it. ### [14:36.375 · The Whole Picture, and What Lies Past It](https://academa.ai/lectures/period-doubling-logistic-map?t=876.3748333333332) Let me put every one of those answers on one picture. Along the bottom, the knob. Up the side, the population, but only where it ends up: the values the orbit visits once the transient is over. For r below three there is one value, and it climbs gently as we turn. That is the fixed point, one minus one over r, drawn once for every setting of the knob. At three it splits. Above three there are two values, the lean year and the crowded year, and the gap between them widens as we go. At three point four four nine each of those splits again, and there are four. You can see the tuning fork shape repeating at a smaller size. And then the splittings come so fast that I cannot draw them. Eight, sixteen, thirty two, all crammed into this sliver, all of them piling up at three point five six nine nine. Past that line, something else entirely. The orbit no longer settles onto any finite list of values at all. It wanders over a whole band, and that band is where the rest of this lecture lives. So let us go inside it, and put the knob at three point nine. Same rule, same construction, same everything. Only r has changed. Twenty years of it. No square. No loop of any size. The staircase covers the box, and shows no sign at all of closing up. And nothing random went into this. Every one of those legs was computed from the one before it by multiplying three numbers together. Which is the thing people find hardest to accept. There is no dice roll anywhere in the logistic map. The unpredictability is manufactured out of arithmetic, and here is the property that does the manufacturing. Take two starting populations that differ in the third decimal place. Nought point four zero zero, and nought point four zero one. Run both for twenty years, and plot them against the year. For the first ten years or so you would not know there were two of them. They rise together and crash together. Then, somewhere around here, they part company. And after that they have nothing to say to each other. One is booming while the other has collapsed. The tiny difference we started with has been magnified until it is the whole picture. It grows by roughly half again every single year. So a difference of one part in a thousand takes about fifteen years to become total, and a difference of one part in a million takes about thirty. Improving your measurement buys you almost nothing. The rule is completely deterministic and completely useless for long term prediction, and those two facts sit together quite comfortably. But the diagram we drew has one more surprise in it, and it is the one that made people take all of this seriously. ### [17:28.741 · Windows of Order](https://academa.ai/lectures/period-doubling-logistic-map?t=1048.7411874999998) Here is the right hand end of that diagram, magnified. Everything from three point seven up to four, and at first glance it is all band. Except that it is not. Look here, a little past three point eight two. Right in the middle of the chaos there is a clean gap. And running through the gap, three sharp curves. That is a window: a stretch of r where the wandering stops dead and the population goes back to a strict repeating schedule, this time of length three. Which is worth a moment of disbelief. Either side of this narrow strip the very same rule produces something with no pattern in it whatsoever. Let us go and look at the orbit itself. Knob at three point eight three, and the same construction we have used all along. Grey for the transient, yellow for what it settles on. Three points. Nought point one five six, then nought point five zero five, then nought point nine five seven, and straight back to the first. A crash, a recovery, a boom, and round again, for ever. That is as orderly as anything we saw at r equal to two point six. Now turn the knob by four hundredths, to three point eight seven, and watch the schedule evaporate. There it is, wandering again. The window really is narrow, and it really is surrounded on both sides by chaos. Now go back to those three curves and follow them to the right hand end of the window. Each of them splits into two, and then into four, and it accumulates and gives way to chaos again, with the very same ratio four point six six nine governing the splittings. So the window is not merely an island of order. It contains a small copy of the entire picture we spent this lecture building. And magnify that copy and you find windows inside it, each holding a smaller copy again. The structure goes all the way down. There is no scale at which it becomes simple. So here is what one innocent equation turned out to hold. For a small growth rate, a single settled population, and a staircase that walks to it because the curve crosses the line gently. As the rate grows, that crossing steepens until it can no longer hold anything, and a two cycle takes over. Then the two cycle fails the same way, and its successor after it, faster and faster, with gaps shrinking by a factor that has nothing to do with our equation. Past the point where those pile up, orbits that never repeat and that forget where they started. And inside the chaos, windows of perfect order, each containing the whole story again in miniature. None of that was put in. All of it was sitting inside r x times one minus x, waiting for somebody to turn the knob. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M14TZCA2S8TZQ5H5XEFF4NZW/0/semantic.json) Record version: 1. Render attempt: 0. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video. ### Scene 1: [One Innocent Equation](https://academa.ai/lectures/period-doubling-logistic-map?t=0) Span: 00:00–03:30.207 (0s–210.2067708333333s). #### Objects - box: an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25) - card: a Title that says "Dynamical Systems — The Logistic Map: From Fixed Points to Chaos via Period-Doubling" - diag: a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)) - head\_rule: a Heading that says "One Rule, Applied Over and Over" - head\_run: a Heading that says "Turning the Handle" - line: a Line \[yellow\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.41600000000000004)) - line\_10: a Line \[yellow\] drawn in box (start=(0.621370746771992, 0.6116997687528234), end=(0.6116997687528234, 0.6116997687528234)) - line\_11: a Line \[yellow\] drawn in box (start=(0.6116997687528234, 0.6116997687528234), end=(0.6116997687528234, 0.617560220317471)) - line\_12: a Line \[yellow\] drawn in box (start=(0.6116997687528234, 0.617560220317471), end=(0.617560220317471, 0.617560220317471)) - line\_13: a Line \[yellow\] drawn in box (start=(0.617560220317471, 0.617560220317471), end=(0.617560220317471, 0.6140669459571599)) - line\_14: a Line \[yellow\] drawn in box (start=(0.617560220317471, 0.6140669459571599), end=(0.6140669459571599, 0.6140669459571599)) - line\_15: a Line \[yellow\] drawn in box (start=(0.6140669459571599, 0.6140669459571599), end=(0.6140669459571599, 0.6161707027840165)) - line\_16: a Line \[yellow\] drawn in box (start=(0.6140669459571599, 0.6161707027840165), end=(0.6161707027840165, 0.6161707027840165)) - line\_17: a Line \[yellow\] drawn in box (start=(0.6161707027840165, 0.6161707027840165), end=(0.6161707027840165, 0.614911356318136)) - line\_18: a Line \[yellow\] drawn in box (start=(0.6161707027840165, 0.614911356318136), end=(0.614911356318136, 0.614911356318136)) - line\_19: a Line \[yellow\] drawn in box (start=(0.614911356318136, 0.614911356318136), end=(0.614911356318136, 0.6156679884917287)) - line\_2: a Line \[yellow\] drawn in box (start=(0.2, 0.41600000000000004), end=(0.41600000000000004, 0.41600000000000004)) - line\_20: a Line \[yellow\] drawn in box (start=(0.614911356318136, 0.6156679884917287), end=(0.6156679884917287, 0.6156679884917287)) - line\_3: a Line \[yellow\] drawn in box (start=(0.41600000000000004, 0.41600000000000004), end=(0.41600000000000004, 0.6316544000000001)) - line\_4: a Line \[yellow\] drawn in box (start=(0.41600000000000004, 0.6316544000000001), end=(0.6316544000000001, 0.6316544000000001)) - line\_5: a Line \[yellow\] drawn in box (start=(0.6316544000000001, 0.6316544000000001), end=(0.6316544000000001, 0.604934509297664)) - line\_6: a Line \[yellow\] drawn in box (start=(0.6316544000000001, 0.604934509297664), end=(0.604934509297664, 0.604934509297664)) - line\_7: a Line \[yellow\] drawn in box (start=(0.604934509297664, 0.604934509297664), end=(0.604934509297664, 0.621370746771992)) - line\_8: a Line \[yellow\] drawn in box (start=(0.604934509297664, 0.621370746771992), end=(0.621370746771992, 0.621370746771992)) - line\_9: a Line \[yellow\] drawn in box (start=(0.621370746771992, 0.621370746771992), end=(0.621370746771992, 0.6116997687528234)) - meaning: a Panel that says "$x\_n$ is the population as a fraction of what the place can hold, so $x\_n$ lies between $0$ and $1$. The number $r$ is a growth rate, and it is the knob we are going to turn." - para: a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)) - point: a Point \[yellow\] drawn in box (location=(0.1, 0.234)) - point\_2: a Point \[yellow\] drawn in box (location=(0.9, 0.234)) - point\_3: a Point \[yellow\] drawn in box (location=(0.5, 0.65)) - rule: a Math \[text\] that says "$x\_(n+1) = r thin x\_n (1 - x\_n)$" - s1: a Math \[text\] that says "$x\_0 = 0.200$" - s2: a Math \[text\] that says "$x\_1 = 2.6 (0.200)(0.800) = 0.416$" - s3: a Math \[text\] that says "$x\_2 = 2.6 (0.416)(0.584) = 0.632$" - star: a Point \[red\] labelled "x^\*" drawn in box (location=(0.615385, 0.615385)) - start\_dot: a Point \[yellow\] labelled "x\_0" drawn in box (location=(0.2, 0.0)) #### Beats ##### [00:00](https://academa.ai/lectures/period-doubling-logistic-map?t=0) Narration: Here is an equation you could write on a biology exam without anybody blinking. Next year's population is this year's, multiplied by a growth rate, multiplied by however much room is left. Nothing about it looks dangerous. And yet if we turn that growth rate up slowly and watch, this little rule stops settling down and starts doing something nobody had any right to expect. Board: Empty. Actions: - [00:00](https://academa.ai/lectures/period-doubling-logistic-map?t=0): card is shown on the screen, written out. - [00:1.5](https://academa.ai/lectures/period-doubling-logistic-map?t=1.5): card: enter:write-left-to-right. - [00:23.185](https://academa.ai/lectures/period-doubling-logistic-map?t=23.185): card is hidden from the screen — left the board. ##### [00:24.385](https://academa.ai/lectures/period-doubling-logistic-map?t=24.384999999999998) Narration: So let us write it down properly. Call x sub n the population in year n, measured not in animals but as a fraction of the most the environment could support. Then next year's is r times x n, times one minus x n. Board: Unchanged from the preceding beat in this scene. Actions: - [00:24.385](https://academa.ai/lectures/period-doubling-logistic-map?t=24.384999999999998): head\_rule is shown on the screen, written out. - [00:35.856](https://academa.ai/lectures/period-doubling-logistic-map?t=35.855999999999995): rule is shown on the screen, written out. ##### [00:42.052](https://academa.ai/lectures/period-doubling-logistic-map?t=42.052) Narration: Read the two factors separately. The first is plain growth: more of them this year means more of them next year. The second is crowding. While the population is small that factor sits near one and hardly bites, and as the population approaches capacity it drags the whole product down towards nothing. Board: rule — a Math \[text\] that says "$x\_(n+1) = r thin x\_n (1 - x\_n)$"; head\_rule — a Heading that says "One Rule, Applied Over and Over" Actions: - [00:45.872](https://academa.ai/lectures/period-doubling-logistic-map?t=45.872): rule (the "r thin x\_n" part) is emphasized. - [00:50.876](https://academa.ai/lectures/period-doubling-logistic-map?t=50.876): rule (the "(1 - x\_n)" part) is emphasized. - [00:50.876](https://academa.ai/lectures/period-doubling-logistic-map?t=50.876): rule (the "r thin x\_n" part) is no longer emphasized. - [01:0.941](https://academa.ai/lectures/period-doubling-logistic-map?t=60.940999999999995): rule (the "(1 - x\_n)" part) is no longer emphasized. ##### [01:2.354](https://academa.ai/lectures/period-doubling-logistic-map?t=62.354) Narration: Two things are fixed for the rest of the lecture. The population x always lies between zero and one, nought being extinction and one being a full house. And r is a knob, which we shall turn from one up to four and no further. Board: Unchanged from the preceding beat in this scene. Actions: - [01:3.399](https://academa.ai/lectures/period-doubling-logistic-map?t=63.399): meaning is shown on the screen, written out. ##### [01:18.779](https://academa.ai/lectures/period-doubling-logistic-map?t=78.77850000000001) Narration: The equation has a name, the logistic map, and the word map is worth noticing. This is not a differential equation and there is no time flowing smoothly along. There is this year, and there is next year, and a rule joining them. Board: rule — a Math \[text\] that says "$x\_(n+1) = r thin x\_n (1 - x\_n)$"; meaning — a Panel that says "$x\_n$ is the population as a fraction of what the place can hold, so $x\_n$ lies between $0$ and $1$. The number $r$ is a growth rate, and it is the knob we are going to turn."; head\_rule — a Heading that says "One Rule, Applied Over and Over" Actions: - None. ##### [01:33.856](https://academa.ai/lectures/period-doubling-logistic-map?t=93.85600000000001) Narration: Here it is drawn as a picture, with the knob set at two point six. Input along the bottom, output up the side. Feed in a population, read off the one that follows it. Board: Unchanged from the preceding beat in this scene. Actions: - [01:33.856](https://academa.ai/lectures/period-doubling-logistic-map?t=93.85600000000001): box is shown on the screen, written out. - [01:35.052](https://academa.ai/lectures/period-doubling-logistic-map?t=95.052): para is shown on the screen, drawn. ##### [01:45.613](https://academa.ai/lectures/period-doubling-logistic-map?t=105.61300000000001) Narration: It is a hump. A small population gives a small answer. A population near capacity gives a small answer too, because there is nowhere to put the young. And the best year of all comes from starting somewhere near the middle. Board: rule — a Math \[text\] that says "$x\_(n+1) = r thin x\_n (1 - x\_n)$"; meaning — a Panel that says "$x\_n$ is the population as a fraction of what the place can hold, so $x\_n$ lies between $0$ and $1$. The number $r$ is a growth rate, and it is the knob we are going to turn."; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_rule — a Heading that says "One Rule, Applied Over and Over"; para — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)) Actions: - [01:47.354](https://academa.ai/lectures/period-doubling-logistic-map?t=107.35400000000001): point is shown on the screen, grown. - [01:49.354](https://academa.ai/lectures/period-doubling-logistic-map?t=109.35400000000001): point is hidden from the screen. - [01:51.209](https://academa.ai/lectures/period-doubling-logistic-map?t=111.20900000000003): point\_2 is shown on the screen, grown. - [01:53.209](https://academa.ai/lectures/period-doubling-logistic-map?t=113.20900000000003): point\_2 is hidden from the screen. - [01:59.092](https://academa.ai/lectures/period-doubling-logistic-map?t=119.09200000000003): point\_3 is shown on the screen, grown. - [01:59.928](https://academa.ai/lectures/period-doubling-logistic-map?t=119.92800000000001): rule moves to a new place on the board. - [01:59.928](https://academa.ai/lectures/period-doubling-logistic-map?t=119.92800000000001): head\_rule is hidden from the screen — left the board. - [01:59.928](https://academa.ai/lectures/period-doubling-logistic-map?t=119.92800000000001): meaning is hidden from the screen — left the board. ##### [02:0.528](https://academa.ai/lectures/period-doubling-logistic-map?t=120.52800000000002) Narration: Now the only thing that ever happens in this lecture is that we apply the rule again, and again, and again. So start somewhere. Say a fifth of capacity, nought point two. Board: rule — a Math \[text\] that says "$x\_(n+1) = r thin x\_n (1 - x\_n)$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); para — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); point\_3 — a Point \[yellow\] drawn in box (location=(0.5, 0.65)) Actions: - [02:0.528](https://academa.ai/lectures/period-doubling-logistic-map?t=120.52800000000002): head\_run is shown on the screen, written out. - [02:1.092](https://academa.ai/lectures/period-doubling-logistic-map?t=121.09200000000003): point\_3 is hidden from the screen. - [02:9.445](https://academa.ai/lectures/period-doubling-logistic-map?t=129.44500000000005): s1 is shown on the screen, written out. - [02:9.781](https://academa.ai/lectures/period-doubling-logistic-map?t=129.78100000000003): start\_dot is shown on the screen, written out. ##### [02:12.46](https://academa.ai/lectures/period-doubling-logistic-map?t=132.45950000000002) Narration: One turn of the handle takes nought point two to nought point four one six. On the picture that is going straight up from nought point two until you hit the curve. There is the answer, measured up the side. Board: rule — a Math \[text\] that says "$x\_(n+1) = r thin x\_n (1 - x\_n)$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); para — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); s1 — a Math \[text\] that says "$x\_0 = 0.200$"; head\_run — a Heading that says "Turning the Handle"; start\_dot — a Point \[yellow\] labelled "x\_0" drawn in box (location=(0.2, 0.0)) Actions: - [02:15.548](https://academa.ai/lectures/period-doubling-logistic-map?t=135.54800000000006): s2 is shown on the screen, written out. - [02:18.671](https://academa.ai/lectures/period-doubling-logistic-map?t=138.67100000000005): line is shown on the screen, written out. ##### [02:25.001](https://academa.ai/lectures/period-doubling-logistic-map?t=145.00050000000002) Narration: And here is the move that makes the whole construction work. That answer has to become next year's input, so somehow it must get from the vertical axis back onto the horizontal one. The green line, y equals x, does exactly that. Board: rule — a Math \[text\] that says "$x\_(n+1) = r thin x\_n (1 - x\_n)$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); para — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); s1 — a Math \[text\] that says "$x\_0 = 0.200$"; s2 — a Math \[text\] that says "$x\_1 = 2.6 (0.200)(0.800) = 0.416$"; head\_run — a Heading that says "Turning the Handle"; start\_dot — a Point \[yellow\] labelled "x\_0" drawn in box (location=(0.2, 0.0)); line — a Line \[yellow\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.41600000000000004)) Actions: - [02:36.372](https://academa.ai/lectures/period-doubling-logistic-map?t=156.37200000000004): diag is shown on the screen, written out. ##### [02:40.536](https://academa.ai/lectures/period-doubling-logistic-map?t=160.536) Narration: Go across to it, and you are now standing above nought point four one six. Then climb to the curve for the year after, and cross back down to the line, and carry on. Each pair of moves is one year. Board: rule — a Math \[text\] that says "$x\_(n+1) = r thin x\_n (1 - x\_n)$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); para — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); s1 — a Math \[text\] that says "$x\_0 = 0.200$"; s2 — a Math \[text\] that says "$x\_1 = 2.6 (0.200)(0.800) = 0.416$"; head\_run — a Heading that says "Turning the Handle"; start\_dot — a Point \[yellow\] labelled "x\_0" drawn in box (location=(0.2, 0.0)); line — a Line \[yellow\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.41600000000000004)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)) Actions: - [02:41.117](https://academa.ai/lectures/period-doubling-logistic-map?t=161.11700000000002): line\_2 is shown on the screen, written out. - [02:46.527](https://academa.ai/lectures/period-doubling-logistic-map?t=166.52700000000002): line\_3 is shown on the screen, written out. - [02:47.618](https://academa.ai/lectures/period-doubling-logistic-map?t=167.61800000000002): s3 is shown on the screen, written out. - [02:48.663](https://academa.ai/lectures/period-doubling-logistic-map?t=168.663): line\_4 is shown on the screen, written out. ##### [02:54.453](https://academa.ai/lectures/period-doubling-logistic-map?t=174.453) Narration: Let it run for ten years and watch where it goes. Board: rule — a Math \[text\] that says "$x\_(n+1) = r thin x\_n (1 - x\_n)$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); para — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); s1 — a Math \[text\] that says "$x\_0 = 0.200$"; s2 — a Math \[text\] that says "$x\_1 = 2.6 (0.200)(0.800) = 0.416$"; s3 — a Math \[text\] that says "$x\_2 = 2.6 (0.416)(0.584) = 0.632$"; head\_run — a Heading that says "Turning the Handle"; start\_dot — a Point \[yellow\] labelled "x\_0" drawn in box (location=(0.2, 0.0)); line — a Line \[yellow\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.41600000000000004)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); line\_2 — a Line \[yellow\] drawn in box (start=(0.2, 0.41600000000000004), end=(0.41600000000000004, 0.41600000000000004)); line\_3 — a Line \[yellow\] drawn in box (start=(0.41600000000000004, 0.41600000000000004), end=(0.41600000000000004, 0.6316544000000001)); line\_4 — a Line \[yellow\] drawn in box (start=(0.41600000000000004, 0.6316544000000001), end=(0.6316544000000001, 0.6316544000000001)) Actions: - [02:55.08](https://academa.ai/lectures/period-doubling-logistic-map?t=175.08000000000007): line\_5 is shown on the screen, written out. - [02:55.33](https://academa.ai/lectures/period-doubling-logistic-map?t=175.33000000000007): line\_6 is shown on the screen, written out. - [02:55.58](https://academa.ai/lectures/period-doubling-logistic-map?t=175.58000000000007): line\_7 is shown on the screen, written out. - [02:55.83](https://academa.ai/lectures/period-doubling-logistic-map?t=175.83000000000007): line\_8 is shown on the screen, written out. - [02:56.08](https://academa.ai/lectures/period-doubling-logistic-map?t=176.08000000000007): line\_9 is shown on the screen, written out. - [02:56.33](https://academa.ai/lectures/period-doubling-logistic-map?t=176.33000000000007): line\_10 is shown on the screen, written out. - [02:56.58](https://academa.ai/lectures/period-doubling-logistic-map?t=176.58000000000007): line\_11 is shown on the screen, written out. - [02:56.83](https://academa.ai/lectures/period-doubling-logistic-map?t=176.83000000000007): line\_12 is shown on the screen, written out. - [02:57.08](https://academa.ai/lectures/period-doubling-logistic-map?t=177.08000000000007): line\_13 is shown on the screen, written out. - [02:57.33](https://academa.ai/lectures/period-doubling-logistic-map?t=177.33000000000007): line\_14 is shown on the screen, written out. - [02:57.58](https://academa.ai/lectures/period-doubling-logistic-map?t=177.58000000000007): line\_15 is shown on the screen, written out. - [02:57.83](https://academa.ai/lectures/period-doubling-logistic-map?t=177.83000000000007): line\_16 is shown on the screen, written out. - [02:58.08](https://academa.ai/lectures/period-doubling-logistic-map?t=178.08000000000007): line\_17 is shown on the screen, written out. - [02:58.33](https://academa.ai/lectures/period-doubling-logistic-map?t=178.33000000000007): line\_18 is shown on the screen, written out. - [02:58.58](https://academa.ai/lectures/period-doubling-logistic-map?t=178.58000000000007): line\_19 is shown on the screen, written out. ##### [02:58.799](https://academa.ai/lectures/period-doubling-logistic-map?t=178.7995) Narration: The staircase spirals inwards and parks itself exactly where the curve crosses the line. A little over six tenths of capacity, and it stays there for ever after. Start anywhere else and it goes to the same place. Board: rule — a Math \[text\] that says "$x\_(n+1) = r thin x\_n (1 - x\_n)$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); para — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); s1 — a Math \[text\] that says "$x\_0 = 0.200$"; s2 — a Math \[text\] that says "$x\_1 = 2.6 (0.200)(0.800) = 0.416$"; s3 — a Math \[text\] that says "$x\_2 = 2.6 (0.416)(0.584) = 0.632$"; head\_run — a Heading that says "Turning the Handle"; start\_dot — a Point \[yellow\] labelled "x\_0" drawn in box (location=(0.2, 0.0)); line — a Line \[yellow\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.41600000000000004)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); line\_2 — a Line \[yellow\] drawn in box (start=(0.2, 0.41600000000000004), end=(0.41600000000000004, 0.41600000000000004)); line\_3 — a Line \[yellow\] drawn in box (start=(0.41600000000000004, 0.41600000000000004), end=(0.41600000000000004, 0.6316544000000001)); line\_4 — a Line \[yellow\] drawn in box (start=(0.41600000000000004, 0.6316544000000001), end=(0.6316544000000001, 0.6316544000000001)); line\_5 — a Line \[yellow\] drawn in box (start=(0.6316544000000001, 0.6316544000000001), end=(0.6316544000000001, 0.604934509297664)); line\_6 — a Line \[yellow\] drawn in box (start=(0.6316544000000001, 0.604934509297664), end=(0.604934509297664, 0.604934509297664)); line\_7 — a Line \[yellow\] drawn in box (start=(0.604934509297664, 0.604934509297664), end=(0.604934509297664, 0.621370746771992)); line\_8 — a Line \[yellow\] drawn in box (start=(0.604934509297664, 0.621370746771992), end=(0.621370746771992, 0.621370746771992)); line\_9 — a Line \[yellow\] drawn in box (start=(0.621370746771992, 0.621370746771992), end=(0.621370746771992, 0.6116997687528234)); line\_10 — a Line \[yellow\] drawn in box (start=(0.621370746771992, 0.6116997687528234), end=(0.6116997687528234, 0.6116997687528234)); line\_11 — a Line \[yellow\] drawn in box (start=(0.6116997687528234, 0.6116997687528234), end=(0.6116997687528234, 0.617560220317471)); line\_12 — a Line \[yellow\] drawn in box (start=(0.6116997687528234, 0.617560220317471), end=(0.617560220317471, 0.617560220317471)); line\_13 — a Line \[yellow\] drawn in box (start=(0.617560220317471, 0.617560220317471), end=(0.617560220317471, 0.6140669459571599)); line\_14 — a Line \[yellow\] drawn in box (start=(0.617560220317471, 0.6140669459571599), end=(0.6140669459571599, 0.6140669459571599)); line\_15 — a Line \[yellow\] drawn in box (start=(0.6140669459571599, 0.6140669459571599), end=(0.6140669459571599, 0.6161707027840165)); line\_16 — a Line \[yellow\] drawn in box (start=(0.6140669459571599, 0.6161707027840165), end=(0.6161707027840165, 0.6161707027840165)); line\_17 — a Line \[yellow\] drawn in box (start=(0.6161707027840165, 0.6161707027840165), end=(0.6161707027840165, 0.614911356318136)); line\_18 — a Line \[yellow\] drawn in box (start=(0.6161707027840165, 0.614911356318136), end=(0.614911356318136, 0.614911356318136)); line\_19 — a Line \[yellow\] drawn in box (start=(0.614911356318136, 0.614911356318136), end=(0.614911356318136, 0.6156679884917287)) Actions: - [02:58.83](https://academa.ai/lectures/period-doubling-logistic-map?t=178.83000000000007): line\_20 is shown on the screen, written out. - [03:2.932](https://academa.ai/lectures/period-doubling-logistic-map?t=182.93200000000004): star is shown on the screen, written out. ##### [03:13.145](https://academa.ai/lectures/period-doubling-logistic-map?t=193.1455) Narration: That crossing is the whole story of the settled state, because a crossing is precisely a year in which next year's population equals this year's. So there are two questions. What is it? And why did the staircase walk towards it rather than away? Board: rule — a Math \[text\] that says "$x\_(n+1) = r thin x\_n (1 - x\_n)$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); para — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); s1 — a Math \[text\] that says "$x\_0 = 0.200$"; s2 — a Math \[text\] that says "$x\_1 = 2.6 (0.200)(0.800) = 0.416$"; s3 — a Math \[text\] that says "$x\_2 = 2.6 (0.416)(0.584) = 0.632$"; head\_run — a Heading that says "Turning the Handle"; start\_dot — a Point \[yellow\] labelled "x\_0" drawn in box (location=(0.2, 0.0)); line — a Line \[yellow\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.41600000000000004)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); line\_2 — a Line \[yellow\] drawn in box (start=(0.2, 0.41600000000000004), end=(0.41600000000000004, 0.41600000000000004)); line\_3 — a Line \[yellow\] drawn in box (start=(0.41600000000000004, 0.41600000000000004), end=(0.41600000000000004, 0.6316544000000001)); line\_4 — a Line \[yellow\] drawn in box (start=(0.41600000000000004, 0.6316544000000001), end=(0.6316544000000001, 0.6316544000000001)); line\_5 — a Line \[yellow\] drawn in box (start=(0.6316544000000001, 0.6316544000000001), end=(0.6316544000000001, 0.604934509297664)); line\_6 — a Line \[yellow\] drawn in box (start=(0.6316544000000001, 0.604934509297664), end=(0.604934509297664, 0.604934509297664)); line\_7 — a Line \[yellow\] drawn in box (start=(0.604934509297664, 0.604934509297664), end=(0.604934509297664, 0.621370746771992)); line\_8 — a Line \[yellow\] drawn in box (start=(0.604934509297664, 0.621370746771992), end=(0.621370746771992, 0.621370746771992)); line\_9 — a Line \[yellow\] drawn in box (start=(0.621370746771992, 0.621370746771992), end=(0.621370746771992, 0.6116997687528234)); line\_10 — a Line \[yellow\] drawn in box (start=(0.621370746771992, 0.6116997687528234), end=(0.6116997687528234, 0.6116997687528234)); line\_11 — a Line \[yellow\] drawn in box (start=(0.6116997687528234, 0.6116997687528234), end=(0.6116997687528234, 0.617560220317471)); line\_12 — a Line \[yellow\] drawn in box (start=(0.6116997687528234, 0.617560220317471), end=(0.617560220317471, 0.617560220317471)); line\_13 — a Line \[yellow\] drawn in box (start=(0.617560220317471, 0.617560220317471), end=(0.617560220317471, 0.6140669459571599)); line\_14 — a Line \[yellow\] drawn in box (start=(0.617560220317471, 0.6140669459571599), end=(0.6140669459571599, 0.6140669459571599)); line\_15 — a Line \[yellow\] drawn in box (start=(0.6140669459571599, 0.6140669459571599), end=(0.6140669459571599, 0.6161707027840165)); line\_16 — a Line \[yellow\] drawn in box (start=(0.6140669459571599, 0.6161707027840165), end=(0.6161707027840165, 0.6161707027840165)); line\_17 — a Line \[yellow\] drawn in box (start=(0.6161707027840165, 0.6161707027840165), end=(0.6161707027840165, 0.614911356318136)); line\_18 — a Line \[yellow\] drawn in box (start=(0.6161707027840165, 0.614911356318136), end=(0.614911356318136, 0.614911356318136)); line\_19 — a Line \[yellow\] drawn in box (start=(0.614911356318136, 0.614911356318136), end=(0.614911356318136, 0.6156679884917287)); line\_20 — a Line \[yellow\] drawn in box (start=(0.614911356318136, 0.6156679884917287), end=(0.6156679884917287, 0.6156679884917287)); star — a Point \[red\] labelled "x^\*" drawn in box (location=(0.615385, 0.615385)) Actions: - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): box is hidden from the screen — left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): para is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): start\_dot is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): diag is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_2 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_3 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_4 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_5 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_6 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_7 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_8 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_9 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_10 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_11 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_12 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_13 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_14 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_15 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_16 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_17 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_18 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_19 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): line\_20 is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): star is hidden from the screen — box left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): head\_run is hidden from the screen — left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): rule is hidden from the screen — left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): s1 is hidden from the screen — left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): s2 is hidden from the screen — left the board. - [03:29.165](https://academa.ai/lectures/period-doubling-logistic-map?t=209.16510416666665): s3 is hidden from the screen — left the board. ### Scene 2: [Why It Holds, and Why It Lets Go](https://academa.ai/lectures/period-doubling-logistic-map?t=210.2067708333333) Span: 03:30.207–07:32.071 (210.2067708333333s–452.07149999999996s). #### Objects - box: an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25) - diag: a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)) - edge\_note: a Panel that says "A fixed point holds whatever is near it while $abs(f'(x^\*)) \< 1$. 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2 x) \\ f'(x^\*) &= r (1 - 2 (1 - 1 / r)) \\ &= 2 - r$" - stable: a Math \[text\] that says "$abs(2 - r) \< 1 quad arrow.l.r.double quad 1 \< r \< 3$" - star26: a Point \[red\] labelled "x^\*" drawn in box (location=(0.615385, 0.615385)) - star29: a Point \[red\] labelled "x^\*" drawn in box (location=(0.655172, 0.655172)) - star32: a Point \[red\] labelled "x^\*" drawn in box (location=(0.6875, 0.6875)) - tangent: a TangentLine \[yellow\] labelled "f'(x^\*) = -0.6" drawn in box (target='para26', x=0.615385, length=0.55) - value: a Math \[text\] that says "$x^\* = 0.615$" #### Beats ##### [03:30.207](https://academa.ai/lectures/period-doubling-logistic-map?t=210.2067708333333) Narration: Start with the settled state, since that is what we just watched happen. A population that has stopped moving is one where next year equals this year. Call that value x star. Then x star must equal r x star times one minus x star. Board: Empty. Actions: - [03:30.207](https://academa.ai/lectures/period-doubling-logistic-map?t=210.2067708333333): head\_fix is shown on the screen, written out. - [03:30.207](https://academa.ai/lectures/period-doubling-logistic-map?t=210.2067708333333): box is shown on the screen, written out. - [03:30.207](https://academa.ai/lectures/period-doubling-logistic-map?t=210.2067708333333): para26 is shown on the screen, written out. - [03:30.207](https://academa.ai/lectures/period-doubling-logistic-map?t=210.2067708333333): diag is shown on the screen, written out. - [03:42.618](https://academa.ai/lectures/period-doubling-logistic-map?t=222.6177708333333): box moves to a new place on the board. - [03:42.618](https://academa.ai/lectures/period-doubling-logistic-map?t=222.6177708333333): fix is shown on the screen, written out. ##### [03:47.594](https://academa.ai/lectures/period-doubling-logistic-map?t=227.59427083333333) Narration: Now x star could of course be zero, and we shall come back to that in a moment. Otherwise divide both sides by it. One equals r times one minus x star. Board: box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_fix — a Heading that says "Where the Population Stops Moving"; para26 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)) Actions: - [03:49.801](https://academa.ai/lectures/period-doubling-logistic-map?t=229.8007708333333): point is shown on the screen, grown. - [03:51.801](https://academa.ai/lectures/period-doubling-logistic-map?t=231.8007708333333): point is hidden from the screen. - [03:53.179](https://academa.ai/lectures/period-doubling-logistic-map?t=233.17877083333332): fix is shown on the screen, written out. ##### [03:58.655](https://academa.ai/lectures/period-doubling-logistic-map?t=238.6552708333333) Narration: Rearrange, and there it is. The settled population is one minus one over r. With the knob at two point six that comes to nought point six one five, which is precisely where our staircase parked. Board: Unchanged from the preceding beat in this scene. Actions: - [03:59.004](https://academa.ai/lectures/period-doubling-logistic-map?t=239.0037708333333): fix is shown on the screen, written out. - [04:5.412](https://academa.ai/lectures/period-doubling-logistic-map?t=245.41177083333332): knob is shown on the screen, written out. - [04:8.628](https://academa.ai/lectures/period-doubling-logistic-map?t=248.6277708333333): value is shown on the screen, written out. - [04:11.821](https://academa.ai/lectures/period-doubling-logistic-map?t=251.8207708333333): star26 is shown on the screen, written out. ##### [04:13.164](https://academa.ai/lectures/period-doubling-logistic-map?t=253.1637708333333) Narration: But now the question this whole lecture turns on. Why did the staircase walk towards that point? Nothing in the algebra we just did says it has to. Zero satisfies the very same equation, and orbits run away from zero as fast as they can go. Board: knob — a Math \[text\] that says "$r = 2.6$"; value — a Math \[text\] that says "$x^\* = 0.615$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_fix — a Heading that says "Where the Population Stops Moving"; para26 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); star26 — a Point \[red\] labelled "x^\*" drawn in box (location=(0.615385, 0.615385)) Actions: - [04:25.749](https://academa.ai/lectures/period-doubling-logistic-map?t=265.7487708333333): line is shown on the screen, written out. - [04:25.929](https://academa.ai/lectures/period-doubling-logistic-map?t=265.9287708333333): line\_2 is shown on the screen, written out. - [04:26.109](https://academa.ai/lectures/period-doubling-logistic-map?t=266.1087708333333): line\_3 is shown on the screen, written out. - [04:26.289](https://academa.ai/lectures/period-doubling-logistic-map?t=266.28877083333333): line\_4 is shown on the screen, written out. - [04:26.469](https://academa.ai/lectures/period-doubling-logistic-map?t=266.46877083333334): line\_5 is shown on the screen, written out. - [04:26.649](https://academa.ai/lectures/period-doubling-logistic-map?t=266.6487708333333): line\_6 is shown on the screen, written out. - [04:26.829](https://academa.ai/lectures/period-doubling-logistic-map?t=266.8287708333333): line\_7 is shown on the screen, written out. - [04:27.009](https://academa.ai/lectures/period-doubling-logistic-map?t=267.0087708333333): line\_8 is shown on the screen, written out. - [04:27.189](https://academa.ai/lectures/period-doubling-logistic-map?t=267.1887708333333): line\_9 is shown on the screen, written out. - [04:27.369](https://academa.ai/lectures/period-doubling-logistic-map?t=267.3687708333333): line\_10 is shown on the screen, written out. - [04:28.292](https://academa.ai/lectures/period-doubling-logistic-map?t=268.2917708333333): knob moves to a new place on the board. - [04:28.292](https://academa.ai/lectures/period-doubling-logistic-map?t=268.2917708333333): fix is hidden from the screen — left the board. - [04:28.292](https://academa.ai/lectures/period-doubling-logistic-map?t=268.2917708333333): head\_fix is hidden from the screen — left the board. - [04:28.292](https://academa.ai/lectures/period-doubling-logistic-map?t=268.2917708333333): value is hidden from the screen — left the board. ##### [04:29.492](https://academa.ai/lectures/period-doubling-logistic-map?t=269.4917708333333) Narration: The answer is the steepness of the curve where it crosses the line. So differentiate the rule. F prime of x is r times one minus two x. Board: knob — a Math \[text\] that says "$r = 2.6$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); para26 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); star26 — a Point \[red\] labelled "x^\*" drawn in box (location=(0.615385, 0.615385)); line — a Line \[magenta\] drawn in box (start=(0.02, 0.0), end=(0.02, 0.050960000000000005)); line\_2 — a Line \[magenta\] drawn in box (start=(0.02, 0.050960000000000005), end=(0.050960000000000005, 0.050960000000000005)); line\_3 — a Line \[magenta\] drawn in box (start=(0.050960000000000005, 0.050960000000000005), end=(0.050960000000000005, 0.12574400384000003)); line\_4 — a Line \[magenta\] drawn in box (start=(0.050960000000000005, 0.12574400384000003), end=(0.12574400384000003, 0.12574400384000003)); line\_5 — a Line \[magenta\] drawn in box (start=(0.12574400384000003, 0.12574400384000003), end=(0.12574400384000003, 0.28582436827954383)); line\_6 — a Line \[magenta\] drawn in box (start=(0.12574400384000003, 0.28582436827954383), end=(0.28582436827954383, 0.28582436827954383)); line\_7 — a Line \[magenta\] drawn in box (start=(0.28582436827954383, 0.28582436827954383), end=(0.28582436827954383, 0.5307348768205732)); line\_8 — a Line \[magenta\] drawn in box (start=(0.28582436827954383, 0.5307348768205732), end=(0.5307348768205732, 0.5307348768205732)); line\_9 — a Line \[magenta\] drawn in box (start=(0.5307348768205732, 0.5307348768205732), end=(0.5307348768205732, 0.6475439551017429)); line\_10 — a Line \[magenta\] drawn in box (start=(0.5307348768205732, 0.6475439551017429), end=(0.6475439551017429, 0.6475439551017429)) Actions: - [04:29.492](https://academa.ai/lectures/period-doubling-logistic-map?t=269.4917708333333): head\_slope is shown on the screen, written out. - [04:29.492](https://academa.ai/lectures/period-doubling-logistic-map?t=269.4917708333333): line is hidden from the screen. - [04:29.492](https://academa.ai/lectures/period-doubling-logistic-map?t=269.4917708333333): line\_2 is hidden from the screen. - [04:29.492](https://academa.ai/lectures/period-doubling-logistic-map?t=269.4917708333333): line\_3 is hidden from the screen. - [04:29.492](https://academa.ai/lectures/period-doubling-logistic-map?t=269.4917708333333): line\_4 is hidden from the screen. - [04:29.492](https://academa.ai/lectures/period-doubling-logistic-map?t=269.4917708333333): line\_5 is hidden from the screen. - [04:29.492](https://academa.ai/lectures/period-doubling-logistic-map?t=269.4917708333333): line\_6 is hidden from the screen. - [04:29.492](https://academa.ai/lectures/period-doubling-logistic-map?t=269.4917708333333): line\_7 is hidden from the screen. - [04:29.492](https://academa.ai/lectures/period-doubling-logistic-map?t=269.4917708333333): line\_8 is hidden from the screen. - [04:29.492](https://academa.ai/lectures/period-doubling-logistic-map?t=269.4917708333333): line\_9 is hidden from the screen. - [04:29.492](https://academa.ai/lectures/period-doubling-logistic-map?t=269.4917708333333): line\_10 is hidden from the screen. - [04:33.915](https://academa.ai/lectures/period-doubling-logistic-map?t=273.9147708333333): slope\_der is shown on the screen, written out. ##### [04:40.111](https://academa.ai/lectures/period-doubling-logistic-map?t=280.1107708333333) Narration: Substitute the fixed point, one minus one over r, and the r cancels beautifully. The slope at the crossing is simply two minus r. At two point six that is nought point six, with a minus sign in front. Board: knob — a Math \[text\] that says "$r = 2.6$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); para26 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); star26 — a Point \[red\] labelled "x^\*" drawn in box (location=(0.615385, 0.615385)); head\_slope — a Heading that says "The Slope Decides" Actions: - [04:40.459](https://academa.ai/lectures/period-doubling-logistic-map?t=280.4587708333333): slope\_der is shown on the screen, written out. - [04:44.36](https://academa.ai/lectures/period-doubling-logistic-map?t=284.3597708333333): slope\_der is shown on the screen, written out. - [04:50.723](https://academa.ai/lectures/period-doubling-logistic-map?t=290.7227708333333): tangent is shown on the screen, written out. ##### [04:54.388](https://academa.ai/lectures/period-doubling-logistic-map?t=294.3877708333333) Narration: Here is what that number does for a living. Suppose you are a small distance away from x star. Very near a crossing the curve is almost exactly its own tangent line, so one turn of the handle takes that distance and multiplies it by the slope. Board: knob — a Math \[text\] that says "$r = 2.6$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); para26 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); star26 — a Point \[red\] labelled "x^\*" drawn in box (location=(0.615385, 0.615385)); head\_slope — a Heading that says "The Slope Decides"; tangent — a TangentLine \[yellow\] labelled "f'(x^\*) = -0.6" drawn in box (target='para26', x=0.615385, length=0.55) Actions: - [05:4.291](https://academa.ai/lectures/period-doubling-logistic-map?t=304.29077083333334): tangent is indicated — a transient flash. ##### [05:10.08](https://academa.ai/lectures/period-doubling-logistic-map?t=310.0802708333333) Narration: Multiply a distance by minus nought point six and it shrinks, and it changes sides. Multiply again and it shrinks and changes sides again. Which is exactly the inward spiral we watched: over, under, over, under, closing in. Board: Unchanged from the preceding beat in this scene. Actions: - [05:21.423](https://academa.ai/lectures/period-doubling-logistic-map?t=321.4227708333333): line\_11 is shown on the screen, written out. - [05:21.723](https://academa.ai/lectures/period-doubling-logistic-map?t=321.7227708333333): line\_12 is shown on the screen, written out. - [05:22.023](https://academa.ai/lectures/period-doubling-logistic-map?t=322.0227708333333): line\_13 is shown on the screen, written out. - [05:22.323](https://academa.ai/lectures/period-doubling-logistic-map?t=322.3227708333333): line\_14 is shown on the screen, written out. - [05:22.623](https://academa.ai/lectures/period-doubling-logistic-map?t=322.62277083333333): line\_15 is shown on the screen, written out. - [05:22.923](https://academa.ai/lectures/period-doubling-logistic-map?t=322.9227708333333): line\_16 is shown on the screen, written out. - [05:23.223](https://academa.ai/lectures/period-doubling-logistic-map?t=323.2227708333333): line\_17 is shown on the screen, written out. - [05:23.523](https://academa.ai/lectures/period-doubling-logistic-map?t=323.5227708333333): line\_18 is shown on the screen, written out. - [05:23.823](https://academa.ai/lectures/period-doubling-logistic-map?t=323.8227708333333): line\_19 is shown on the screen, written out. - [05:24.123](https://academa.ai/lectures/period-doubling-logistic-map?t=324.12277083333333): line\_20 is shown on the screen, written out. - [05:24.423](https://academa.ai/lectures/period-doubling-logistic-map?t=324.4227708333333): line\_21 is shown on the screen, written out. - [05:24.723](https://academa.ai/lectures/period-doubling-logistic-map?t=324.7227708333333): line\_22 is shown on the screen, written out. ##### [05:28.223](https://academa.ai/lectures/period-doubling-logistic-map?t=328.2227708333333) Narration: And notice there are two flavours. A positive slope smaller than one shrinks distances without changing sides, and the staircase climbs to the crossing like an ordinary flight of stairs. A negative one gives the spiral. Either way, less than one in size means the point holds. Board: knob — a Math \[text\] that says "$r = 2.6$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); para26 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); star26 — a Point \[red\] labelled "x^\*" drawn in box (location=(0.615385, 0.615385)); head\_slope — a Heading that says "The Slope Decides"; tangent — a TangentLine \[yellow\] labelled "f'(x^\*) = -0.6" drawn in box (target='para26', x=0.615385, length=0.55); line\_11 — a Line \[yellow\] drawn in box (start=(0.45, 0.0), end=(0.45, 0.6435000000000002)); line\_12 — a Line \[yellow\] drawn in box (start=(0.45, 0.6435000000000002), end=(0.6435000000000002, 0.6435000000000002)); line\_13 — a Line \[yellow\] drawn in box (start=(0.6435000000000002, 0.6435000000000002), end=(0.6435000000000002, 0.5964601499999999)); line\_14 — a Line \[yellow\] drawn in box (start=(0.6435000000000002, 0.5964601499999999), end=(0.5964601499999999, 0.5964601499999999)); line\_15 — a Line \[yellow\] drawn in box (start=(0.5964601499999999, 0.5964601499999999), end=(0.5964601499999999, 0.6258081426011416)); line\_16 — a Line \[yellow\] drawn in box (start=(0.5964601499999999, 0.6258081426011416), end=(0.6258081426011416, 0.6258081426011416)); line\_17 — a Line \[yellow\] drawn in box (start=(0.6258081426011416, 0.6258081426011416), end=(0.6258081426011416, 0.6088480092636521)); line\_18 — a Line \[yellow\] drawn in box (start=(0.6258081426011416, 0.6088480092636521), end=(0.6088480092636521, 0.6088480092636521)); line\_19 — a Line \[yellow\] drawn in box (start=(0.6088480092636521, 0.6088480092636521), end=(0.6088480092636521, 0.6191954882862837)); line\_20 — a Line \[yellow\] drawn in box (start=(0.6088480092636521, 0.6191954882862837), end=(0.6191954882862837, 0.6191954882862837)); line\_21 — a Line \[yellow\] drawn in box (start=(0.6191954882862837, 0.6191954882862837), end=(0.6191954882862837, 0.6130603324877055)); line\_22 — a Line \[yellow\] drawn in box (start=(0.6191954882862837, 0.6130603324877055), end=(0.6130603324877055, 0.6130603324877055)) Actions: - None. ##### [05:47.655](https://academa.ai/lectures/period-doubling-logistic-map?t=347.65477083333326) Narration: So the condition is this, and it is the only piece of theory in the whole lecture. The fixed point holds on as long as the size of two minus r is less than one, and that is the same as saying r lies between one and three. Board: Unchanged from the preceding beat in this scene. Actions: - [05:48.317](https://academa.ai/lectures/period-doubling-logistic-map?t=348.3167708333333): stable is shown on the screen, written out. - [06:1.587](https://academa.ai/lectures/period-doubling-logistic-map?t=361.5867708333333): knob moves to a new place on the board. - [06:1.587](https://academa.ai/lectures/period-doubling-logistic-map?t=361.5867708333333): stable moves to a new place on the board. - [06:1.587](https://academa.ai/lectures/period-doubling-logistic-map?t=361.5867708333333): head\_slope is hidden from the screen — left the board. - [06:1.587](https://academa.ai/lectures/period-doubling-logistic-map?t=361.5867708333333): slope\_der is hidden from the screen — left the board. ##### [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333) Narration: Watch it get feeble. Turn the knob up to two point nine. The crossing has shifted a little to the right, and the slope there is now minus nought point nine. Board: knob — a Math \[text\] that says "$r = 2.6$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); para26 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); star26 — a Point \[red\] labelled "x^\*" drawn in box (location=(0.615385, 0.615385)); stable — a Math \[text\] that says "$abs(2 - r) \< 1 quad arrow.l.r.double quad 1 \< r \< 3$"; tangent — a TangentLine \[yellow\] labelled "f'(x^\*) = -0.6" drawn in box (target='para26', x=0.615385, length=0.55); line\_11 — a Line \[yellow\] drawn in box (start=(0.45, 0.0), end=(0.45, 0.6435000000000002)); line\_12 — a Line \[yellow\] drawn in box (start=(0.45, 0.6435000000000002), end=(0.6435000000000002, 0.6435000000000002)); line\_13 — a Line \[yellow\] drawn in box (start=(0.6435000000000002, 0.6435000000000002), end=(0.6435000000000002, 0.5964601499999999)); line\_14 — a Line \[yellow\] drawn in box (start=(0.6435000000000002, 0.5964601499999999), end=(0.5964601499999999, 0.5964601499999999)); line\_15 — a Line \[yellow\] drawn in box (start=(0.5964601499999999, 0.5964601499999999), end=(0.5964601499999999, 0.6258081426011416)); line\_16 — a Line \[yellow\] drawn in box (start=(0.5964601499999999, 0.6258081426011416), end=(0.6258081426011416, 0.6258081426011416)); line\_17 — a Line \[yellow\] drawn in box (start=(0.6258081426011416, 0.6258081426011416), end=(0.6258081426011416, 0.6088480092636521)); line\_18 — a Line \[yellow\] drawn in box (start=(0.6258081426011416, 0.6088480092636521), end=(0.6088480092636521, 0.6088480092636521)); line\_19 — a Line \[yellow\] drawn in box (start=(0.6088480092636521, 0.6088480092636521), end=(0.6088480092636521, 0.6191954882862837)); line\_20 — a Line \[yellow\] drawn in box (start=(0.6088480092636521, 0.6191954882862837), end=(0.6191954882862837, 0.6191954882862837)); line\_21 — a Line \[yellow\] drawn in box (start=(0.6191954882862837, 0.6191954882862837), end=(0.6191954882862837, 0.6130603324877055)); line\_22 — a Line \[yellow\] drawn in box (start=(0.6191954882862837, 0.6130603324877055), end=(0.6130603324877055, 0.6130603324877055)) Actions: - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): head\_edge is shown on the screen, written out. - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): line\_11 is hidden from the screen. - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): line\_12 is hidden from the screen. - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): line\_13 is hidden from the screen. - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): line\_14 is hidden from the screen. - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): line\_15 is hidden from the screen. - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): line\_16 is hidden from the screen. - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): line\_17 is hidden from the screen. - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): line\_18 is hidden from the screen. - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): line\_19 is hidden from the screen. - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): line\_20 is hidden from the screen. - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): line\_21 is hidden from the screen. - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): line\_22 is hidden from the screen. - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): para26 is hidden from the screen. - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): star26 is hidden from the screen. - [06:2.787](https://academa.ai/lectures/period-doubling-logistic-map?t=362.7867708333333): tangent is hidden from the screen. - [06:4.981](https://academa.ai/lectures/period-doubling-logistic-map?t=364.9807708333333): para29 is shown on the screen, written out. - [06:6.456](https://academa.ai/lectures/period-doubling-logistic-map?t=366.4557708333333): knob becomes "$r = 2.9$". - [06:7.861](https://academa.ai/lectures/period-doubling-logistic-map?t=367.8607708333333): star29 is shown on the screen, written out. ##### [06:14.115](https://academa.ai/lectures/period-doubling-logistic-map?t=374.1152708333333) Narration: Start from a fifth of capacity again. Each year the distance from the crossing shrinks by only a tenth of itself, so the staircase takes an age. But a tenth of a tenth of a tenth is nothing in the end, and it still parks. Board: knob — a Math \[text\] that says "$r = 2.6$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); stable — a Math \[text\] that says "$abs(2 - r) \< 1 quad arrow.l.r.double quad 1 \< r \< 3$"; head\_edge — a Heading that says "Turning the Knob to the Edge"; para29 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); star29 — a Point \[red\] labelled "x^\*" drawn in box (location=(0.655172, 0.655172)) Actions: - [06:14.522](https://academa.ai/lectures/period-doubling-logistic-map?t=374.5217708333333): line\_23 is shown on the screen, written out. - [06:14.722](https://academa.ai/lectures/period-doubling-logistic-map?t=374.72177083333327): line\_24 is shown on the screen, written out. - [06:14.922](https://academa.ai/lectures/period-doubling-logistic-map?t=374.9217708333333): line\_25 is shown on the screen, written out. - [06:15.122](https://academa.ai/lectures/period-doubling-logistic-map?t=375.12177083333324): line\_26 is shown on the screen, written out. - [06:15.322](https://academa.ai/lectures/period-doubling-logistic-map?t=375.3217708333333): line\_27 is shown on the screen, written out. - [06:15.522](https://academa.ai/lectures/period-doubling-logistic-map?t=375.5217708333333): line\_28 is shown on the screen, written out. - [06:15.722](https://academa.ai/lectures/period-doubling-logistic-map?t=375.72177083333327): line\_29 is shown on the screen, written out. - [06:15.922](https://academa.ai/lectures/period-doubling-logistic-map?t=375.9217708333333): line\_30 is shown on the screen, written out. - [06:16.122](https://academa.ai/lectures/period-doubling-logistic-map?t=376.12177083333324): line\_31 is shown on the screen, written out. - [06:16.322](https://academa.ai/lectures/period-doubling-logistic-map?t=376.3217708333333): line\_32 is shown on the screen, written out. - [06:16.522](https://academa.ai/lectures/period-doubling-logistic-map?t=376.5217708333333): line\_33 is shown on the screen, written out. - [06:16.722](https://academa.ai/lectures/period-doubling-logistic-map?t=376.72177083333327): line\_34 is shown on the screen, written out. - [06:16.922](https://academa.ai/lectures/period-doubling-logistic-map?t=376.9217708333333): line\_35 is shown on the screen, written out. - [06:17.122](https://academa.ai/lectures/period-doubling-logistic-map?t=377.12177083333324): line\_36 is shown on the screen, written out. - [06:17.322](https://academa.ai/lectures/period-doubling-logistic-map?t=377.3217708333333): line\_37 is shown on the screen, written out. - [06:17.522](https://academa.ai/lectures/period-doubling-logistic-map?t=377.5217708333333): line\_38 is shown on the screen, written out. - [06:17.722](https://academa.ai/lectures/period-doubling-logistic-map?t=377.72177083333327): line\_39 is shown on the screen, written out. - [06:17.922](https://academa.ai/lectures/period-doubling-logistic-map?t=377.9217708333333): line\_40 is shown on the screen, written out. - [06:18.122](https://academa.ai/lectures/period-doubling-logistic-map?t=378.12177083333324): line\_41 is shown on the screen, written out. - [06:18.322](https://academa.ai/lectures/period-doubling-logistic-map?t=378.3217708333333): line\_42 is shown on the screen, written out. - [06:18.522](https://academa.ai/lectures/period-doubling-logistic-map?t=378.5217708333333): line\_43 is shown on the screen, written out. - [06:18.722](https://academa.ai/lectures/period-doubling-logistic-map?t=378.72177083333327): line\_44 is shown on the screen, written out. - [06:18.922](https://academa.ai/lectures/period-doubling-logistic-map?t=378.9217708333333): line\_45 is shown on the screen, written out. - [06:19.122](https://academa.ai/lectures/period-doubling-logistic-map?t=379.12177083333324): line\_46 is shown on the screen, written out. - [06:19.322](https://academa.ai/lectures/period-doubling-logistic-map?t=379.3217708333333): line\_47 is shown on the screen, written out. - [06:19.522](https://academa.ai/lectures/period-doubling-logistic-map?t=379.5217708333333): line\_48 is shown on the screen, written out. - [06:19.722](https://academa.ai/lectures/period-doubling-logistic-map?t=379.72177083333327): line\_49 is shown on the screen, written out. - [06:19.922](https://academa.ai/lectures/period-doubling-logistic-map?t=379.9217708333333): line\_50 is shown on the screen, written out. ##### [06:28.101](https://academa.ai/lectures/period-doubling-logistic-map?t=388.1007708333333) Narration: At r exactly three, the slope at the crossing is exactly minus one. Distances are no longer shrunk at all. They are simply reflected, over and back, over and back, at the same size for ever. The fixed point has stopped pulling, and it has not yet started pushing. Board: knob — a Math \[text\] that says "$r = 2.6$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); stable — a Math \[text\] that says "$abs(2 - r) \< 1 quad arrow.l.r.double quad 1 \< r \< 3$"; head\_edge — a Heading that says "Turning the Knob to the Edge"; para29 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); star29 — a Point \[red\] labelled "x^\*" drawn in box (location=(0.655172, 0.655172)); line\_23 — a Line \[yellow\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.46399999999999997)); line\_24 — a Line \[yellow\] drawn in box (start=(0.2, 0.46399999999999997), end=(0.46399999999999997, 0.46399999999999997)); line\_25 — a Line \[yellow\] drawn in box (start=(0.46399999999999997, 0.46399999999999997), end=(0.46399999999999997, 0.7212416)); line\_26 — a Line \[yellow\] drawn in box (start=(0.46399999999999997, 0.7212416), end=(0.7212416, 0.7212416)); line\_27 — a Line \[yellow\] drawn in box (start=(0.7212416, 0.7212416), end=(0.7212416, 0.583051247845376)); line\_28 — a Line \[yellow\] drawn in box (start=(0.7212416, 0.583051247845376), end=(0.583051247845376, 0.583051247845376)); line\_29 — a Line \[yellow\] drawn in box (start=(0.583051247845376, 0.583051247845376), end=(0.583051247845376, 0.7049972216708452)); line\_30 — a Line \[yellow\] drawn in box (start=(0.583051247845376, 0.7049972216708452), end=(0.7049972216708452, 0.7049972216708452)); line\_31 — a Line \[yellow\] drawn in box (start=(0.7049972216708452, 0.7049972216708452), end=(0.7049972216708452, 0.6031308034109796)); line\_32 — a Line \[yellow\] drawn in box (start=(0.7049972216708452, 0.6031308034109796), end=(0.6031308034109796, 0.6031308034109796)); line\_33 — a Line \[yellow\] drawn in box (start=(0.6031308034109796, 0.6031308034109796), end=(0.6031308034109796, 0.694155708424637)); line\_34 — a Line \[yellow\] drawn in box (start=(0.6031308034109796, 0.694155708424637), end=(0.694155708424637, 0.694155708424637)); line\_35 — a Line \[yellow\] drawn in box (start=(0.694155708424637, 0.694155708424637), end=(0.694155708424637, 0.6156803265697692)); line\_36 — a Line \[yellow\] drawn in box (start=(0.694155708424637, 0.6156803265697692), end=(0.6156803265697692, 0.6156803265697692)); line\_37 — a Line \[yellow\] drawn in box (start=(0.6156803265697692, 0.6156803265697692), end=(0.6156803265697692, 0.6861923799296634)); line\_38 — a Line \[yellow\] drawn in box (start=(0.6156803265697692, 0.6861923799296634), end=(0.6861923799296634, 0.6861923799296634)); line\_39 — a Line \[yellow\] drawn in box (start=(0.6861923799296634, 0.6861923799296634), end=(0.6861923799296634, 0.6244639532027708)); line\_40 — a Line \[yellow\] drawn in box (start=(0.6861923799296634, 0.6244639532027708), end=(0.6244639532027708, 0.6244639532027708)); line\_41 — a Line \[yellow\] drawn in box (start=(0.6244639532027708, 0.6244639532027708), end=(0.6244639532027708, 0.6800753006241016)); line\_42 — a Line \[yellow\] drawn in box (start=(0.6244639532027708, 0.6800753006241016), end=(0.6800753006241016, 0.6800753006241016)); line\_43 — a Line \[yellow\] drawn in box (start=(0.6800753006241016, 0.6800753006241016), end=(0.6800753006241016, 0.6309613697049044)); line\_44 — a Line \[yellow\] drawn in box (start=(0.6800753006241016, 0.6309613697049044), end=(0.6309613697049044, 0.6309613697049044)); line\_45 — a Line \[yellow\] drawn in box (start=(0.6309613697049044, 0.6309613697049044), end=(0.6309613697049044, 0.6752624469705445)); line\_46 — a Line \[yellow\] drawn in box (start=(0.6309613697049044, 0.6752624469705445), end=(0.6752624469705445, 0.6752624469705445)); line\_47 — a Line \[yellow\] drawn in box (start=(0.6752624469705445, 0.6752624469705445), end=(0.6752624469705445, 0.6359209165775015)); line\_48 — a Line \[yellow\] drawn in box (start=(0.6752624469705445, 0.6359209165775015), end=(0.6359209165775015, 0.6359209165775015)); line\_49 — a Line \[yellow\] drawn in box (start=(0.6359209165775015, 0.6359209165775015), end=(0.6359209165775015, 0.6714239628665225)); line\_50 — a Line \[yellow\] drawn in box (start=(0.6359209165775015, 0.6714239628665225), end=(0.6714239628665225, 0.6714239628665225)) Actions: - None. ##### [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333) Narration: So go past it. Three point two. The slope at the crossing is minus one point two, and each year the distance from it is multiplied by one point two. Board: Unchanged from the preceding beat in this scene. Actions: - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_23 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_24 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_25 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_26 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_27 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_28 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_29 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_30 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_31 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_32 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_33 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_34 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_35 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_36 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_37 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_38 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_39 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_40 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_41 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_42 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_43 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_44 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_45 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_46 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_47 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_48 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_49 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): line\_50 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): para29 is hidden from the screen. - [06:45.883](https://academa.ai/lectures/period-doubling-logistic-map?t=405.8832708333333): star29 is hidden from the screen. - [06:47.729](https://academa.ai/lectures/period-doubling-logistic-map?t=407.7287708333333): para32 is shown on the screen, written out. - [06:48.043](https://academa.ai/lectures/period-doubling-logistic-map?t=408.0427708333333): knob becomes "$r = 3.2$". - [06:49.738](https://academa.ai/lectures/period-doubling-logistic-map?t=409.73777083333323): star32 is shown on the screen, written out. ##### [06:56.143](https://academa.ai/lectures/period-doubling-logistic-map?t=416.1427708333333) Narration: Now start almost exactly on the fixed point, a hair above it, and watch what the construction does. Board: knob — a Math \[text\] that says "$r = 2.6$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); stable — a Math \[text\] that says "$abs(2 - r) \< 1 quad arrow.l.r.double quad 1 \< r \< 3$"; head\_edge — a Heading that says "Turning the Knob to the Edge"; para32 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); star32 — a Point \[red\] labelled "x^\*" drawn in box (location=(0.6875, 0.6875)) Actions: - [07:0.392](https://academa.ai/lectures/period-doubling-logistic-map?t=420.3917708333332): line\_51 is shown on the screen, written out. - [07:0.672](https://academa.ai/lectures/period-doubling-logistic-map?t=420.67177083333326): line\_52 is shown on the screen, written out. - [07:0.952](https://academa.ai/lectures/period-doubling-logistic-map?t=420.9517708333333): line\_53 is shown on the screen, written out. - [07:1.232](https://academa.ai/lectures/period-doubling-logistic-map?t=421.23177083333326): line\_54 is shown on the screen, written out. - [07:1.512](https://academa.ai/lectures/period-doubling-logistic-map?t=421.51177083333323): line\_55 is shown on the screen, written out. - [07:1.792](https://academa.ai/lectures/period-doubling-logistic-map?t=421.79177083333326): line\_56 is shown on the screen, written out. - [07:2.072](https://academa.ai/lectures/period-doubling-logistic-map?t=422.0717708333333): line\_57 is shown on the screen, written out. - [07:2.352](https://academa.ai/lectures/period-doubling-logistic-map?t=422.35177083333326): line\_58 is shown on the screen, written out. - [07:2.632](https://academa.ai/lectures/period-doubling-logistic-map?t=422.63177083333323): line\_59 is shown on the screen, written out. ##### [07:2.85](https://academa.ai/lectures/period-doubling-logistic-map?t=422.8497708333333) Narration: It leaves. The staircase spirals outwards, away from a value that satisfies the fixed point equation perfectly well. The point has not gone anywhere. It has simply lost the ability to hold on to anything, and everything near it is now on its way out. Board: knob — a Math \[text\] that says "$r = 2.6$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); stable — a Math \[text\] that says "$abs(2 - r) \< 1 quad arrow.l.r.double quad 1 \< r \< 3$"; head\_edge — a Heading that says "Turning the Knob to the Edge"; para32 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); star32 — a Point \[red\] labelled "x^\*" drawn in box (location=(0.6875, 0.6875)); line\_51 — a Line \[yellow\] drawn in box (start=(0.72, 0.0), end=(0.72, 0.64512)); line\_52 — a Line \[yellow\] drawn in box (start=(0.72, 0.64512), end=(0.64512, 0.64512)); line\_53 — a Line \[yellow\] drawn in box (start=(0.64512, 0.64512), end=(0.64512, 0.73260859392)); line\_54 — a Line \[yellow\] drawn in box (start=(0.64512, 0.73260859392), end=(0.73260859392, 0.73260859392)); line\_55 — a Line \[yellow\] drawn in box (start=(0.73260859392, 0.73260859392), end=(0.73260859392, 0.6268583745105938)); line\_56 — a Line \[yellow\] drawn in box (start=(0.73260859392, 0.6268583745105938), end=(0.6268583745105938, 0.6268583745105938)); line\_57 — a Line \[yellow\] drawn in box (start=(0.6268583745105938, 0.6268583745105938), end=(0.6268583745105938, 0.7485022490128957)); line\_58 — a Line \[yellow\] drawn in box (start=(0.6268583745105938, 0.7485022490128957), end=(0.7485022490128957, 0.7485022490128957)); line\_59 — a Line \[yellow\] drawn in box (start=(0.7485022490128957, 0.7485022490128957), end=(0.7485022490128957, 0.6023892231537048)) Actions: - [07:2.912](https://academa.ai/lectures/period-doubling-logistic-map?t=422.91177083333326): line\_60 is shown on the screen, written out. - [07:3.192](https://academa.ai/lectures/period-doubling-logistic-map?t=423.1917708333333): line\_61 is shown on the screen, written out. - [07:3.472](https://academa.ai/lectures/period-doubling-logistic-map?t=423.47177083333327): line\_62 is shown on the screen, written out. - [07:3.752](https://academa.ai/lectures/period-doubling-logistic-map?t=423.75177083333324): line\_63 is shown on the screen, written out. - [07:4.032](https://academa.ai/lectures/period-doubling-logistic-map?t=424.0317708333332): line\_64 is shown on the screen, written out. - [07:4.312](https://academa.ai/lectures/period-doubling-logistic-map?t=424.31177083333324): line\_65 is shown on the screen, written out. - [07:4.592](https://academa.ai/lectures/period-doubling-logistic-map?t=424.59177083333327): line\_66 is shown on the screen, written out. - [07:4.872](https://academa.ai/lectures/period-doubling-logistic-map?t=424.87177083333324): line\_67 is shown on the screen, written out. - [07:5.152](https://academa.ai/lectures/period-doubling-logistic-map?t=425.1517708333332): line\_68 is shown on the screen, written out. - [07:14.251](https://academa.ai/lectures/period-doubling-logistic-map?t=434.25077083333326): edge\_note is shown on the screen, written out. ##### [07:19.75](https://academa.ai/lectures/period-doubling-logistic-map?t=439.7497708333333) Narration: Which leaves an obvious gap in the story. The population is not going to the fixed point. It is trapped between zero and one so it cannot run off. Where on earth is it going? Board: knob — a Math \[text\] that says "$r = 2.6$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); stable — a Math \[text\] that says "$abs(2 - r) \< 1 quad arrow.l.r.double quad 1 \< r \< 3$"; edge\_note — a Panel that says "A fixed point holds whatever is near it while $abs(f'(x^\*)) \< 1$. At $abs(f'(x^\*)) = 1$ it stops holding, and beyond that it is still a solution and still on the picture, but nothing near it stays near it."; head\_edge — a Heading that says "Turning the Knob to the Edge"; para32 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); star32 — a Point \[red\] labelled "x^\*" drawn in box (location=(0.6875, 0.6875)); line\_51 — a Line \[yellow\] drawn in box (start=(0.72, 0.0), end=(0.72, 0.64512)); line\_52 — a Line \[yellow\] drawn in box (start=(0.72, 0.64512), end=(0.64512, 0.64512)); line\_53 — a Line \[yellow\] drawn in box (start=(0.64512, 0.64512), end=(0.64512, 0.73260859392)); line\_54 — a Line \[yellow\] drawn in box (start=(0.64512, 0.73260859392), end=(0.73260859392, 0.73260859392)); line\_55 — a Line \[yellow\] drawn in box (start=(0.73260859392, 0.73260859392), end=(0.73260859392, 0.6268583745105938)); line\_56 — a Line \[yellow\] drawn in box (start=(0.73260859392, 0.6268583745105938), end=(0.6268583745105938, 0.6268583745105938)); line\_57 — a Line \[yellow\] drawn in box (start=(0.6268583745105938, 0.6268583745105938), end=(0.6268583745105938, 0.7485022490128957)); line\_58 — a Line \[yellow\] drawn in box (start=(0.6268583745105938, 0.7485022490128957), end=(0.7485022490128957, 0.7485022490128957)); line\_59 — a Line \[yellow\] drawn in box (start=(0.7485022490128957, 0.7485022490128957), end=(0.7485022490128957, 0.6023892231537048)); line\_60 — a Line \[yellow\] drawn in box (start=(0.7485022490128957, 0.6023892231537048), end=(0.6023892231537048, 0.6023892231537048)); line\_61 — a Line \[yellow\] drawn in box (start=(0.6023892231537048, 0.6023892231537048), end=(0.6023892231537048, 0.7664526303423387)); line\_62 — a Line \[yellow\] drawn in box (start=(0.6023892231537048, 0.7664526303423387), end=(0.7664526303423387, 0.7664526303423387)); line\_63 — a Line \[yellow\] drawn in box (start=(0.7664526303423387, 0.7664526303423387), end=(0.7664526303423387, 0.5728095865076769)); line\_64 — a Line \[yellow\] drawn in box (start=(0.7664526303423387, 0.5728095865076769), end=(0.5728095865076769, 0.5728095865076769)); line\_65 — a Line \[yellow\] drawn in box (start=(0.5728095865076769, 0.5728095865076769), end=(0.5728095865076769, 0.7830360451602596)); line\_66 — a Line \[yellow\] drawn in box (start=(0.5728095865076769, 0.7830360451602596), end=(0.7830360451602596, 0.7830360451602596)); line\_67 — a Line \[yellow\] drawn in box (start=(0.7830360451602596, 0.7830360451602596), end=(0.7830360451602596, 0.5436499108481264)); line\_68 — a Line \[yellow\] drawn in box (start=(0.7830360451602596, 0.5436499108481264), end=(0.5436499108481264, 0.5436499108481264)) Actions: - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): box is hidden from the screen — left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): diag is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): para32 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): star32 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_51 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_52 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_53 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_54 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_55 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_56 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_57 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_58 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_59 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_60 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_61 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_62 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_63 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_64 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_65 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_66 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_67 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): line\_68 is hidden from the screen — box left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): edge\_note is hidden from the screen — left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): head\_edge is hidden from the screen — left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): knob is hidden from the screen — left the board. - [07:31.03](https://academa.ai/lectures/period-doubling-logistic-map?t=451.0298333333333): stable is hidden from the screen — left the board. ### Scene 3: [The First Split](https://academa.ai/lectures/period-doubling-logistic-map?t=452.07149999999996) Span: 07:32.071–11:1.154 (452.07149999999996s–661.1541874999999s). #### Objects - box3: an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25) - box4: an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25) - compose: a Math \[text\] that says "$x\_(n+2) = f(f(x\_n))$" - count: a Text \[text\] that says "Four crossings: $0$ and $x^\*$ carried over from $f$, and two new ones, $p$ and $q$." - cycle\_eq: a Math \[text\] that says "$q = f(p), quad p = f(q)$" - diag3: a Line \[green\] labelled "y = x" drawn in box3 (end=(1.0, 1.0)) - diag4: a Line \[green\] labelled "y = x" drawn in box4 (end=(1.0, 1.0)) - head\_double: a Heading that says "Applying the Rule Twice" - head\_law: a Heading that says "What Happens at a Doubling" - head\_two: a Heading that says "A Square Instead of a Point" - knob3: a Math \[text\] that says "$r = 3.2$" - law: a Math \[text\] that says "$f'(x^\*) = -1$" - law\_note: a Panel that says "When the slope at an attracting point reaches $-1$, the point does not disappear. It goes on existing, it stops attracting, and an attracting cycle of twice the period appears around it and takes over." - line: a Line \[gray\] drawn in box3 (start=(0.2, 0.0), end=(0.2, 0.5120000000000001)) - line\_10: a Line \[gray\] drawn in box3 (start=(0.7994688034800593, 0.5130189943751092), end=(0.5130189943751092, 0.5130189943751092)) - line\_11: a Line \[gray\] drawn in box3 (start=(0.5130189943751092, 0.5130189943751092), end=(0.5130189943751092, 0.7994576185134749)) - line\_12: a Line \[gray\] drawn in box3 (start=(0.5130189943751092, 0.7994576185134749), end=(0.7994576185134749, 0.7994576185134749)) - line\_13: a Line \[gray\] drawn in box3 (start=(0.7994576185134749, 0.7994576185134749), end=(0.7994576185134749, 0.5130404310855622)) - line\_14: a Line \[gray\] drawn in box3 (start=(0.7994576185134749, 0.5130404310855622), end=(0.5130404310855622, 0.5130404310855622)) - line\_15: a Line \[gray\] drawn in box3 (start=(0.5130404310855622, 0.5130404310855622), end=(0.5130404310855622, 0.7994558309027286)) - line\_16: a Line \[gray\] drawn in box3 (start=(0.5130404310855622, 0.7994558309027286), end=(0.7994558309027286, 0.7994558309027286)) - line\_17: a Line \[gray\] drawn in box3 (start=(0.7994558309027286, 0.7994558309027286), end=(0.7994558309027286, 0.5130438570827405)) - line\_18: a Line \[gray\] drawn in box3 (start=(0.7994558309027286, 0.5130438570827405), end=(0.5130438570827405, 0.5130438570827405)) - line\_19: a Line \[gray\] drawn in box3 (start=(0.5130438570827405, 0.5130438570827405), end=(0.5130438570827405, 0.7994555449356961)) - line\_2: a Line \[gray\] drawn in box3 (start=(0.2, 0.5120000000000001), end=(0.5120000000000001, 0.5120000000000001)) - line\_20: a Line \[gray\] drawn in box3 (start=(0.5130438570827405, 0.7994555449356961), end=(0.7994555449356961, 0.7994555449356961)) - line\_21: a Line \[yellow\] drawn in box3 (start=(0.513045, 0.513045), end=(0.513045, 0.7994554495200001)) - line\_22: a Line \[yellow\] drawn in box3 (start=(0.513045, 0.7994554495200001), end=(0.7994554495200001, 0.7994554495200001)) - line\_23: a Line \[yellow\] drawn in box3 (start=(0.7994554495200001, 0.7994554495200001), end=(0.7994554495200001, 0.5130445880088789)) - line\_24: a Line \[yellow\] drawn in box3 (start=(0.7994554495200001, 0.5130445880088789), end=(0.5130445880088789, 0.5130445880088789)) - line\_3: a Line \[gray\] drawn in box3 (start=(0.5120000000000001, 0.5120000000000001), end=(0.5120000000000001, 0.7995392)) - line\_4: a Line \[gray\] drawn in box3 (start=(0.5120000000000001, 0.7995392), end=(0.7995392, 0.7995392)) - line\_5: a Line \[gray\] drawn in box3 (start=(0.7995392, 0.7995392), end=(0.7995392, 0.512884056522752)) - line\_6: a Line \[gray\] drawn in box3 (start=(0.7995392, 0.512884056522752), end=(0.512884056522752, 0.512884056522752)) - line\_7: a Line \[gray\] drawn in box3 (start=(0.512884056522752, 0.512884056522752), end=(0.512884056522752, 0.7994688034800593)) - line\_8: a Line \[gray\] drawn in box3 (start=(0.512884056522752, 0.7994688034800593), end=(0.7994688034800593, 0.7994688034800593)) - line\_9: a Line \[gray\] drawn in box3 (start=(0.7994688034800593, 0.7994688034800593), end=(0.7994688034800593, 0.5130189943751092)) - p\_dot: a Point \[red\] labelled "p" drawn in box3 (location=(0.513045, 0.513045)) - para3: a FunctionPlot \[blue\] drawn in box3 (function=\, x\_range=(0.0, 1.0)) - para\_a: a FunctionPlot \[blue\] labelled "f" drawn in box4 (function=\, x\_range=(0.0, 1.0)) - para\_b: a FunctionPlot \[red\] drawn in box4 (function=\, x\_range=(0.0, 1.0)) - para\_b28: a FunctionPlot \[gray\] drawn in box4 (function=\, x\_range=(0.0, 1.0)) - point: a Point \[yellow\] drawn in box4 - point\_2: a Point \[yellow\] drawn in box4 (location=(0.513045, 0.513045)) - point\_3: a Point \[yellow\] drawn in box4 (location=(0.799455, 0.799455)) - q\_dot: a Point \[red\] labelled "q" drawn in box3 (location=(0.799455, 0.799455)) - star4: a Point \[gray\] labelled "x^\*" drawn in box4 (location=(0.6875, 0.6875)) - vals2: a Math \[text\] that says "$p = 0.513, quad q = 0.800$" #### Beats ##### [07:32.071](https://academa.ai/lectures/period-doubling-logistic-map?t=452.07149999999996) Narration: So here is the knob at three point two again, and here is an orbit starting from a fifth of capacity. I have drawn the first ten years in grey, because they are only the transient: the population finding its feet. Board: Empty. Actions: - [07:32.071](https://academa.ai/lectures/period-doubling-logistic-map?t=452.07149999999996): head\_two is shown on the screen, written out. - [07:32.071](https://academa.ai/lectures/period-doubling-logistic-map?t=452.07149999999996): box3 is shown on the screen, written out. - [07:32.071](https://academa.ai/lectures/period-doubling-logistic-map?t=452.07149999999996): para3 is shown on the screen, written out. - [07:32.071](https://academa.ai/lectures/period-doubling-logistic-map?t=452.07149999999996): diag3 is shown on the screen, written out. - [07:32.779](https://academa.ai/lectures/period-doubling-logistic-map?t=452.7795): knob3 is shown on the screen, written out. - [07:40.419](https://academa.ai/lectures/period-doubling-logistic-map?t=460.41949999999997): line is shown on the screen, written out. - [07:40.579](https://academa.ai/lectures/period-doubling-logistic-map?t=460.57949999999994): line\_2 is shown on the screen, written out. - [07:40.739](https://academa.ai/lectures/period-doubling-logistic-map?t=460.73949999999996): line\_3 is shown on the screen, written out. - [07:40.899](https://academa.ai/lectures/period-doubling-logistic-map?t=460.89949999999993): line\_4 is shown on the screen, written out. - [07:41.059](https://academa.ai/lectures/period-doubling-logistic-map?t=461.05949999999996): line\_5 is shown on the screen, written out. - [07:41.219](https://academa.ai/lectures/period-doubling-logistic-map?t=461.2195): line\_6 is shown on the screen, written out. - [07:41.379](https://academa.ai/lectures/period-doubling-logistic-map?t=461.37949999999995): line\_7 is shown on the screen, written out. - [07:41.539](https://academa.ai/lectures/period-doubling-logistic-map?t=461.5395): line\_8 is shown on the screen, written out. - [07:41.699](https://academa.ai/lectures/period-doubling-logistic-map?t=461.69949999999994): line\_9 is shown on the screen, written out. - [07:41.859](https://academa.ai/lectures/period-doubling-logistic-map?t=461.85949999999997): line\_10 is shown on the screen, written out. - [07:42.019](https://academa.ai/lectures/period-doubling-logistic-map?t=462.01949999999994): line\_11 is shown on the screen, written out. - [07:42.179](https://academa.ai/lectures/period-doubling-logistic-map?t=462.17949999999996): line\_12 is shown on the screen, written out. - [07:42.339](https://academa.ai/lectures/period-doubling-logistic-map?t=462.33949999999993): line\_13 is shown on the screen, written out. - [07:42.499](https://academa.ai/lectures/period-doubling-logistic-map?t=462.49949999999995): line\_14 is shown on the screen, written out. - [07:42.659](https://academa.ai/lectures/period-doubling-logistic-map?t=462.6595): line\_15 is shown on the screen, written out. - [07:42.819](https://academa.ai/lectures/period-doubling-logistic-map?t=462.81949999999995): line\_16 is shown on the screen, written out. - [07:42.979](https://academa.ai/lectures/period-doubling-logistic-map?t=462.9795): line\_17 is shown on the screen, written out. - [07:43.139](https://academa.ai/lectures/period-doubling-logistic-map?t=463.13949999999994): line\_18 is shown on the screen, written out. - [07:43.299](https://academa.ai/lectures/period-doubling-logistic-map?t=463.29949999999997): line\_19 is shown on the screen, written out. - [07:43.459](https://academa.ai/lectures/period-doubling-logistic-map?t=463.45949999999993): line\_20 is shown on the screen, written out. ##### [07:45.814](https://academa.ai/lectures/period-doubling-logistic-map?t=465.81399999999996) Narration: And here is what it finds. Not a point at all. A square. Board: knob3 — a Math \[text\] that says "$r = 3.2$"; box3 — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_two — a Heading that says "A Square Instead of a Point"; para3 — a FunctionPlot \[blue\] drawn in box3 (function=\, x\_range=(0.0, 1.0)); diag3 — a Line \[green\] labelled "y = x" drawn in box3 (end=(1.0, 1.0)); line — a Line \[gray\] drawn in box3 (start=(0.2, 0.0), end=(0.2, 0.5120000000000001)); line\_2 — a Line \[gray\] drawn in box3 (start=(0.2, 0.5120000000000001), end=(0.5120000000000001, 0.5120000000000001)); line\_3 — a Line \[gray\] drawn in box3 (start=(0.5120000000000001, 0.5120000000000001), end=(0.5120000000000001, 0.7995392)); line\_4 — a Line \[gray\] drawn in box3 (start=(0.5120000000000001, 0.7995392), end=(0.7995392, 0.7995392)); line\_5 — a Line \[gray\] drawn in box3 (start=(0.7995392, 0.7995392), end=(0.7995392, 0.512884056522752)); line\_6 — a Line \[gray\] drawn in box3 (start=(0.7995392, 0.512884056522752), end=(0.512884056522752, 0.512884056522752)); line\_7 — a Line \[gray\] drawn in box3 (start=(0.512884056522752, 0.512884056522752), end=(0.512884056522752, 0.7994688034800593)); line\_8 — a Line \[gray\] drawn in box3 (start=(0.512884056522752, 0.7994688034800593), end=(0.7994688034800593, 0.7994688034800593)); line\_9 — a Line \[gray\] drawn in box3 (start=(0.7994688034800593, 0.7994688034800593), end=(0.7994688034800593, 0.5130189943751092)); line\_10 — a Line \[gray\] drawn in box3 (start=(0.7994688034800593, 0.5130189943751092), end=(0.5130189943751092, 0.5130189943751092)); line\_11 — a Line \[gray\] drawn in box3 (start=(0.5130189943751092, 0.5130189943751092), end=(0.5130189943751092, 0.7994576185134749)); line\_12 — a Line \[gray\] drawn in box3 (start=(0.5130189943751092, 0.7994576185134749), end=(0.7994576185134749, 0.7994576185134749)); line\_13 — a Line \[gray\] drawn in box3 (start=(0.7994576185134749, 0.7994576185134749), end=(0.7994576185134749, 0.5130404310855622)); line\_14 — a Line \[gray\] drawn in box3 (start=(0.7994576185134749, 0.5130404310855622), end=(0.5130404310855622, 0.5130404310855622)); line\_15 — a Line \[gray\] drawn in box3 (start=(0.5130404310855622, 0.5130404310855622), end=(0.5130404310855622, 0.7994558309027286)); line\_16 — a Line \[gray\] drawn in box3 (start=(0.5130404310855622, 0.7994558309027286), end=(0.7994558309027286, 0.7994558309027286)); line\_17 — a Line \[gray\] drawn in box3 (start=(0.7994558309027286, 0.7994558309027286), end=(0.7994558309027286, 0.5130438570827405)); line\_18 — a Line \[gray\] drawn in box3 (start=(0.7994558309027286, 0.5130438570827405), end=(0.5130438570827405, 0.5130438570827405)); line\_19 — a Line \[gray\] drawn in box3 (start=(0.5130438570827405, 0.5130438570827405), end=(0.5130438570827405, 0.7994555449356961)); line\_20 — a Line \[gray\] drawn in box3 (start=(0.5130438570827405, 0.7994555449356961), end=(0.7994555449356961, 0.7994555449356961)) Actions: - [07:49.691](https://academa.ai/lectures/period-doubling-logistic-map?t=469.69149999999996): line\_21 is shown on the screen, written out. - [07:50.141](https://academa.ai/lectures/period-doubling-logistic-map?t=470.14149999999995): line\_22 is shown on the screen, written out. - [07:50.591](https://academa.ai/lectures/period-doubling-logistic-map?t=470.59149999999994): line\_23 is shown on the screen, written out. - [07:51.041](https://academa.ai/lectures/period-doubling-logistic-map?t=471.0414999999999): line\_24 is shown on the screen, written out. ##### [07:51.174](https://academa.ai/lectures/period-doubling-logistic-map?t=471.174) Narration: Read that square as a timetable. Up to the curve and across to the line is one year. Then again is the next year. And the fourth move brings you back exactly where you started, so year three is a repeat of year one. Board: knob3 — a Math \[text\] that says "$r = 3.2$"; box3 — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_two — a Heading that says "A Square Instead of a Point"; para3 — a FunctionPlot \[blue\] drawn in box3 (function=\, x\_range=(0.0, 1.0)); diag3 — a Line \[green\] labelled "y = x" drawn in box3 (end=(1.0, 1.0)); line — a Line \[gray\] drawn in box3 (start=(0.2, 0.0), end=(0.2, 0.5120000000000001)); line\_2 — a Line \[gray\] drawn in box3 (start=(0.2, 0.5120000000000001), end=(0.5120000000000001, 0.5120000000000001)); line\_3 — a Line \[gray\] drawn in box3 (start=(0.5120000000000001, 0.5120000000000001), end=(0.5120000000000001, 0.7995392)); line\_4 — a Line \[gray\] drawn in box3 (start=(0.5120000000000001, 0.7995392), end=(0.7995392, 0.7995392)); line\_5 — a Line \[gray\] drawn in box3 (start=(0.7995392, 0.7995392), end=(0.7995392, 0.512884056522752)); line\_6 — a Line \[gray\] drawn in box3 (start=(0.7995392, 0.512884056522752), end=(0.512884056522752, 0.512884056522752)); line\_7 — a Line \[gray\] drawn in box3 (start=(0.512884056522752, 0.512884056522752), end=(0.512884056522752, 0.7994688034800593)); line\_8 — a Line \[gray\] drawn in box3 (start=(0.512884056522752, 0.7994688034800593), end=(0.7994688034800593, 0.7994688034800593)); line\_9 — a Line \[gray\] drawn in box3 (start=(0.7994688034800593, 0.7994688034800593), end=(0.7994688034800593, 0.5130189943751092)); line\_10 — a Line \[gray\] drawn in box3 (start=(0.7994688034800593, 0.5130189943751092), end=(0.5130189943751092, 0.5130189943751092)); line\_11 — a Line \[gray\] drawn in box3 (start=(0.5130189943751092, 0.5130189943751092), end=(0.5130189943751092, 0.7994576185134749)); line\_12 — a Line \[gray\] drawn in box3 (start=(0.5130189943751092, 0.7994576185134749), end=(0.7994576185134749, 0.7994576185134749)); line\_13 — a Line \[gray\] drawn in box3 (start=(0.7994576185134749, 0.7994576185134749), end=(0.7994576185134749, 0.5130404310855622)); line\_14 — a Line \[gray\] drawn in box3 (start=(0.7994576185134749, 0.5130404310855622), end=(0.5130404310855622, 0.5130404310855622)); line\_15 — a Line \[gray\] drawn in box3 (start=(0.5130404310855622, 0.5130404310855622), end=(0.5130404310855622, 0.7994558309027286)); line\_16 — a Line \[gray\] drawn in box3 (start=(0.5130404310855622, 0.7994558309027286), end=(0.7994558309027286, 0.7994558309027286)); line\_17 — a Line \[gray\] drawn in box3 (start=(0.7994558309027286, 0.7994558309027286), end=(0.7994558309027286, 0.5130438570827405)); line\_18 — a Line \[gray\] drawn in box3 (start=(0.7994558309027286, 0.5130438570827405), end=(0.5130438570827405, 0.5130438570827405)); line\_19 — a Line \[gray\] drawn in box3 (start=(0.5130438570827405, 0.5130438570827405), end=(0.5130438570827405, 0.7994555449356961)); line\_20 — a Line \[gray\] drawn in box3 (start=(0.5130438570827405, 0.7994555449356961), end=(0.7994555449356961, 0.7994555449356961)); line\_21 — a Line \[yellow\] drawn in box3 (start=(0.513045, 0.513045), end=(0.513045, 0.7994554495200001)); line\_22 — a Line \[yellow\] drawn in box3 (start=(0.513045, 0.7994554495200001), end=(0.7994554495200001, 0.7994554495200001)); line\_23 — a Line \[yellow\] drawn in box3 (start=(0.7994554495200001, 0.7994554495200001), end=(0.7994554495200001, 0.5130445880088789)); line\_24 — a Line \[yellow\] drawn in box3 (start=(0.7994554495200001, 0.5130445880088789), end=(0.5130445880088789, 0.5130445880088789)) Actions: - [07:53.96](https://academa.ai/lectures/period-doubling-logistic-map?t=473.96049999999997): line\_21 is indicated — a transient flash. - [07:54.819](https://academa.ai/lectures/period-doubling-logistic-map?t=474.81949999999995): line\_22 is indicated — a transient flash. - [07:57.246](https://academa.ai/lectures/period-doubling-logistic-map?t=477.24649999999997): line\_23 is indicated — a transient flash. - [08:0.775](https://academa.ai/lectures/period-doubling-logistic-map?t=480.77549999999997): line\_24 is indicated — a transient flash. ##### [08:5.392](https://academa.ai/lectures/period-doubling-logistic-map?t=485.39199999999994) Narration: So the population never settles. It alternates. A lean year, a crowded year, a lean year, a crowded year, without end. Nought point five one three, then nought point eight, then nought point five one three again. Board: Unchanged from the preceding beat in this scene. Actions: - [08:9.92](https://academa.ai/lectures/period-doubling-logistic-map?t=489.92049999999995): p\_dot is shown on the screen, written out. - [08:10.813](https://academa.ai/lectures/period-doubling-logistic-map?t=490.8135): q\_dot is shown on the screen, written out. - [08:16.375](https://academa.ai/lectures/period-doubling-logistic-map?t=496.3755): vals2 is shown on the screen, written out. ##### [08:21.758](https://academa.ai/lectures/period-doubling-logistic-map?t=501.7579999999999) Narration: And notice how far apart those two values are. This is not the old fixed point with a small wobble on top. The population is swinging from half capacity to four fifths and back, every single year, and it will do that for ever. Board: knob3 — a Math \[text\] that says "$r = 3.2$"; vals2 — a Math \[text\] that says "$p = 0.513, quad q = 0.800$"; box3 — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_two — a Heading that says "A Square Instead of a Point"; para3 — a FunctionPlot \[blue\] drawn in box3 (function=\, x\_range=(0.0, 1.0)); diag3 — a Line \[green\] labelled "y = x" drawn in box3 (end=(1.0, 1.0)); line — a Line \[gray\] drawn in box3 (start=(0.2, 0.0), end=(0.2, 0.5120000000000001)); line\_2 — a Line \[gray\] drawn in box3 (start=(0.2, 0.5120000000000001), end=(0.5120000000000001, 0.5120000000000001)); line\_3 — a Line \[gray\] drawn in box3 (start=(0.5120000000000001, 0.5120000000000001), end=(0.5120000000000001, 0.7995392)); line\_4 — a Line \[gray\] drawn in box3 (start=(0.5120000000000001, 0.7995392), end=(0.7995392, 0.7995392)); line\_5 — a Line \[gray\] drawn in box3 (start=(0.7995392, 0.7995392), end=(0.7995392, 0.512884056522752)); line\_6 — a Line \[gray\] drawn in box3 (start=(0.7995392, 0.512884056522752), end=(0.512884056522752, 0.512884056522752)); line\_7 — a Line \[gray\] drawn in box3 (start=(0.512884056522752, 0.512884056522752), end=(0.512884056522752, 0.7994688034800593)); line\_8 — a Line \[gray\] drawn in box3 (start=(0.512884056522752, 0.7994688034800593), end=(0.7994688034800593, 0.7994688034800593)); line\_9 — a Line \[gray\] drawn in box3 (start=(0.7994688034800593, 0.7994688034800593), end=(0.7994688034800593, 0.5130189943751092)); line\_10 — a Line \[gray\] drawn in box3 (start=(0.7994688034800593, 0.5130189943751092), end=(0.5130189943751092, 0.5130189943751092)); line\_11 — a Line \[gray\] drawn in box3 (start=(0.5130189943751092, 0.5130189943751092), end=(0.5130189943751092, 0.7994576185134749)); line\_12 — a Line \[gray\] drawn in box3 (start=(0.5130189943751092, 0.7994576185134749), end=(0.7994576185134749, 0.7994576185134749)); line\_13 — a Line \[gray\] drawn in box3 (start=(0.7994576185134749, 0.7994576185134749), end=(0.7994576185134749, 0.5130404310855622)); line\_14 — a Line \[gray\] drawn in box3 (start=(0.7994576185134749, 0.5130404310855622), end=(0.5130404310855622, 0.5130404310855622)); line\_15 — a Line \[gray\] drawn in box3 (start=(0.5130404310855622, 0.5130404310855622), end=(0.5130404310855622, 0.7994558309027286)); line\_16 — a Line \[gray\] drawn in box3 (start=(0.5130404310855622, 0.7994558309027286), end=(0.7994558309027286, 0.7994558309027286)); line\_17 — a Line \[gray\] drawn in box3 (start=(0.7994558309027286, 0.7994558309027286), end=(0.7994558309027286, 0.5130438570827405)); line\_18 — a Line \[gray\] drawn in box3 (start=(0.7994558309027286, 0.5130438570827405), end=(0.5130438570827405, 0.5130438570827405)); line\_19 — a Line \[gray\] drawn in box3 (start=(0.5130438570827405, 0.5130438570827405), end=(0.5130438570827405, 0.7994555449356961)); line\_20 — a Line \[gray\] drawn in box3 (start=(0.5130438570827405, 0.7994555449356961), end=(0.7994555449356961, 0.7994555449356961)); line\_21 — a Line \[yellow\] drawn in box3 (start=(0.513045, 0.513045), end=(0.513045, 0.7994554495200001)); line\_22 — a Line \[yellow\] drawn in box3 (start=(0.513045, 0.7994554495200001), end=(0.7994554495200001, 0.7994554495200001)); line\_23 — a Line \[yellow\] drawn in box3 (start=(0.7994554495200001, 0.7994554495200001), end=(0.7994554495200001, 0.5130445880088789)); line\_24 — a Line \[yellow\] drawn in box3 (start=(0.7994554495200001, 0.5130445880088789), end=(0.5130445880088789, 0.5130445880088789)); p\_dot — a Point \[red\] labelled "p" drawn in box3 (location=(0.513045, 0.513045)); q\_dot — a Point \[red\] labelled "q" drawn in box3 (location=(0.799455, 0.799455)) Actions: - None. ##### [08:36.51](https://academa.ai/lectures/period-doubling-logistic-map?t=516.5105) Narration: The two values are locked to each other. F of p is q, and f of q is p, so applying the rule twice brings each of them home. Board: Unchanged from the preceding beat in this scene. Actions: - [08:37.822](https://academa.ai/lectures/period-doubling-logistic-map?t=517.8225): cycle\_eq is shown on the screen, written out. ##### [08:46.619](https://academa.ai/lectures/period-doubling-logistic-map?t=526.6194999999999) Narration: This is called a two cycle, and it appeared at exactly the value of r where the fixed point lost its grip. That is no coincidence, and there is a clean way to see why. Board: knob3 — a Math \[text\] that says "$r = 3.2$"; cycle\_eq — a Math \[text\] that says "$q = f(p), quad p = f(q)$"; vals2 — a Math \[text\] that says "$p = 0.513, quad q = 0.800$"; box3 — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_two — a Heading that says "A Square Instead of a Point"; para3 — a FunctionPlot \[blue\] drawn in box3 (function=\, x\_range=(0.0, 1.0)); diag3 — a Line \[green\] labelled "y = x" drawn in box3 (end=(1.0, 1.0)); line — a Line \[gray\] drawn in box3 (start=(0.2, 0.0), end=(0.2, 0.5120000000000001)); line\_2 — a Line \[gray\] drawn in box3 (start=(0.2, 0.5120000000000001), end=(0.5120000000000001, 0.5120000000000001)); line\_3 — a Line \[gray\] drawn in box3 (start=(0.5120000000000001, 0.5120000000000001), end=(0.5120000000000001, 0.7995392)); line\_4 — a Line \[gray\] drawn in box3 (start=(0.5120000000000001, 0.7995392), end=(0.7995392, 0.7995392)); line\_5 — a Line \[gray\] drawn in box3 (start=(0.7995392, 0.7995392), end=(0.7995392, 0.512884056522752)); line\_6 — a Line \[gray\] drawn in box3 (start=(0.7995392, 0.512884056522752), end=(0.512884056522752, 0.512884056522752)); line\_7 — a Line \[gray\] drawn in box3 (start=(0.512884056522752, 0.512884056522752), end=(0.512884056522752, 0.7994688034800593)); line\_8 — a Line \[gray\] drawn in box3 (start=(0.512884056522752, 0.7994688034800593), end=(0.7994688034800593, 0.7994688034800593)); line\_9 — a Line \[gray\] drawn in box3 (start=(0.7994688034800593, 0.7994688034800593), end=(0.7994688034800593, 0.5130189943751092)); line\_10 — a Line \[gray\] drawn in box3 (start=(0.7994688034800593, 0.5130189943751092), end=(0.5130189943751092, 0.5130189943751092)); line\_11 — a Line \[gray\] drawn in box3 (start=(0.5130189943751092, 0.5130189943751092), end=(0.5130189943751092, 0.7994576185134749)); line\_12 — a Line \[gray\] drawn in box3 (start=(0.5130189943751092, 0.7994576185134749), end=(0.7994576185134749, 0.7994576185134749)); line\_13 — a Line \[gray\] drawn in box3 (start=(0.7994576185134749, 0.7994576185134749), end=(0.7994576185134749, 0.5130404310855622)); line\_14 — a Line \[gray\] drawn in box3 (start=(0.7994576185134749, 0.5130404310855622), end=(0.5130404310855622, 0.5130404310855622)); line\_15 — a Line \[gray\] drawn in box3 (start=(0.5130404310855622, 0.5130404310855622), end=(0.5130404310855622, 0.7994558309027286)); line\_16 — a Line \[gray\] drawn in box3 (start=(0.5130404310855622, 0.7994558309027286), end=(0.7994558309027286, 0.7994558309027286)); line\_17 — a Line \[gray\] drawn in box3 (start=(0.7994558309027286, 0.7994558309027286), end=(0.7994558309027286, 0.5130438570827405)); line\_18 — a Line \[gray\] drawn in box3 (start=(0.7994558309027286, 0.5130438570827405), end=(0.5130438570827405, 0.5130438570827405)); line\_19 — a Line \[gray\] drawn in box3 (start=(0.5130438570827405, 0.5130438570827405), end=(0.5130438570827405, 0.7994555449356961)); line\_20 — a Line \[gray\] drawn in box3 (start=(0.5130438570827405, 0.7994555449356961), end=(0.7994555449356961, 0.7994555449356961)); line\_21 — a Line \[yellow\] drawn in box3 (start=(0.513045, 0.513045), end=(0.513045, 0.7994554495200001)); line\_22 — a Line \[yellow\] drawn in box3 (start=(0.513045, 0.7994554495200001), end=(0.7994554495200001, 0.7994554495200001)); line\_23 — a Line \[yellow\] drawn in box3 (start=(0.7994554495200001, 0.7994554495200001), end=(0.7994554495200001, 0.5130445880088789)); line\_24 — a Line \[yellow\] drawn in box3 (start=(0.7994554495200001, 0.5130445880088789), end=(0.5130445880088789, 0.5130445880088789)); p\_dot — a Point \[red\] labelled "p" drawn in box3 (location=(0.513045, 0.513045)); q\_dot — a Point \[red\] labelled "q" drawn in box3 (location=(0.799455, 0.799455)) Actions: - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): box3 is hidden from the screen — left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): para3 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): diag3 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_2 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_3 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_4 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_5 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_6 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_7 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_8 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_9 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_10 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_11 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_12 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_13 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_14 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_15 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_16 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_17 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_18 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_19 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_20 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_21 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_22 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_23 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): line\_24 is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): p\_dot is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): q\_dot is hidden from the screen — box3 left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): cycle\_eq is hidden from the screen — left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): head\_two is hidden from the screen — left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): knob3 is hidden from the screen — left the board. - [08:57.289](https://academa.ai/lectures/period-doubling-logistic-map?t=537.289): vals2 is hidden from the screen — left the board. ##### [08:58.489](https://academa.ai/lectures/period-doubling-logistic-map?t=538.4889999999999) Narration: If a population repeats every two years, then applying the rule twice must bring it back to itself. So let us look at the map that takes this year straight to the year after next. Board: Empty. Actions: - [08:58.489](https://academa.ai/lectures/period-doubling-logistic-map?t=538.4889999999999): head\_double is shown on the screen, written out. - [08:58.489](https://academa.ai/lectures/period-doubling-logistic-map?t=538.4889999999999): box4 is shown on the screen, written out. - [08:58.489](https://academa.ai/lectures/period-doubling-logistic-map?t=538.4889999999999): para\_a is shown on the screen, written out. - [08:58.489](https://academa.ai/lectures/period-doubling-logistic-map?t=538.4889999999999): diag4 is shown on the screen, written out. - [09:6.116](https://academa.ai/lectures/period-doubling-logistic-map?t=546.1165): box4 moves to a new place on the board. - [09:6.116](https://academa.ai/lectures/period-doubling-logistic-map?t=546.1165): compose is shown on the screen, written out. ##### [09:9.654](https://academa.ai/lectures/period-doubling-logistic-map?t=549.654) Narration: That is f of f of x, and here it is in red, sitting on top of the original hump. It has two humps of its own, and it meets the line y equals x in four places. Board: compose — a Math \[text\] that says "$x\_(n+2) = f(f(x\_n))$"; box4 — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_double — a Heading that says "Applying the Rule Twice"; para\_a — a FunctionPlot \[blue\] labelled "f" drawn in box4 (function=\, x\_range=(0.0, 1.0)); diag4 — a Line \[green\] labelled "y = x" drawn in box4 (end=(1.0, 1.0)) Actions: - [09:12.696](https://academa.ai/lectures/period-doubling-logistic-map?t=552.6964999999999): para\_b is shown on the screen, drawn. - [09:19.801](https://academa.ai/lectures/period-doubling-logistic-map?t=559.8014999999999): count is shown on the screen, written out. ##### [09:21.748](https://academa.ai/lectures/period-doubling-logistic-map?t=561.7484999999999) Narration: Two of those crossings are old friends. Zero is fixed under one step, so it is certainly fixed under two. And so is x star, the fixed point we already had, which is still sitting there at nought point six eight seven five. Board: compose — a Math \[text\] that says "$x\_(n+2) = f(f(x\_n))$"; count — a Text \[text\] that says "Four crossings: $0$ and $x^\*$ carried over from $f$, and two new ones, $p$ and $q$."; box4 — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_double — a Heading that says "Applying the Rule Twice"; para\_a — a FunctionPlot \[blue\] labelled "f" drawn in box4 (function=\, x\_range=(0.0, 1.0)); diag4 — a Line \[green\] labelled "y = x" drawn in box4 (end=(1.0, 1.0)); para\_b — a FunctionPlot \[red\] drawn in box4 (function=\, x\_range=(0.0, 1.0)) Actions: - [09:24.963](https://academa.ai/lectures/period-doubling-logistic-map?t=564.9635): point is shown on the screen, grown. - [09:26.963](https://academa.ai/lectures/period-doubling-logistic-map?t=566.9635): point is hidden from the screen. - [09:30.304](https://academa.ai/lectures/period-doubling-logistic-map?t=570.3045): star4 is shown on the screen, written out. ##### [09:37.173](https://academa.ai/lectures/period-doubling-logistic-map?t=577.1734999999999) Narration: The other two are new, and they are our lean year and our crowded year. Each is a perfectly good fixed point of this doubled map. And look at how the red curve meets the line at those two. It crosses gently. A shallow slope means attracting. Board: compose — a Math \[text\] that says "$x\_(n+2) = f(f(x\_n))$"; count — a Text \[text\] that says "Four crossings: $0$ and $x^\*$ carried over from $f$, and two new ones, $p$ and $q$."; box4 — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_double — a Heading that says "Applying the Rule Twice"; para\_a — a FunctionPlot \[blue\] labelled "f" drawn in box4 (function=\, x\_range=(0.0, 1.0)); diag4 — a Line \[green\] labelled "y = x" drawn in box4 (end=(1.0, 1.0)); para\_b — a FunctionPlot \[red\] drawn in box4 (function=\, x\_range=(0.0, 1.0)); star4 — a Point \[gray\] labelled "x^\*" drawn in box4 (location=(0.6875, 0.6875)) Actions: - [09:39.797](https://academa.ai/lectures/period-doubling-logistic-map?t=579.7974999999999): point\_2 is shown on the screen, grown. - [09:40.575](https://academa.ai/lectures/period-doubling-logistic-map?t=580.5754999999999): point\_3 is shown on the screen, grown. - [09:41.797](https://academa.ai/lectures/period-doubling-logistic-map?t=581.7974999999999): point\_2 is hidden from the screen. - [09:42.575](https://academa.ai/lectures/period-doubling-logistic-map?t=582.5754999999999): point\_3 is hidden from the screen. - [09:51.314](https://academa.ai/lectures/period-doubling-logistic-map?t=591.3145): para\_b is indicated — a transient flash. ##### [09:54.248](https://academa.ai/lectures/period-doubling-logistic-map?t=594.2479999999999) Narration: Now here is the event at r equal to three. Below three the red curve looked like this: tucked up against the line, touching it only at x star. As r passed three it pushed straight through, and made two new crossings, one on each side. Board: Unchanged from the preceding beat in this scene. Actions: - [09:59.612](https://academa.ai/lectures/period-doubling-logistic-map?t=599.6125): para\_b28 is shown on the screen, written out. - [10:6.775](https://academa.ai/lectures/period-doubling-logistic-map?t=606.7755): para\_b28 is hidden from the screen. ##### [10:11.718](https://academa.ai/lectures/period-doubling-logistic-map?t=611.718) Narration: One solution became three, and the middle one was the one that failed. It is worth saying plainly what did not happen. Nothing was destroyed and no solution went missing. The fixed point is exactly where it always was, and no population will ever go to it again. Board: Unchanged from the preceding beat in this scene. Actions: - [10:23.409](https://academa.ai/lectures/period-doubling-logistic-map?t=623.4095): star4 is indicated — a transient flash. - [10:27.821](https://academa.ai/lectures/period-doubling-logistic-map?t=627.8214999999999): box4 is hidden from the screen — left the board. - [10:27.821](https://academa.ai/lectures/period-doubling-logistic-map?t=627.8214999999999): para\_a is hidden from the screen — box4 left the board. - [10:27.821](https://academa.ai/lectures/period-doubling-logistic-map?t=627.8214999999999): diag4 is hidden from the screen — box4 left the board. - [10:27.821](https://academa.ai/lectures/period-doubling-logistic-map?t=627.8214999999999): para\_b is hidden from the screen — box4 left the board. - [10:27.821](https://academa.ai/lectures/period-doubling-logistic-map?t=627.8214999999999): star4 is hidden from the screen — box4 left the board. - [10:27.821](https://academa.ai/lectures/period-doubling-logistic-map?t=627.8214999999999): compose is hidden from the screen — left the board. - [10:27.821](https://academa.ai/lectures/period-doubling-logistic-map?t=627.8214999999999): count is hidden from the screen — left the board. - [10:27.821](https://academa.ai/lectures/period-doubling-logistic-map?t=627.8214999999999): head\_double is hidden from the screen — left the board. ##### [10:29.021](https://academa.ai/lectures/period-doubling-logistic-map?t=629.0215) Narration: So here is the pattern, and it is about to happen over and over. A point holds on while the slope there is shallower than one in size. When the slope reaches minus one, the point does not vanish. It stays, it turns repelling, and a cycle of twice the period is born around it. Board: Empty. Actions: - [10:29.021](https://academa.ai/lectures/period-doubling-logistic-map?t=629.0215): head\_law is shown on the screen, written out. - [10:39.597](https://academa.ai/lectures/period-doubling-logistic-map?t=639.5975): law is shown on the screen, written out. - [10:43.127](https://academa.ai/lectures/period-doubling-logistic-map?t=643.1274999999999): law\_note is shown on the screen, written out. ##### [10:48.952](https://academa.ai/lectures/period-doubling-logistic-map?t=648.952) Narration: Which invites the obvious question. Our new two cycle has a slope of its own, measured on that red curve. What do you suppose happens when that one reaches minus one? Board: law — a Math \[text\] that says "$f'(x^\*) = -1$"; law\_note — a Panel that says "When the slope at an attracting point reaches $-1$, the point does not disappear. It goes on existing, it stops attracting, and an attracting cycle of twice the period appears around it and takes over."; head\_law — a Heading that says "What Happens at a Doubling" Actions: - [10:53.979](https://academa.ai/lectures/period-doubling-logistic-map?t=653.9794999999999): A box is drawn around law. - [11:0.113](https://academa.ai/lectures/period-doubling-logistic-map?t=660.1125208333333): head\_law is hidden from the screen — left the board. - [11:0.113](https://academa.ai/lectures/period-doubling-logistic-map?t=660.1125208333333): law is hidden from the screen — left the board. - [11:0.113](https://academa.ai/lectures/period-doubling-logistic-map?t=660.1125208333333): law\_note is hidden from the screen — left the board. ### Scene 4: [Again, and Again, and Again](https://academa.ai/lectures/period-doubling-logistic-map?t=661.1541874999999) Span: 11:1.154–14:36.375 (661.1541874999999s–876.3748333333332s). #### Objects - box: an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25) - delta: a Math \[text\] that says "$delta = 4.669 thin 201 thin 609 dots$" - diag: a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)) - dots35: a Point \[red\] drawn in box (location=(0.38282, 0.38282), marker\_radius=0.012) - dots35\_2: a Point \[red\] drawn in box (location=(0.826941, 0.826941), marker\_radius=0.012) - dots35\_3: a Point \[red\] drawn in box (location=(0.500884, 0.500884), marker\_radius=0.012) - dots35\_4: a Point \[red\] drawn in box (location=(0.874997, 0.874997), marker\_radius=0.012) - eight\_note: a Text \[text\] that says "Eight values now, and the knob moved only a twentieth as far as it did the first time." - four\_vals: a Math \[text\] that says "$0.383 arrow.r 0.827 arrow.r 0.501 arrow.r 0.875$" - head\_delta: a Heading that says "A Number That Belongs to Nobody" - head\_eight: a Heading that says "And Then Eight" - head\_four: a Heading that says "Four Kinds of Year" - head\_ratio: a Heading that says "How Fast the Splittings Come" - knob: a Math \[text\] that says "$r = 3.5$" - line: a Line \[gray\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.56)) - line\_10: a Line \[gray\] drawn in box (start=(0.8499095593877504, 0.4464715508717464), end=(0.4464715508717464, 0.4464715508717464)) - line\_11: a Line \[gray\] drawn in box (start=(0.4464715508717464, 0.4464715508717464), end=(0.4464715508717464, 0.8649714679687339)) - line\_12: a Line \[gray\] drawn in box (start=(0.4464715508717464, 0.8649714679687339), end=(0.8649714679687339, 0.8649714679687339)) - line\_13: a Line \[gray\] drawn in box (start=(0.8649714679687339, 0.8649714679687339), end=(0.8649714679687339, 0.408785396490616)) - line\_14: a Line \[gray\] drawn in box (start=(0.8649714679687339, 0.408785396490616), end=(0.408785396490616, 0.408785396490616)) - line\_15: a Line \[gray\] drawn in box (start=(0.408785396490616, 0.408785396490616), end=(0.408785396490616, 0.8458796363731905)) - line\_16: a Line \[gray\] drawn in box (start=(0.408785396490616, 0.8458796363731905), end=(0.8458796363731905, 0.8458796363731905)) - line\_17: a Line \[gray\] drawn in box (start=(0.8458796363731905, 0.8458796363731905), end=(0.8458796363731905, 0.4562854699982232)) - line\_18: a Line \[gray\] drawn in box (start=(0.8458796363731905, 0.4562854699982232), end=(0.4562854699982232, 0.4562854699982232)) - line\_19: a Line \[gray\] drawn in box (start=(0.4562854699982232, 0.4562854699982232), end=(0.4562854699982232, 0.8683116395335331)) - line\_2: a Line \[gray\] drawn in box (start=(0.2, 0.56), end=(0.56, 0.56)) - line\_20: a Line \[gray\] drawn in box (start=(0.4562854699982232, 0.8683116395335331), end=(0.8683116395335331, 0.8683116395335331)) - line\_21: a Line \[yellow\] drawn in box (start=(0.38282, 0.38282), end=(0.38282, 0.8269409666)) - line\_22: a Line \[yellow\] drawn in box (start=(0.38282, 0.8269409666), end=(0.8269409666, 0.8269409666)) - line\_23: a Line \[yellow\] drawn in box (start=(0.8269409666, 0.8269409666), end=(0.8269409666, 0.5008836152553018)) - line\_24: a Line \[yellow\] drawn in box (start=(0.8269409666, 0.5008836152553018), end=(0.5008836152553018, 0.5008836152553018)) - 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line\_7: a Line \[gray\] drawn in box (start=(0.4153318400000001, 0.4153318400000001), end=(0.4153318400000001, 0.8499095593877504)) - line\_8: a Line \[gray\] drawn in box (start=(0.4153318400000001, 0.8499095593877504), end=(0.8499095593877504, 0.8499095593877504)) - line\_9: a Line \[gray\] drawn in box (start=(0.8499095593877504, 0.8499095593877504), end=(0.8499095593877504, 0.4464715508717464)) - para35: a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)) - para355: a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)) - r\_inf: a Math \[text\] that says "$r\_infinity = 3.569 thin 946 dots$" - ratio\_note: a Text \[text\] that says "Each gap is the distance to the next doubling. Each ratio compares one gap with the one after it." - split\_note: a Text \[text\] that says "Each value of the two cycle has split into two nearby values." - tbl: a Table \[text\] that says "Doubling $r$ Gap Ratio $1 arrow.r 2$ 3.000000 0.449490 4.751 $2 arrow.r 4$ 3.449490 0.094601 4.656 $4 arrow.r 8$ 3.544090 0.020317 4.668 $8 arrow.r 16$ 3.564407 0.004352 4.669 $16 arrow.r 32$ 3.568759 0.000932 4.669" (rows=(('Doubling', '$r$', 'Gap', 'Ratio'), ('$1 arrow.r 2$', '3.0000…, header=True) - univ\_note: a Panel that says "The same $delta$ turns up in the doubling cascade of a sine arch, or of almost any smooth map with a single rounded maximum. It belongs to the doubling, not to this equation." #### Beats ##### [11:1.154](https://academa.ai/lectures/period-doubling-logistic-map?t=661.1541874999999) Narration: Turn the knob again. At r equal to three point four four nine the two cycle fails, in exactly the way the fixed point failed. It is the same mechanism one level down. Board: Empty. Actions: - [11:1.154](https://academa.ai/lectures/period-doubling-logistic-map?t=661.1541874999999): head\_four is shown on the screen, written out. - [11:1.154](https://academa.ai/lectures/period-doubling-logistic-map?t=661.1541874999999): box is shown on the screen, written out. - [11:1.154](https://academa.ai/lectures/period-doubling-logistic-map?t=661.1541874999999): para35 is shown on the screen, written out. - [11:1.154](https://academa.ai/lectures/period-doubling-logistic-map?t=661.1541874999999): diag is shown on the screen, written out. ##### [11:13.109](https://academa.ai/lectures/period-doubling-logistic-map?t=673.1086875) Narration: The two cycle is a fixed point of the doubled map, so it has a slope of its own, and that slope walks down through minus one as we turn. When it crosses, each of the two values sheds a pair, and we have four. Board: box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_four — a Heading that says "Four Kinds of Year"; para35 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)) Actions: - None. ##### [11:26.933](https://academa.ai/lectures/period-doubling-logistic-map?t=686.9326874999999) Narration: Here is the knob at three point five. Grey for the transient again, and then the settled orbit in yellow. Board: Unchanged from the preceding beat in this scene. Actions: - [11:27.745](https://academa.ai/lectures/period-doubling-logistic-map?t=687.7451874999999): box moves to a new place on the board. - [11:27.745](https://academa.ai/lectures/period-doubling-logistic-map?t=687.7451874999999): knob is shown on the screen, written out. - [11:29.881](https://academa.ai/lectures/period-doubling-logistic-map?t=689.8811874999999): line is shown on the screen, written out. - [11:30.041](https://academa.ai/lectures/period-doubling-logistic-map?t=690.0411875): line\_2 is shown on the screen, written out. - [11:30.201](https://academa.ai/lectures/period-doubling-logistic-map?t=690.2011875): line\_3 is shown on the screen, written out. - [11:30.361](https://academa.ai/lectures/period-doubling-logistic-map?t=690.3611874999999): line\_4 is shown on the screen, written out. - [11:30.521](https://academa.ai/lectures/period-doubling-logistic-map?t=690.5211874999999): line\_5 is shown on the screen, written out. - [11:30.681](https://academa.ai/lectures/period-doubling-logistic-map?t=690.6811875): line\_6 is shown on the screen, written out. - [11:30.841](https://academa.ai/lectures/period-doubling-logistic-map?t=690.8411874999999): line\_7 is shown on the screen, written out. - [11:31.001](https://academa.ai/lectures/period-doubling-logistic-map?t=691.0011874999999): line\_8 is shown on the screen, written out. - [11:31.161](https://academa.ai/lectures/period-doubling-logistic-map?t=691.1611874999999): line\_9 is shown on the screen, written out. - [11:31.321](https://academa.ai/lectures/period-doubling-logistic-map?t=691.3211875): line\_10 is shown on the screen, written out. - [11:31.481](https://academa.ai/lectures/period-doubling-logistic-map?t=691.4811874999999): line\_11 is shown on the screen, written out. - [11:31.641](https://academa.ai/lectures/period-doubling-logistic-map?t=691.6411874999999): line\_12 is shown on the screen, written out. - [11:31.801](https://academa.ai/lectures/period-doubling-logistic-map?t=691.8011875): line\_13 is shown on the screen, written out. - [11:31.961](https://academa.ai/lectures/period-doubling-logistic-map?t=691.9611874999999): line\_14 is shown on the screen, written out. - [11:32.121](https://academa.ai/lectures/period-doubling-logistic-map?t=692.1211874999999): line\_15 is shown on the screen, written out. - [11:32.181](https://academa.ai/lectures/period-doubling-logistic-map?t=692.1811875): line\_21 is shown on the screen, written out. - [11:32.281](https://academa.ai/lectures/period-doubling-logistic-map?t=692.2811874999999): line\_16 is shown on the screen, written out. - [11:32.441](https://academa.ai/lectures/period-doubling-logistic-map?t=692.4411875): line\_17 is shown on the screen, written out. - [11:32.481](https://academa.ai/lectures/period-doubling-logistic-map?t=692.4811874999999): line\_22 is shown on the screen, written out. - [11:32.601](https://academa.ai/lectures/period-doubling-logistic-map?t=692.6011874999999): line\_18 is shown on the screen, written out. - [11:32.761](https://academa.ai/lectures/period-doubling-logistic-map?t=692.7611874999999): line\_19 is shown on the screen, written out. - [11:32.781](https://academa.ai/lectures/period-doubling-logistic-map?t=692.7811874999999): line\_23 is shown on the screen, written out. - [11:32.921](https://academa.ai/lectures/period-doubling-logistic-map?t=692.9211875): line\_20 is shown on the screen, written out. - [11:33.081](https://academa.ai/lectures/period-doubling-logistic-map?t=693.0811874999999): line\_24 is shown on the screen, written out. - [11:33.381](https://academa.ai/lectures/period-doubling-logistic-map?t=693.3811874999999): line\_25 is shown on the screen, written out. - [11:33.681](https://academa.ai/lectures/period-doubling-logistic-map?t=693.6811875): line\_26 is shown on the screen, written out. - [11:33.981](https://academa.ai/lectures/period-doubling-logistic-map?t=693.9811874999999): line\_27 is shown on the screen, written out. - [11:34.281](https://academa.ai/lectures/period-doubling-logistic-map?t=694.2811874999999): line\_28 is shown on the screen, written out. ##### [11:34.499](https://academa.ai/lectures/period-doubling-logistic-map?t=694.4991875) Narration: Four points, and the population visits them in a fixed order. Nought point three eight three, then nought point eight two seven, then nought point five zero one, then nought point eight seven five, and then back to the first. Board: knob — a Math \[text\] that says "$r = 3.5$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_four — a Heading that says "Four Kinds of Year"; para35 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); line — a Line \[gray\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.56)); line\_2 — a Line \[gray\] drawn in box (start=(0.2, 0.56), end=(0.56, 0.56)); line\_3 — a Line \[gray\] drawn in box (start=(0.56, 0.56), end=(0.56, 0.8623999999999999)); line\_4 — a Line \[gray\] drawn in box (start=(0.56, 0.8623999999999999), end=(0.8623999999999999, 0.8623999999999999)); line\_5 — a Line \[gray\] drawn in box (start=(0.8623999999999999, 0.8623999999999999), end=(0.8623999999999999, 0.4153318400000001)); line\_6 — a Line \[gray\] drawn in box (start=(0.8623999999999999, 0.4153318400000001), end=(0.4153318400000001, 0.4153318400000001)); line\_7 — a Line \[gray\] drawn in box (start=(0.4153318400000001, 0.4153318400000001), end=(0.4153318400000001, 0.8499095593877504)); line\_8 — a Line \[gray\] drawn in box (start=(0.4153318400000001, 0.8499095593877504), end=(0.8499095593877504, 0.8499095593877504)); line\_9 — a Line \[gray\] drawn in box (start=(0.8499095593877504, 0.8499095593877504), end=(0.8499095593877504, 0.4464715508717464)); line\_10 — a Line \[gray\] drawn in box (start=(0.8499095593877504, 0.4464715508717464), end=(0.4464715508717464, 0.4464715508717464)); line\_11 — a Line \[gray\] drawn in box (start=(0.4464715508717464, 0.4464715508717464), end=(0.4464715508717464, 0.8649714679687339)); line\_12 — a Line \[gray\] drawn in box (start=(0.4464715508717464, 0.8649714679687339), end=(0.8649714679687339, 0.8649714679687339)); line\_13 — a Line \[gray\] drawn in box (start=(0.8649714679687339, 0.8649714679687339), end=(0.8649714679687339, 0.408785396490616)); line\_14 — a Line \[gray\] drawn in box (start=(0.8649714679687339, 0.408785396490616), end=(0.408785396490616, 0.408785396490616)); line\_15 — a Line \[gray\] drawn in box (start=(0.408785396490616, 0.408785396490616), end=(0.408785396490616, 0.8458796363731905)); line\_16 — a Line \[gray\] drawn in box (start=(0.408785396490616, 0.8458796363731905), end=(0.8458796363731905, 0.8458796363731905)); line\_17 — a Line \[gray\] drawn in box (start=(0.8458796363731905, 0.8458796363731905), end=(0.8458796363731905, 0.4562854699982232)); line\_18 — a Line \[gray\] drawn in box (start=(0.8458796363731905, 0.4562854699982232), end=(0.4562854699982232, 0.4562854699982232)); line\_19 — a Line \[gray\] drawn in box (start=(0.4562854699982232, 0.4562854699982232), end=(0.4562854699982232, 0.8683116395335331)); line\_20 — a Line \[gray\] drawn in box (start=(0.4562854699982232, 0.8683116395335331), end=(0.8683116395335331, 0.8683116395335331)); line\_21 — a Line \[yellow\] drawn in box (start=(0.38282, 0.38282), end=(0.38282, 0.8269409666)); line\_22 — a Line \[yellow\] drawn in box (start=(0.38282, 0.8269409666), end=(0.8269409666, 0.8269409666)); line\_23 — a Line \[yellow\] drawn in box (start=(0.8269409666, 0.8269409666), end=(0.8269409666, 0.5008836152553018)); line\_24 — a Line \[yellow\] drawn in box (start=(0.8269409666, 0.5008836152553018), end=(0.5008836152553018, 0.5008836152553018)); line\_25 — a Line \[yellow\] drawn in box (start=(0.5008836152553018, 0.5008836152553018), end=(0.5008836152553018, 0.8749972672842821)); line\_26 — a Line \[yellow\] drawn in box (start=(0.5008836152553018, 0.8749972672842821), end=(0.8749972672842821, 0.8749972672842821)); line\_27 — a Line \[yellow\] drawn in box (start=(0.8749972672842821, 0.8749972672842821), end=(0.8749972672842821, 0.3828196733526225)); line\_28 — a Line \[yellow\] drawn in box (start=(0.8749972672842821, 0.3828196733526225), end=(0.3828196733526225, 0.3828196733526225)) Actions: - [11:37.994](https://academa.ai/lectures/period-doubling-logistic-map?t=697.9941875): four\_vals is shown on the screen, written out. - [11:37.994](https://academa.ai/lectures/period-doubling-logistic-map?t=697.9941875): dots35 is shown on the screen, written out. - [11:38.144](https://academa.ai/lectures/period-doubling-logistic-map?t=698.1441874999999): dots35\_2 is shown on the screen, written out. - [11:38.294](https://academa.ai/lectures/period-doubling-logistic-map?t=698.2941874999999): dots35\_3 is shown on the screen, written out. - [11:38.444](https://academa.ai/lectures/period-doubling-logistic-map?t=698.4441874999999): dots35\_4 is shown on the screen, written out. ##### [11:50.936](https://academa.ai/lectures/period-doubling-logistic-map?t=710.9356874999999) Narration: So the years no longer come in two kinds. They come in four, and the pattern closes only every fourth year. A good year, a terrible year, a middling year, a very good year, and only then a repeat. Board: knob — a Math \[text\] that says "$r = 3.5$"; four\_vals — a Math \[text\] that says "$0.383 arrow.r 0.827 arrow.r 0.501 arrow.r 0.875$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_four — a Heading that says "Four Kinds of Year"; para35 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); line — a Line \[gray\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.56)); line\_2 — a Line \[gray\] drawn in box (start=(0.2, 0.56), end=(0.56, 0.56)); line\_3 — a Line \[gray\] drawn in box (start=(0.56, 0.56), end=(0.56, 0.8623999999999999)); line\_4 — a Line \[gray\] drawn in box (start=(0.56, 0.8623999999999999), end=(0.8623999999999999, 0.8623999999999999)); line\_5 — a Line \[gray\] drawn in box (start=(0.8623999999999999, 0.8623999999999999), end=(0.8623999999999999, 0.4153318400000001)); line\_6 — a Line \[gray\] drawn in box (start=(0.8623999999999999, 0.4153318400000001), end=(0.4153318400000001, 0.4153318400000001)); line\_7 — a Line \[gray\] drawn in box (start=(0.4153318400000001, 0.4153318400000001), end=(0.4153318400000001, 0.8499095593877504)); line\_8 — a Line \[gray\] drawn in box (start=(0.4153318400000001, 0.8499095593877504), end=(0.8499095593877504, 0.8499095593877504)); line\_9 — a Line \[gray\] drawn in box (start=(0.8499095593877504, 0.8499095593877504), end=(0.8499095593877504, 0.4464715508717464)); line\_10 — a Line \[gray\] drawn in box (start=(0.8499095593877504, 0.4464715508717464), end=(0.4464715508717464, 0.4464715508717464)); line\_11 — a Line \[gray\] drawn in box (start=(0.4464715508717464, 0.4464715508717464), end=(0.4464715508717464, 0.8649714679687339)); line\_12 — a Line \[gray\] drawn in box (start=(0.4464715508717464, 0.8649714679687339), end=(0.8649714679687339, 0.8649714679687339)); line\_13 — a Line \[gray\] drawn in box (start=(0.8649714679687339, 0.8649714679687339), end=(0.8649714679687339, 0.408785396490616)); line\_14 — a Line \[gray\] drawn in box (start=(0.8649714679687339, 0.408785396490616), end=(0.408785396490616, 0.408785396490616)); line\_15 — a Line \[gray\] drawn in box (start=(0.408785396490616, 0.408785396490616), end=(0.408785396490616, 0.8458796363731905)); line\_16 — a Line \[gray\] drawn in box (start=(0.408785396490616, 0.8458796363731905), end=(0.8458796363731905, 0.8458796363731905)); line\_17 — a Line \[gray\] drawn in box (start=(0.8458796363731905, 0.8458796363731905), end=(0.8458796363731905, 0.4562854699982232)); line\_18 — a Line \[gray\] drawn in box (start=(0.8458796363731905, 0.4562854699982232), end=(0.4562854699982232, 0.4562854699982232)); line\_19 — a Line \[gray\] drawn in box (start=(0.4562854699982232, 0.4562854699982232), end=(0.4562854699982232, 0.8683116395335331)); line\_20 — a Line \[gray\] drawn in box (start=(0.4562854699982232, 0.8683116395335331), end=(0.8683116395335331, 0.8683116395335331)); line\_21 — a Line \[yellow\] drawn in box (start=(0.38282, 0.38282), end=(0.38282, 0.8269409666)); line\_22 — a Line \[yellow\] drawn in box (start=(0.38282, 0.8269409666), end=(0.8269409666, 0.8269409666)); line\_23 — a Line \[yellow\] drawn in box (start=(0.8269409666, 0.8269409666), end=(0.8269409666, 0.5008836152553018)); line\_24 — a Line \[yellow\] drawn in box (start=(0.8269409666, 0.5008836152553018), end=(0.5008836152553018, 0.5008836152553018)); line\_25 — a Line \[yellow\] drawn in box (start=(0.5008836152553018, 0.5008836152553018), end=(0.5008836152553018, 0.8749972672842821)); line\_26 — a Line \[yellow\] drawn in box (start=(0.5008836152553018, 0.8749972672842821), end=(0.8749972672842821, 0.8749972672842821)); line\_27 — a Line \[yellow\] drawn in box (start=(0.8749972672842821, 0.8749972672842821), end=(0.8749972672842821, 0.3828196733526225)); line\_28 — a Line \[yellow\] drawn in box (start=(0.8749972672842821, 0.3828196733526225), end=(0.3828196733526225, 0.3828196733526225)); dots35 — a Point \[red\] drawn in box (location=(0.38282, 0.38282), marker\_radius=0.012); dots35\_2 — a Point \[red\] drawn in box (location=(0.826941, 0.826941), marker\_radius=0.012); dots35\_3 — a Point \[red\] drawn in box (location=(0.500884, 0.500884), marker\_radius=0.012); dots35\_4 — a Point \[red\] drawn in box (location=(0.874997, 0.874997), marker\_radius=0.012) Actions: - [11:55.127](https://academa.ai/lectures/period-doubling-logistic-map?t=715.1271874999999): split\_note is shown on the screen, written out. - [12:5.854](https://academa.ai/lectures/period-doubling-logistic-map?t=725.8541874999999): knob moves to a new place on the board. - [12:5.854](https://academa.ai/lectures/period-doubling-logistic-map?t=725.8541874999999): four\_vals is hidden from the screen — left the board. - [12:5.854](https://academa.ai/lectures/period-doubling-logistic-map?t=725.8541874999999): head\_four is hidden from the screen — left the board. - [12:5.854](https://academa.ai/lectures/period-doubling-logistic-map?t=725.8541874999999): split\_note is hidden from the screen — left the board. ##### [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999) Narration: Push the knob a little further, to three point five five, and every one of those four splits again. The orbit now needs eight years to close. Board: knob — a Math \[text\] that says "$r = 3.5$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); para35 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); line — a Line \[gray\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.56)); line\_2 — a Line \[gray\] drawn in box (start=(0.2, 0.56), end=(0.56, 0.56)); line\_3 — a Line \[gray\] drawn in box (start=(0.56, 0.56), end=(0.56, 0.8623999999999999)); line\_4 — a Line \[gray\] drawn in box (start=(0.56, 0.8623999999999999), end=(0.8623999999999999, 0.8623999999999999)); line\_5 — a Line \[gray\] drawn in box (start=(0.8623999999999999, 0.8623999999999999), end=(0.8623999999999999, 0.4153318400000001)); line\_6 — a Line \[gray\] drawn in box (start=(0.8623999999999999, 0.4153318400000001), end=(0.4153318400000001, 0.4153318400000001)); line\_7 — a Line \[gray\] drawn in box (start=(0.4153318400000001, 0.4153318400000001), end=(0.4153318400000001, 0.8499095593877504)); line\_8 — a Line \[gray\] drawn in box (start=(0.4153318400000001, 0.8499095593877504), end=(0.8499095593877504, 0.8499095593877504)); line\_9 — a Line \[gray\] drawn in box (start=(0.8499095593877504, 0.8499095593877504), end=(0.8499095593877504, 0.4464715508717464)); line\_10 — a Line \[gray\] drawn in box (start=(0.8499095593877504, 0.4464715508717464), end=(0.4464715508717464, 0.4464715508717464)); line\_11 — a Line \[gray\] drawn in box (start=(0.4464715508717464, 0.4464715508717464), end=(0.4464715508717464, 0.8649714679687339)); line\_12 — a Line \[gray\] drawn in box (start=(0.4464715508717464, 0.8649714679687339), end=(0.8649714679687339, 0.8649714679687339)); line\_13 — a Line \[gray\] drawn in box (start=(0.8649714679687339, 0.8649714679687339), end=(0.8649714679687339, 0.408785396490616)); line\_14 — a Line \[gray\] drawn in box (start=(0.8649714679687339, 0.408785396490616), end=(0.408785396490616, 0.408785396490616)); line\_15 — a Line \[gray\] drawn in box (start=(0.408785396490616, 0.408785396490616), end=(0.408785396490616, 0.8458796363731905)); line\_16 — a Line \[gray\] drawn in box (start=(0.408785396490616, 0.8458796363731905), end=(0.8458796363731905, 0.8458796363731905)); line\_17 — a Line \[gray\] drawn in box (start=(0.8458796363731905, 0.8458796363731905), end=(0.8458796363731905, 0.4562854699982232)); line\_18 — a Line \[gray\] drawn in box (start=(0.8458796363731905, 0.4562854699982232), end=(0.4562854699982232, 0.4562854699982232)); line\_19 — a Line \[gray\] drawn in box (start=(0.4562854699982232, 0.4562854699982232), end=(0.4562854699982232, 0.8683116395335331)); line\_20 — a Line \[gray\] drawn in box (start=(0.4562854699982232, 0.8683116395335331), end=(0.8683116395335331, 0.8683116395335331)); line\_21 — a Line \[yellow\] drawn in box (start=(0.38282, 0.38282), end=(0.38282, 0.8269409666)); line\_22 — a Line \[yellow\] drawn in box (start=(0.38282, 0.8269409666), end=(0.8269409666, 0.8269409666)); line\_23 — a Line \[yellow\] drawn in box (start=(0.8269409666, 0.8269409666), end=(0.8269409666, 0.5008836152553018)); line\_24 — a Line \[yellow\] drawn in box (start=(0.8269409666, 0.5008836152553018), end=(0.5008836152553018, 0.5008836152553018)); line\_25 — a Line \[yellow\] drawn in box (start=(0.5008836152553018, 0.5008836152553018), end=(0.5008836152553018, 0.8749972672842821)); line\_26 — a Line \[yellow\] drawn in box (start=(0.5008836152553018, 0.8749972672842821), end=(0.8749972672842821, 0.8749972672842821)); line\_27 — a Line \[yellow\] drawn in box (start=(0.8749972672842821, 0.8749972672842821), end=(0.8749972672842821, 0.3828196733526225)); line\_28 — a Line \[yellow\] drawn in box (start=(0.8749972672842821, 0.3828196733526225), end=(0.3828196733526225, 0.3828196733526225)); dots35 — a Point \[red\] drawn in box (location=(0.38282, 0.38282), marker\_radius=0.012); dots35\_2 — a Point \[red\] drawn in box (location=(0.826941, 0.826941), marker\_radius=0.012); dots35\_3 — a Point \[red\] drawn in box (location=(0.500884, 0.500884), marker\_radius=0.012); dots35\_4 — a Point \[red\] drawn in box (location=(0.874997, 0.874997), marker\_radius=0.012) Actions: - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): head\_eight is shown on the screen, written out. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_2 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_3 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_4 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_5 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_6 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_7 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_8 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_9 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_10 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_11 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_12 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_13 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_14 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_15 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_16 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_17 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_18 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_19 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_20 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_21 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_22 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_23 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_24 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_25 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_26 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_27 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): line\_28 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): para35 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): dots35 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): dots35\_2 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): dots35\_3 is hidden from the screen. - [12:6.454](https://academa.ai/lectures/period-doubling-logistic-map?t=726.4541874999999): dots35\_4 is hidden from the screen. - [12:6.93](https://academa.ai/lectures/period-doubling-logistic-map?t=726.9301874999999): para355 is shown on the screen, written out. - [12:9.182](https://academa.ai/lectures/period-doubling-logistic-map?t=729.1821874999999): knob becomes "$r = 3.55$". - [12:13.664](https://academa.ai/lectures/period-doubling-logistic-map?t=733.6641874999999): line\_29 is shown on the screen, written out. - [12:13.864](https://academa.ai/lectures/period-doubling-logistic-map?t=733.8641875): line\_30 is shown on the screen, written out. - [12:14.064](https://academa.ai/lectures/period-doubling-logistic-map?t=734.0641874999999): line\_31 is shown on the screen, written out. - [12:14.264](https://academa.ai/lectures/period-doubling-logistic-map?t=734.2641874999999): line\_32 is shown on the screen, written out. - [12:14.464](https://academa.ai/lectures/period-doubling-logistic-map?t=734.4641875): line\_33 is shown on the screen, written out. - [12:14.664](https://academa.ai/lectures/period-doubling-logistic-map?t=734.6641874999999): line\_34 is shown on the screen, written out. - [12:14.864](https://academa.ai/lectures/period-doubling-logistic-map?t=734.8641875): line\_35 is shown on the screen, written out. - [12:15.064](https://academa.ai/lectures/period-doubling-logistic-map?t=735.0641874999999): line\_36 is shown on the screen, written out. - [12:15.264](https://academa.ai/lectures/period-doubling-logistic-map?t=735.2641874999999): line\_37 is shown on the screen, written out. - [12:15.464](https://academa.ai/lectures/period-doubling-logistic-map?t=735.4641875): line\_38 is shown on the screen, written out. - [12:15.664](https://academa.ai/lectures/period-doubling-logistic-map?t=735.6641874999999): line\_39 is shown on the screen, written out. - [12:15.864](https://academa.ai/lectures/period-doubling-logistic-map?t=735.8641875): line\_40 is shown on the screen, written out. ##### [12:15.878](https://academa.ai/lectures/period-doubling-logistic-map?t=735.8776874999999) Narration: The picture is already becoming hard to read, and that is exactly the point. So look instead at how little I had to turn the knob. From three to three point four five was a long way. From three point four five to three point five four was a fifth of that. From there to three point five six, shorter still. Board: knob — a Math \[text\] that says "$r = 3.5$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); head\_eight — a Heading that says "And Then Eight"; para355 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); line\_29 — a Line \[yellow\] drawn in box (start=(0.37032556106597414, 0.37032556106597414), end=(0.37032556106597414, 0.8278051165993668)); line\_30 — a Line \[yellow\] drawn in box (start=(0.37032556106597414, 0.8278051165993668), end=(0.8278051165993668, 0.8278051165993668)); line\_31 — a Line \[yellow\] drawn in box (start=(0.8278051165993668, 0.8278051165993668), end=(0.8278051165993668, 0.506030509636028)); line\_32 — a Line \[yellow\] drawn in box (start=(0.8278051165993668, 0.506030509636028), end=(0.506030509636028, 0.506030509636028)); line\_33 — a Line \[yellow\] drawn in box (start=(0.506030509636028, 0.506030509636028), end=(0.506030509636028, 0.8873708969850306)); line\_34 — a Line \[yellow\] drawn in box (start=(0.506030509636028, 0.8873708969850306), end=(0.8873708969850306, 0.8873708969850306)); line\_35 — a Line \[yellow\] drawn in box (start=(0.8873708969850306, 0.8873708969850306), end=(0.8873708969850306, 0.3548004479999955)); line\_36 — a Line \[yellow\] drawn in box (start=(0.8873708969850306, 0.3548004479999955), end=(0.3548004479999955, 0.3548004479999955)); line\_37 — a Line \[yellow\] drawn in box (start=(0.3548004479999955, 0.3548004479999955), end=(0.3548004479999955, 0.8126556698514428)); line\_38 — a Line \[yellow\] drawn in box (start=(0.3548004479999955, 0.8126556698514428), end=(0.8126556698514428, 0.8126556698514428)); line\_39 — a Line \[yellow\] drawn in box (start=(0.8126556698514428, 0.8126556698514428), end=(0.8126556698514428, 0.540474833989597)); line\_40 — a Line \[yellow\] drawn in box (start=(0.8126556698514428, 0.540474833989597), end=(0.540474833989597, 0.540474833989597)) Actions: - [12:16.064](https://academa.ai/lectures/period-doubling-logistic-map?t=736.0641874999999): line\_41 is shown on the screen, written out. - [12:16.264](https://academa.ai/lectures/period-doubling-logistic-map?t=736.2641874999999): line\_42 is shown on the screen, written out. - [12:16.464](https://academa.ai/lectures/period-doubling-logistic-map?t=736.4641875): line\_43 is shown on the screen, written out. - [12:16.664](https://academa.ai/lectures/period-doubling-logistic-map?t=736.6641874999999): line\_44 is shown on the screen, written out. - [12:22.67](https://academa.ai/lectures/period-doubling-logistic-map?t=742.6701874999999): eight\_note is shown on the screen, written out. ##### [12:36.749](https://academa.ai/lectures/period-doubling-logistic-map?t=756.7491875) Narration: The doublings are arriving faster and faster. Which means we should stop looking at pictures for a moment and look at the numbers. Board: knob — a Math \[text\] that says "$r = 3.5$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); eight\_note — a Text \[text\] that says "Eight values now, and the knob moved only a twentieth as far as it did the first time."; head\_eight — a Heading that says "And Then Eight"; para355 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); line\_29 — a Line \[yellow\] drawn in box (start=(0.37032556106597414, 0.37032556106597414), end=(0.37032556106597414, 0.8278051165993668)); line\_30 — a Line \[yellow\] drawn in box (start=(0.37032556106597414, 0.8278051165993668), end=(0.8278051165993668, 0.8278051165993668)); line\_31 — a Line \[yellow\] drawn in box (start=(0.8278051165993668, 0.8278051165993668), end=(0.8278051165993668, 0.506030509636028)); line\_32 — a Line \[yellow\] drawn in box (start=(0.8278051165993668, 0.506030509636028), end=(0.506030509636028, 0.506030509636028)); line\_33 — a Line \[yellow\] drawn in box (start=(0.506030509636028, 0.506030509636028), end=(0.506030509636028, 0.8873708969850306)); line\_34 — a Line \[yellow\] drawn in box (start=(0.506030509636028, 0.8873708969850306), end=(0.8873708969850306, 0.8873708969850306)); line\_35 — a Line \[yellow\] drawn in box (start=(0.8873708969850306, 0.8873708969850306), end=(0.8873708969850306, 0.3548004479999955)); line\_36 — a Line \[yellow\] drawn in box (start=(0.8873708969850306, 0.3548004479999955), end=(0.3548004479999955, 0.3548004479999955)); line\_37 — a Line \[yellow\] drawn in box (start=(0.3548004479999955, 0.3548004479999955), end=(0.3548004479999955, 0.8126556698514428)); line\_38 — a Line \[yellow\] drawn in box (start=(0.3548004479999955, 0.8126556698514428), end=(0.8126556698514428, 0.8126556698514428)); line\_39 — a Line \[yellow\] drawn in box (start=(0.8126556698514428, 0.8126556698514428), end=(0.8126556698514428, 0.540474833989597)); line\_40 — a Line \[yellow\] drawn in box (start=(0.8126556698514428, 0.540474833989597), end=(0.540474833989597, 0.540474833989597)); line\_41 — a Line \[yellow\] drawn in box (start=(0.540474833989597, 0.540474833989597), end=(0.540474833989597, 0.8816843467379767)); line\_42 — a Line \[yellow\] drawn in box (start=(0.540474833989597, 0.8816843467379767), end=(0.8816843467379767, 0.8816843467379767)); line\_43 — a Line \[yellow\] drawn in box (start=(0.8816843467379767, 0.8816843467379767), end=(0.8816843467379767, 0.37032556106597414)); line\_44 — a Line \[yellow\] drawn in box (start=(0.8816843467379767, 0.37032556106597414), end=(0.37032556106597414, 0.37032556106597414)) Actions: - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): box is hidden from the screen — left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): diag is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): para355 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_29 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_30 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_31 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_32 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_33 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_34 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_35 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_36 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_37 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_38 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_39 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_40 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_41 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_42 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_43 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): line\_44 is hidden from the screen — box left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): eight\_note is hidden from the screen — left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): head\_eight is hidden from the screen — left the board. - [12:43.843](https://academa.ai/lectures/period-doubling-logistic-map?t=763.8431874999999): knob is hidden from the screen — left the board. ##### [12:45.043](https://academa.ai/lectures/period-doubling-logistic-map?t=765.0431874999999) Narration: Here are the values of r at which the period doubles, the gap from each one to the next, and the ratio of one gap to the one after it. Board: Empty. Actions: - [12:45.043](https://academa.ai/lectures/period-doubling-logistic-map?t=765.0431874999999): head\_ratio is shown on the screen, written out. - [12:46.041](https://academa.ai/lectures/period-doubling-logistic-map?t=766.0411874999999): tbl is shown on the screen, written out. - [12:48.015](https://academa.ai/lectures/period-doubling-logistic-map?t=768.0151874999999): tbl is shown on the screen, written out. - [12:48.944](https://academa.ai/lectures/period-doubling-logistic-map?t=768.9441874999999): tbl is shown on the screen, written out. - [12:50.998](https://academa.ai/lectures/period-doubling-logistic-map?t=770.9981874999999): ratio\_note is shown on the screen, written out. ##### [12:54.199](https://academa.ai/lectures/period-doubling-logistic-map?t=774.1991874999999) Narration: Two to four, then four to eight, then eight to sixteen, then sixteen to thirty two. The gaps collapse: nearly a half at the start, and under a thousandth four steps later. Board: ratio\_note — a Text \[text\] that says "Each gap is the distance to the next doubling. Each ratio compares one gap with the one after it."; head\_ratio — a Heading that says "How Fast the Splittings Come" Actions: - [12:55.906](https://academa.ai/lectures/period-doubling-logistic-map?t=775.9061874999999): tbl is shown on the screen, written out. - [12:58.495](https://academa.ai/lectures/period-doubling-logistic-map?t=778.4951874999999): tbl is shown on the screen, written out. - [13:1.235](https://academa.ai/lectures/period-doubling-logistic-map?t=781.2351874999999): tbl (the "column=3" part) is emphasized. - [13:4.451](https://academa.ai/lectures/period-doubling-logistic-map?t=784.4511875): tbl (the "column=3" part) is no longer emphasized. ##### [13:7.164](https://academa.ai/lectures/period-doubling-logistic-map?t=787.1636874999999) Narration: But the ratio does not collapse with them. Look down the last column. Four point seven five, four point six six, four point six seven, and then it simply stops moving. Board: Unchanged from the preceding beat in this scene. Actions: - [13:11.366](https://academa.ai/lectures/period-doubling-logistic-map?t=791.3661874999999): tbl (the "column=4" part) is emphasized. - [13:18.321](https://academa.ai/lectures/period-doubling-logistic-map?t=798.3211875): tbl (the "column=4" part) is no longer emphasized. - [13:19.157](https://academa.ai/lectures/period-doubling-logistic-map?t=799.1566874999999): head\_ratio is hidden from the screen — left the board. - [13:19.157](https://academa.ai/lectures/period-doubling-logistic-map?t=799.1566874999999): ratio\_note is hidden from the screen — left the board. - [13:19.157](https://academa.ai/lectures/period-doubling-logistic-map?t=799.1566874999999): tbl is hidden from the screen — left the board. ##### [13:20.357](https://academa.ai/lectures/period-doubling-logistic-map?t=800.3566874999999) Narration: Carry the cascade on and that ratio converges on a definite number. Four point six six nine two zero one, and on it goes. It is called the Feigenbaum constant, and it is written delta. Board: Empty. Actions: - [13:20.357](https://academa.ai/lectures/period-doubling-logistic-map?t=800.3566874999999): head\_delta is shown on the screen, written out. - [13:24.316](https://academa.ai/lectures/period-doubling-logistic-map?t=804.3161874999998): delta is shown on the screen, written out. ##### [13:34.553](https://academa.ai/lectures/period-doubling-logistic-map?t=814.5526874999999) Narration: Two things about it deserve a pause, and the first is only arithmetic. Because the gaps shrink by a fixed factor every time, they behave like a geometric series, and a geometric series with ratio bigger than one in the denominator adds up to something finite. Board: delta — a Math \[text\] that says "$delta = 4.669 thin 201 thin 609 dots$"; head\_delta — a Heading that says "A Number That Belongs to Nobody" Actions: - None. ##### [13:51.894](https://academa.ai/lectures/period-doubling-logistic-map?t=831.8936874999999) Narration: So the doublings do not go on for ever in r. They pile up at a finite value, three point five six nine nine and a bit, and by the time you arrive there the period has doubled infinitely often. Board: Unchanged from the preceding beat in this scene. Actions: - [13:55.272](https://academa.ai/lectures/period-doubling-logistic-map?t=835.2721875): r\_inf is shown on the screen, written out. ##### [14:4.139](https://academa.ai/lectures/period-doubling-logistic-map?t=844.1391874999999) Narration: The second thing is much stranger. That number has nothing whatever to do with our particular hump. Replace r x times one minus x with a sine arch, or with almost any smooth curve that has one rounded maximum, run the same cascade, and out comes the same constant. Board: delta — a Math \[text\] that says "$delta = 4.669 thin 201 thin 609 dots$"; r\_inf — a Math \[text\] that says "$r\_infinity = 3.569 thin 946 dots$"; head\_delta — a Heading that says "A Number That Belongs to Nobody" Actions: - [14:5.892](https://academa.ai/lectures/period-doubling-logistic-map?t=845.8921874999999): univ\_note is shown on the screen, written out. ##### [14:23.373](https://academa.ai/lectures/period-doubling-logistic-map?t=863.3726874999999) Narration: Mitchell Feigenbaum found that in nineteen seventy five, on a pocket calculator, and it took him three years to get it published. Which leaves the place where the doublings pile up, and everything past it. Board: delta — a Math \[text\] that says "$delta = 4.669 thin 201 thin 609 dots$"; r\_inf — a Math \[text\] that says "$r\_infinity = 3.569 thin 946 dots$"; univ\_note — a Panel that says "The same $delta$ turns up in the doubling cascade of a sine arch, or of almost any smooth map with a single rounded maximum. It belongs to the doubling, not to this equation."; head\_delta — a Heading that says "A Number That Belongs to Nobody" Actions: - [14:27.796](https://academa.ai/lectures/period-doubling-logistic-map?t=867.7961874999999): A box is drawn around delta. - [14:35.333](https://academa.ai/lectures/period-doubling-logistic-map?t=875.3331666666666): delta is hidden from the screen — left the board. - [14:35.333](https://academa.ai/lectures/period-doubling-logistic-map?t=875.3331666666666): head\_delta is hidden from the screen — left the board. - [14:35.333](https://academa.ai/lectures/period-doubling-logistic-map?t=875.3331666666666): r\_inf is hidden from the screen — left the board. - [14:35.333](https://academa.ai/lectures/period-doubling-logistic-map?t=875.3331666666666): univ\_note is hidden from the screen — left the board. ### Scene 5: [The Whole Picture, and What Lies Past It](https://academa.ai/lectures/period-doubling-logistic-map?t=876.3748333333332) Span: 14:36.375–17:28.741 (876.3748333333332s–1048.7411874999998s). #### Objects - bif: an Axes (x\_range=(2.8, 4.0), x\_ticks\_every=0.2, y\_ticks\_every=0.25) - box: an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25) - c1: a Tex \[text\] that says "$r \< 3$: one settled population" - c2: a Tex \[text\] that says "$3 \< r \< 3.449$: two" - c3: a Tex \[text\] that says "$3.449 \< r \< 3.544$: four" - c4: a Tex \[text\] that says "$r \> 3.5699$: never repeats" - chaos: a Polygon \[red\] drawn in bif (vertices=((3.5699, 0.892475), (3.5804025, 0.895100625), (3.5909050000000…, fill\_opacity=0.28) - chaos\_note: a Text \[text\] that says "Twenty years at $r = 3.9$, and no leg ever retraces another." - diag: a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)) - function\_plot: a FunctionPlot \[blue\] drawn in bif (function=\, x\_range=(2.8, 2.995)) - function\_plot\_2: a FunctionPlot \[green\] drawn in bif (function=\, x\_range=(3.005, 3.444)) - function\_plot\_3: a FunctionPlot \[green\] drawn in bif (function=\, x\_range=(3.005, 3.444)) - function\_plot\_4: a FunctionPlot \[yellow\] drawn in bif (function=\, x\_range=(3.455, 3.542)) - function\_plot\_5: a FunctionPlot \[yellow\] drawn in bif (function=\, x\_range=(3.455, 3.542)) - function\_plot\_6: a FunctionPlot \[yellow\] drawn in bif (function=\, x\_range=(3.455, 3.542)) - function\_plot\_7: a FunctionPlot \[yellow\] drawn in bif (function=\, x\_range=(3.455, 3.542)) - head\_inside: a Heading that says "Inside the Band" - head\_map: a Heading that says "Every Ending, All at Once" - head\_sens: a Heading that says "Two Populations, Almost the Same" - line: a Line \[yellow\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.6240000000000001)) - line\_10: a Line \[yellow\] drawn in box (start=(0.8239731430433209, 0.5656614700878645), end=(0.5656614700878645, 0.5656614700878645)) - line\_11: a Line \[yellow\] drawn in box (start=(0.5656614700878645, 0.5656614700878645), end=(0.5656614700878645, 0.9581854282490118)) - line\_12: a Line \[yellow\] drawn in box (start=(0.5656614700878645, 0.9581854282490118), end=(0.9581854282490118, 0.9581854282490118)) - line\_13: a Line \[yellow\] drawn in box (start=(0.9581854282490118, 0.9581854282490118), end=(0.9581854282490118, 0.1562578420270518)) - line\_14: a Line \[yellow\] drawn in box (start=(0.9581854282490118, 0.1562578420270518), end=(0.1562578420270518, 0.1562578420270518)) - line\_15: a Line \[yellow\] drawn in box (start=(0.1562578420270518, 0.1562578420270518), end=(0.1562578420270518, 0.5141811824451928)) - line\_16: a Line \[yellow\] drawn in box (start=(0.1562578420270518, 0.5141811824451928), end=(0.5141811824451928, 0.5141811824451928)) - line\_17: a Line \[yellow\] drawn in box (start=(0.5141811824451928, 0.5141811824451928), end=(0.5141811824451928, 0.9742156868513789)) - line\_18: a Line \[yellow\] drawn in box (start=(0.5141811824451928, 0.9742156868513789), end=(0.9742156868513789, 0.9742156868513789)) - line\_19: a Line \[yellow\] drawn in box (start=(0.9742156868513789, 0.9742156868513789), end=(0.9742156868513789, 0.09796598114189214)) - line\_2: a Line \[yellow\] drawn in box (start=(0.2, 0.6240000000000001), end=(0.6240000000000001, 0.6240000000000001)) - line\_20: a Line \[yellow\] drawn in box (start=(0.9742156868513789, 0.09796598114189214), end=(0.09796598114189214, 0.09796598114189214)) - 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line\_29: a Line \[yellow\] drawn in box (start=(0.9428998465038291, 0.9428998465038291), end=(0.9428998465038291, 0.20997493127085012)) - line\_3: a Line \[yellow\] drawn in box (start=(0.6240000000000001, 0.6240000000000001), end=(0.6240000000000001, 0.9150335999999998)) - line\_30: a Line \[yellow\] drawn in box (start=(0.9428998465038291, 0.20997493127085012), end=(0.20997493127085012, 0.20997493127085012)) - line\_31: a Line \[yellow\] drawn in box (start=(0.20997493127085012, 0.20997493127085012), end=(0.20997493127085012, 0.6469532920837424)) - line\_32: a Line \[yellow\] drawn in box (start=(0.20997493127085012, 0.6469532920837424), end=(0.6469532920837424, 0.6469532920837424)) - line\_33: a Line \[yellow\] drawn in box (start=(0.6469532920837424, 0.6469532920837424), end=(0.6469532920837424, 0.8907784467884261)) - line\_34: a Line \[yellow\] drawn in box (start=(0.6469532920837424, 0.8907784467884261), end=(0.8907784467884261, 0.8907784467884261)) - line\_35: a Line \[yellow\] drawn in box (start=(0.8907784467884261, 0.8907784467884261), end=(0.8907784467884261, 0.3794396015499384)) - line\_36: a Line \[yellow\] drawn in box (start=(0.8907784467884261, 0.3794396015499384), end=(0.3794396015499384, 0.3794396015499384)) - line\_37: a Line \[yellow\] drawn in box (start=(0.3794396015499384, 0.3794396015499384), end=(0.3794396015499384, 0.9183142422696933)) - line\_38: a Line \[yellow\] drawn in box (start=(0.3794396015499384, 0.9183142422696933), end=(0.9183142422696933, 0.9183142422696933)) - line\_39: a Line \[yellow\] drawn in box (start=(0.9183142422696933, 0.9183142422696933), end=(0.9183142422696933, 0.2925514593858962)) - line\_4: a Line \[yellow\] drawn in box (start=(0.6240000000000001, 0.9150335999999998), end=(0.9150335999999998, 0.9150335999999998)) - line\_40: a Line \[yellow\] drawn in box (start=(0.9183142422696933, 0.2925514593858962), end=(0.2925514593858962, 0.2925514593858962)) - line\_5: a Line \[yellow\] drawn in box (start=(0.9150335999999998, 0.9150335999999998), end=(0.9150335999999998, 0.30321373239705673)) - line\_6: a Line \[yellow\] drawn in box (start=(0.9150335999999998, 0.30321373239705673), end=(0.30321373239705673, 0.30321373239705673)) - line\_7: a Line \[yellow\] drawn in box (start=(0.30321373239705673, 0.30321373239705673), end=(0.30321373239705673, 0.8239731430433209)) - line\_8: a Line \[yellow\] drawn in box (start=(0.30321373239705673, 0.8239731430433209), end=(0.8239731430433209, 0.8239731430433209)) - line\_9: a Line \[yellow\] drawn in box (start=(0.8239731430433209, 0.8239731430433209), end=(0.8239731430433209, 0.5656614700878645)) - para39: a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)) - runA: a ParametricCurve \[blue\] labelled "x\_0 = 0.400" drawn in tl (function=\, t\_range=(0.0, 20.0), breakpoints=(1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0,…) - runB: a ParametricCurve \[red\] labelled "x\_0 = 0.401" drawn in tl (function=\, t\_range=(0.0, 20.0), breakpoints=(1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0,…) - sens\_note: a Text \[text\] that says "A difference of one part in four hundred, and after fifteen years the two have nothing to do with each other." - sliver: a Polygon \[magenta\] drawn in bif (vertices=((3.545, 0.31), (3.5699, 0.31), (3.5699, 0.9), (3.545, 0.9)), fill\_opacity=0.32) - tl: an Axes (x\_range=(0.0, 20.0), x\_ticks\_every=5.0, y\_ticks\_every=0.25) #### Beats ##### [14:36.375](https://academa.ai/lectures/period-doubling-logistic-map?t=876.3748333333332) Narration: Let me put every one of those answers on one picture. Along the bottom, the knob. Up the side, the population, but only where it ends up: the values the orbit visits once the transient is over. Board: Empty. Actions: - [14:36.375](https://academa.ai/lectures/period-doubling-logistic-map?t=876.3748333333332): head\_map is shown on the screen, written out. - [14:36.375](https://academa.ai/lectures/period-doubling-logistic-map?t=876.3748333333332): bif is shown on the screen, written out. ##### [14:49.606](https://academa.ai/lectures/period-doubling-logistic-map?t=889.6063333333332) Narration: For r below three there is one value, and it climbs gently as we turn. That is the fixed point, one minus one over r, drawn once for every setting of the knob. Board: bif — an Axes (x\_range=(2.8, 4.0), x\_ticks\_every=0.2, y\_ticks\_every=0.25); head\_map — a Heading that says "Every Ending, All at Once" Actions: - [14:53.194](https://academa.ai/lectures/period-doubling-logistic-map?t=893.1938333333331): function\_plot is shown on the screen, drawn. - [14:55.458](https://academa.ai/lectures/period-doubling-logistic-map?t=895.4578333333332): bif moves to a new place on the board. - [14:55.458](https://academa.ai/lectures/period-doubling-logistic-map?t=895.4578333333332): c1 is shown on the screen, written out. ##### [15:0.958](https://academa.ai/lectures/period-doubling-logistic-map?t=900.9578333333332) Narration: At three it splits. Above three there are two values, the lean year and the crowded year, and the gap between them widens as we go. Board: c1 — a Tex \[text\] that says "$r \< 3$: one settled population"; bif — an Axes (x\_range=(2.8, 4.0), x\_ticks\_every=0.2, y\_ticks\_every=0.25); head\_map — a Heading that says "Every Ending, All at Once"; function\_plot — a FunctionPlot \[blue\] drawn in bif (function=\, x\_range=(2.8, 2.995)) Actions: - [15:1.933](https://academa.ai/lectures/period-doubling-logistic-map?t=901.9328333333332): function\_plot\_2 is shown on the screen, drawn. - [15:2.233](https://academa.ai/lectures/period-doubling-logistic-map?t=902.2328333333331): function\_plot\_3 is shown on the screen, drawn. - [15:8.47](https://academa.ai/lectures/period-doubling-logistic-map?t=908.4698333333332): c2 is shown on the screen, written out. ##### [15:10.451](https://academa.ai/lectures/period-doubling-logistic-map?t=910.4513333333332) Narration: At three point four four nine each of those splits again, and there are four. You can see the tuning fork shape repeating at a smaller size. Board: c1 — a Tex \[text\] that says "$r \< 3$: one settled population"; c2 — a Tex \[text\] that says "$3 \< r \< 3.449$: two"; bif — an Axes (x\_range=(2.8, 4.0), x\_ticks\_every=0.2, y\_ticks\_every=0.25); head\_map — a Heading that says "Every Ending, All at Once"; function\_plot — a FunctionPlot \[blue\] drawn in bif (function=\, x\_range=(2.8, 2.995)); function\_plot\_2 — a FunctionPlot \[green\] drawn in bif (function=\, x\_range=(3.005, 3.444)); function\_plot\_3 — a FunctionPlot \[green\] drawn in bif (function=\, x\_range=(3.005, 3.444)) Actions: - [15:11.496](https://academa.ai/lectures/period-doubling-logistic-map?t=911.4958333333332): c3 is shown on the screen, written out. - [15:12.959](https://academa.ai/lectures/period-doubling-logistic-map?t=912.9588333333331): function\_plot\_4 is shown on the screen, drawn. - [15:13.209](https://academa.ai/lectures/period-doubling-logistic-map?t=913.2088333333331): function\_plot\_5 is shown on the screen, drawn. - [15:13.459](https://academa.ai/lectures/period-doubling-logistic-map?t=913.4588333333331): function\_plot\_6 is shown on the screen, drawn. - [15:13.709](https://academa.ai/lectures/period-doubling-logistic-map?t=913.7088333333331): function\_plot\_7 is shown on the screen, drawn. ##### [15:19.317](https://academa.ai/lectures/period-doubling-logistic-map?t=919.3173333333332) Narration: And then the splittings come so fast that I cannot draw them. Eight, sixteen, thirty two, all crammed into this sliver, all of them piling up at three point five six nine nine. Board: c1 — a Tex \[text\] that says "$r \< 3$: one settled population"; c2 — a Tex \[text\] that says "$3 \< r \< 3.449$: two"; c3 — a Tex \[text\] that says "$3.449 \< r \< 3.544$: four"; bif — an Axes (x\_range=(2.8, 4.0), x\_ticks\_every=0.2, y\_ticks\_every=0.25); head\_map — a Heading that says "Every Ending, All at Once"; function\_plot — a FunctionPlot \[blue\] drawn in bif (function=\, x\_range=(2.8, 2.995)); function\_plot\_2 — a FunctionPlot \[green\] drawn in bif (function=\, x\_range=(3.005, 3.444)); function\_plot\_3 — a FunctionPlot \[green\] drawn in bif (function=\, x\_range=(3.005, 3.444)); function\_plot\_4 — a FunctionPlot \[yellow\] drawn in bif (function=\, x\_range=(3.455, 3.542)); function\_plot\_5 — a FunctionPlot \[yellow\] drawn in bif (function=\, x\_range=(3.455, 3.542)); function\_plot\_6 — a FunctionPlot \[yellow\] drawn in bif (function=\, x\_range=(3.455, 3.542)); function\_plot\_7 — a FunctionPlot \[yellow\] drawn in bif (function=\, x\_range=(3.455, 3.542)) Actions: - [15:26.26](https://academa.ai/lectures/period-doubling-logistic-map?t=926.2598333333332): sliver is shown on the screen, written out. ##### [15:30.529](https://academa.ai/lectures/period-doubling-logistic-map?t=930.5288333333332) Narration: Past that line, something else entirely. The orbit no longer settles onto any finite list of values at all. It wanders over a whole band, and that band is where the rest of this lecture lives. Board: c1 — a Tex \[text\] that says "$r \< 3$: one settled population"; c2 — a Tex \[text\] that says "$3 \< r \< 3.449$: two"; c3 — a Tex \[text\] that says "$3.449 \< r \< 3.544$: four"; bif — an Axes (x\_range=(2.8, 4.0), x\_ticks\_every=0.2, y\_ticks\_every=0.25); head\_map — a Heading that says "Every Ending, All at Once"; function\_plot — a FunctionPlot \[blue\] drawn in bif (function=\, x\_range=(2.8, 2.995)); function\_plot\_2 — a FunctionPlot \[green\] drawn in bif (function=\, x\_range=(3.005, 3.444)); function\_plot\_3 — a FunctionPlot \[green\] drawn in bif (function=\, x\_range=(3.005, 3.444)); function\_plot\_4 — a FunctionPlot \[yellow\] drawn in bif (function=\, x\_range=(3.455, 3.542)); function\_plot\_5 — a FunctionPlot \[yellow\] drawn in bif (function=\, x\_range=(3.455, 3.542)); function\_plot\_6 — a FunctionPlot \[yellow\] drawn in bif (function=\, x\_range=(3.455, 3.542)); function\_plot\_7 — a FunctionPlot \[yellow\] drawn in bif (function=\, x\_range=(3.455, 3.542)); sliver — a Polygon \[magenta\] drawn in bif (vertices=((3.545, 0.31), (3.5699, 0.31), (3.5699, 0.9), (3.545, 0.9)), fill\_opacity=0.32) Actions: - [15:37.704](https://academa.ai/lectures/period-doubling-logistic-map?t=937.7038333333331): chaos is shown on the screen, written out. - [15:38.656](https://academa.ai/lectures/period-doubling-logistic-map?t=938.6558333333331): c4 is shown on the screen, written out. - [15:42.011](https://academa.ai/lectures/period-doubling-logistic-map?t=942.0108333333332): bif is hidden from the screen — left the board. - [15:42.011](https://academa.ai/lectures/period-doubling-logistic-map?t=942.0108333333332): function\_plot is hidden from the screen — bif left the board. - [15:42.011](https://academa.ai/lectures/period-doubling-logistic-map?t=942.0108333333332): function\_plot\_2 is hidden from the screen — bif left the board. - [15:42.011](https://academa.ai/lectures/period-doubling-logistic-map?t=942.0108333333332): function\_plot\_3 is hidden from the screen — bif left the board. - [15:42.011](https://academa.ai/lectures/period-doubling-logistic-map?t=942.0108333333332): function\_plot\_4 is hidden from the screen — bif left the board. - [15:42.011](https://academa.ai/lectures/period-doubling-logistic-map?t=942.0108333333332): function\_plot\_5 is hidden from the screen — bif left the board. - [15:42.011](https://academa.ai/lectures/period-doubling-logistic-map?t=942.0108333333332): function\_plot\_6 is hidden from the screen — bif left the board. - [15:42.011](https://academa.ai/lectures/period-doubling-logistic-map?t=942.0108333333332): function\_plot\_7 is hidden from the screen — bif left the board. - [15:42.011](https://academa.ai/lectures/period-doubling-logistic-map?t=942.0108333333332): sliver is hidden from the screen — bif left the board. - [15:42.011](https://academa.ai/lectures/period-doubling-logistic-map?t=942.0108333333332): chaos is hidden from the screen — bif left the board. - [15:42.011](https://academa.ai/lectures/period-doubling-logistic-map?t=942.0108333333332): c1 is hidden from the screen — left the board. - [15:42.011](https://academa.ai/lectures/period-doubling-logistic-map?t=942.0108333333332): c2 is hidden from the screen — left the board. - [15:42.011](https://academa.ai/lectures/period-doubling-logistic-map?t=942.0108333333332): c3 is hidden from the screen — left the board. - [15:42.011](https://academa.ai/lectures/period-doubling-logistic-map?t=942.0108333333332): c4 is hidden from the screen — left the board. - [15:42.011](https://academa.ai/lectures/period-doubling-logistic-map?t=942.0108333333332): head\_map is hidden from the screen — left the board. ##### [15:43.211](https://academa.ai/lectures/period-doubling-logistic-map?t=943.2108333333332) Narration: So let us go inside it, and put the knob at three point nine. Same rule, same construction, same everything. Only r has changed. Board: Empty. Actions: - [15:43.211](https://academa.ai/lectures/period-doubling-logistic-map?t=943.2108333333332): head\_inside is shown on the screen, written out. - [15:43.211](https://academa.ai/lectures/period-doubling-logistic-map?t=943.2108333333332): box is shown on the screen, written out. - [15:43.211](https://academa.ai/lectures/period-doubling-logistic-map?t=943.2108333333332): para39 is shown on the screen, written out. - [15:43.211](https://academa.ai/lectures/period-doubling-logistic-map?t=943.2108333333332): diag is shown on the screen, written out. ##### [15:53.691](https://academa.ai/lectures/period-doubling-logistic-map?t=953.6908333333332) Narration: Twenty years of it. Board: box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_inside — a Heading that says "Inside the Band"; para39 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)) Actions: - [15:54.504](https://academa.ai/lectures/period-doubling-logistic-map?t=954.5038333333332): line is shown on the screen, written out. - [15:54.664](https://academa.ai/lectures/period-doubling-logistic-map?t=954.6638333333332): line\_2 is shown on the screen, written out. - [15:54.824](https://academa.ai/lectures/period-doubling-logistic-map?t=954.8238333333331): line\_3 is shown on the screen, written out. - [15:54.984](https://academa.ai/lectures/period-doubling-logistic-map?t=954.9838333333332): line\_4 is shown on the screen, written out. - [15:55.144](https://academa.ai/lectures/period-doubling-logistic-map?t=955.1438333333332): line\_5 is shown on the screen, written out. - [15:55.304](https://academa.ai/lectures/period-doubling-logistic-map?t=955.3038333333332): line\_6 is shown on the screen, written out. - [15:55.464](https://academa.ai/lectures/period-doubling-logistic-map?t=955.4638333333332): line\_7 is shown on the screen, written out. - [15:55.624](https://academa.ai/lectures/period-doubling-logistic-map?t=955.6238333333332): line\_8 is shown on the screen, written out. - [15:55.784](https://academa.ai/lectures/period-doubling-logistic-map?t=955.7838333333332): line\_9 is shown on the screen, written out. - [15:55.944](https://academa.ai/lectures/period-doubling-logistic-map?t=955.9438333333331): line\_10 is shown on the screen, written out. - [15:56.104](https://academa.ai/lectures/period-doubling-logistic-map?t=956.1038333333332): line\_11 is shown on the screen, written out. - [15:56.264](https://academa.ai/lectures/period-doubling-logistic-map?t=956.2638333333332): line\_12 is shown on the screen, written out. - [15:56.424](https://academa.ai/lectures/period-doubling-logistic-map?t=956.4238333333332): line\_13 is shown on the screen, written out. - [15:56.584](https://academa.ai/lectures/period-doubling-logistic-map?t=956.5838333333331): line\_14 is shown on the screen, written out. - [15:56.744](https://academa.ai/lectures/period-doubling-logistic-map?t=956.7438333333332): line\_15 is shown on the screen, written out. - [15:56.904](https://academa.ai/lectures/period-doubling-logistic-map?t=956.9038333333332): line\_16 is shown on the screen, written out. ##### [15:56.945](https://academa.ai/lectures/period-doubling-logistic-map?t=956.9448333333332) Narration: No square. No loop of any size. The staircase covers the box, and shows no sign at all of closing up. And nothing random went into this. Every one of those legs was computed from the one before it by multiplying three numbers together. Board: box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_inside — a Heading that says "Inside the Band"; para39 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); line — a Line \[yellow\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.6240000000000001)); line\_2 — a Line \[yellow\] drawn in box (start=(0.2, 0.6240000000000001), end=(0.6240000000000001, 0.6240000000000001)); line\_3 — a Line \[yellow\] drawn in box (start=(0.6240000000000001, 0.6240000000000001), end=(0.6240000000000001, 0.9150335999999998)); line\_4 — a Line \[yellow\] drawn in box (start=(0.6240000000000001, 0.9150335999999998), end=(0.9150335999999998, 0.9150335999999998)); line\_5 — a Line \[yellow\] drawn in box (start=(0.9150335999999998, 0.9150335999999998), end=(0.9150335999999998, 0.30321373239705673)); line\_6 — a Line \[yellow\] drawn in box (start=(0.9150335999999998, 0.30321373239705673), end=(0.30321373239705673, 0.30321373239705673)); line\_7 — a Line \[yellow\] drawn in box (start=(0.30321373239705673, 0.30321373239705673), end=(0.30321373239705673, 0.8239731430433209)); line\_8 — a Line \[yellow\] drawn in box (start=(0.30321373239705673, 0.8239731430433209), end=(0.8239731430433209, 0.8239731430433209)); line\_9 — a Line \[yellow\] drawn in box (start=(0.8239731430433209, 0.8239731430433209), end=(0.8239731430433209, 0.5656614700878645)); line\_10 — a Line \[yellow\] drawn in box (start=(0.8239731430433209, 0.5656614700878645), end=(0.5656614700878645, 0.5656614700878645)); line\_11 — a Line \[yellow\] drawn in box (start=(0.5656614700878645, 0.5656614700878645), end=(0.5656614700878645, 0.9581854282490118)); line\_12 — a Line \[yellow\] drawn in box (start=(0.5656614700878645, 0.9581854282490118), end=(0.9581854282490118, 0.9581854282490118)); line\_13 — a Line \[yellow\] drawn in box (start=(0.9581854282490118, 0.9581854282490118), end=(0.9581854282490118, 0.1562578420270518)); line\_14 — a Line \[yellow\] drawn in box (start=(0.9581854282490118, 0.1562578420270518), end=(0.1562578420270518, 0.1562578420270518)); line\_15 — a Line \[yellow\] drawn in box (start=(0.1562578420270518, 0.1562578420270518), end=(0.1562578420270518, 0.5141811824451928)); line\_16 — a Line \[yellow\] drawn in box (start=(0.1562578420270518, 0.5141811824451928), end=(0.5141811824451928, 0.5141811824451928)) Actions: - [15:57.064](https://academa.ai/lectures/period-doubling-logistic-map?t=957.0638333333332): line\_17 is shown on the screen, written out. - [15:57.224](https://academa.ai/lectures/period-doubling-logistic-map?t=957.2238333333332): line\_18 is shown on the screen, written out. - [15:57.384](https://academa.ai/lectures/period-doubling-logistic-map?t=957.3838333333332): line\_19 is shown on the screen, written out. - [15:57.544](https://academa.ai/lectures/period-doubling-logistic-map?t=957.5438333333332): line\_20 is shown on the screen, written out. - [15:57.704](https://academa.ai/lectures/period-doubling-logistic-map?t=957.7038333333333): line\_21 is shown on the screen, written out. - [15:57.864](https://academa.ai/lectures/period-doubling-logistic-map?t=957.8638333333332): line\_22 is shown on the screen, written out. - [15:58.024](https://academa.ai/lectures/period-doubling-logistic-map?t=958.0238333333332): line\_23 is shown on the screen, written out. - [15:58.184](https://academa.ai/lectures/period-doubling-logistic-map?t=958.1838333333332): line\_24 is shown on the screen, written out. - [15:58.344](https://academa.ai/lectures/period-doubling-logistic-map?t=958.3438333333332): line\_25 is shown on the screen, written out. - [15:58.504](https://academa.ai/lectures/period-doubling-logistic-map?t=958.5038333333332): line\_26 is shown on the screen, written out. - [15:58.664](https://academa.ai/lectures/period-doubling-logistic-map?t=958.6638333333332): line\_27 is shown on the screen, written out. - [15:58.824](https://academa.ai/lectures/period-doubling-logistic-map?t=958.8238333333331): line\_28 is shown on the screen, written out. - [15:58.984](https://academa.ai/lectures/period-doubling-logistic-map?t=958.9838333333332): line\_29 is shown on the screen, written out. - [15:59.144](https://academa.ai/lectures/period-doubling-logistic-map?t=959.1438333333332): line\_30 is shown on the screen, written out. - [15:59.304](https://academa.ai/lectures/period-doubling-logistic-map?t=959.3038333333332): line\_31 is shown on the screen, written out. - [15:59.464](https://academa.ai/lectures/period-doubling-logistic-map?t=959.4638333333332): line\_32 is shown on the screen, written out. - [15:59.624](https://academa.ai/lectures/period-doubling-logistic-map?t=959.6238333333332): line\_33 is shown on the screen, written out. - [15:59.784](https://academa.ai/lectures/period-doubling-logistic-map?t=959.7838333333332): line\_34 is shown on the screen, written out. - [15:59.944](https://academa.ai/lectures/period-doubling-logistic-map?t=959.9438333333331): line\_35 is shown on the screen, written out. - [16:0.104](https://academa.ai/lectures/period-doubling-logistic-map?t=960.1038333333332): line\_36 is shown on the screen, written out. - [16:0.264](https://academa.ai/lectures/period-doubling-logistic-map?t=960.2638333333332): line\_37 is shown on the screen, written out. - [16:0.424](https://academa.ai/lectures/period-doubling-logistic-map?t=960.4238333333332): line\_38 is shown on the screen, written out. - [16:0.584](https://academa.ai/lectures/period-doubling-logistic-map?t=960.5838333333331): line\_39 is shown on the screen, written out. - [16:0.744](https://academa.ai/lectures/period-doubling-logistic-map?t=960.7438333333332): line\_40 is shown on the screen, written out. - [16:5.281](https://academa.ai/lectures/period-doubling-logistic-map?t=965.2808333333332): chaos\_note is shown on the screen, written out. ##### [16:12.359](https://academa.ai/lectures/period-doubling-logistic-map?t=972.3593333333332) Narration: Which is the thing people find hardest to accept. There is no dice roll anywhere in the logistic map. The unpredictability is manufactured out of arithmetic, and here is the property that does the manufacturing. Board: box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); chaos\_note — a Text \[text\] that says "Twenty years at $r = 3.9$, and no leg ever retraces another."; head\_inside — a Heading that says "Inside the Band"; para39 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); line — a Line \[yellow\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.6240000000000001)); line\_2 — a Line \[yellow\] drawn in box (start=(0.2, 0.6240000000000001), end=(0.6240000000000001, 0.6240000000000001)); line\_3 — a Line \[yellow\] drawn in box (start=(0.6240000000000001, 0.6240000000000001), end=(0.6240000000000001, 0.9150335999999998)); line\_4 — a Line \[yellow\] drawn in box (start=(0.6240000000000001, 0.9150335999999998), end=(0.9150335999999998, 0.9150335999999998)); line\_5 — a Line \[yellow\] drawn in box (start=(0.9150335999999998, 0.9150335999999998), end=(0.9150335999999998, 0.30321373239705673)); line\_6 — a Line \[yellow\] drawn in box (start=(0.9150335999999998, 0.30321373239705673), end=(0.30321373239705673, 0.30321373239705673)); line\_7 — a Line \[yellow\] drawn in box (start=(0.30321373239705673, 0.30321373239705673), end=(0.30321373239705673, 0.8239731430433209)); line\_8 — a Line \[yellow\] drawn in box (start=(0.30321373239705673, 0.8239731430433209), end=(0.8239731430433209, 0.8239731430433209)); line\_9 — a Line \[yellow\] drawn in box (start=(0.8239731430433209, 0.8239731430433209), end=(0.8239731430433209, 0.5656614700878645)); line\_10 — a Line \[yellow\] drawn in box (start=(0.8239731430433209, 0.5656614700878645), end=(0.5656614700878645, 0.5656614700878645)); line\_11 — a Line \[yellow\] drawn in box (start=(0.5656614700878645, 0.5656614700878645), end=(0.5656614700878645, 0.9581854282490118)); line\_12 — a Line \[yellow\] drawn in box (start=(0.5656614700878645, 0.9581854282490118), end=(0.9581854282490118, 0.9581854282490118)); line\_13 — a Line \[yellow\] drawn in box (start=(0.9581854282490118, 0.9581854282490118), end=(0.9581854282490118, 0.1562578420270518)); line\_14 — a Line \[yellow\] drawn in box (start=(0.9581854282490118, 0.1562578420270518), end=(0.1562578420270518, 0.1562578420270518)); line\_15 — a Line \[yellow\] drawn in box (start=(0.1562578420270518, 0.1562578420270518), end=(0.1562578420270518, 0.5141811824451928)); line\_16 — a Line \[yellow\] drawn in box (start=(0.1562578420270518, 0.5141811824451928), end=(0.5141811824451928, 0.5141811824451928)); line\_17 — a Line \[yellow\] drawn in box (start=(0.5141811824451928, 0.5141811824451928), end=(0.5141811824451928, 0.9742156868513789)); line\_18 — a Line \[yellow\] drawn in box (start=(0.5141811824451928, 0.9742156868513789), end=(0.9742156868513789, 0.9742156868513789)); line\_19 — a Line \[yellow\] drawn in box (start=(0.9742156868513789, 0.9742156868513789), end=(0.9742156868513789, 0.09796598114189214)); line\_20 — a Line \[yellow\] drawn in box (start=(0.9742156868513789, 0.09796598114189214), end=(0.09796598114189214, 0.09796598114189214)); line\_21 — a Line \[yellow\] drawn in box (start=(0.09796598114189214, 0.09796598114189214), end=(0.09796598114189214, 0.34463772595511444)); line\_22 — a Line \[yellow\] drawn in box (start=(0.09796598114189214, 0.34463772595511444), end=(0.34463772595511444, 0.34463772595511444)); line\_23 — a Line \[yellow\] drawn in box (start=(0.34463772595511444, 0.34463772595511444), end=(0.34463772595511444, 0.8808639988340473)); line\_24 — a Line \[yellow\] drawn in box (start=(0.34463772595511444, 0.8808639988340473), end=(0.8808639988340473, 0.8808639988340473)); line\_25 — a Line \[yellow\] drawn in box (start=(0.8808639988340473, 0.8808639988340473), end=(0.8808639988340473, 0.40927619612934135)); line\_26 — a Line \[yellow\] drawn in box (start=(0.8808639988340473, 0.40927619612934135), end=(0.40927619612934135, 0.40927619612934135)); line\_27 — a Line \[yellow\] drawn in box (start=(0.40927619612934135, 0.40927619612934135), end=(0.40927619612934135, 0.9428998465038291)); line\_28 — a Line \[yellow\] drawn in box (start=(0.40927619612934135, 0.9428998465038291), end=(0.9428998465038291, 0.9428998465038291)); line\_29 — a Line \[yellow\] drawn in box (start=(0.9428998465038291, 0.9428998465038291), end=(0.9428998465038291, 0.20997493127085012)); line\_30 — a Line \[yellow\] drawn in box (start=(0.9428998465038291, 0.20997493127085012), end=(0.20997493127085012, 0.20997493127085012)); line\_31 — a Line \[yellow\] drawn in box (start=(0.20997493127085012, 0.20997493127085012), end=(0.20997493127085012, 0.6469532920837424)); line\_32 — a Line \[yellow\] drawn in box (start=(0.20997493127085012, 0.6469532920837424), end=(0.6469532920837424, 0.6469532920837424)); line\_33 — a Line \[yellow\] drawn in box (start=(0.6469532920837424, 0.6469532920837424), end=(0.6469532920837424, 0.8907784467884261)); line\_34 — a Line \[yellow\] drawn in box (start=(0.6469532920837424, 0.8907784467884261), end=(0.8907784467884261, 0.8907784467884261)); line\_35 — a Line \[yellow\] drawn in box (start=(0.8907784467884261, 0.8907784467884261), end=(0.8907784467884261, 0.3794396015499384)); line\_36 — a Line \[yellow\] drawn in box (start=(0.8907784467884261, 0.3794396015499384), end=(0.3794396015499384, 0.3794396015499384)); line\_37 — a Line \[yellow\] drawn in box (start=(0.3794396015499384, 0.3794396015499384), end=(0.3794396015499384, 0.9183142422696933)); line\_38 — a Line \[yellow\] drawn in box (start=(0.3794396015499384, 0.9183142422696933), end=(0.9183142422696933, 0.9183142422696933)); line\_39 — a Line \[yellow\] drawn in box (start=(0.9183142422696933, 0.9183142422696933), end=(0.9183142422696933, 0.2925514593858962)); line\_40 — a Line \[yellow\] drawn in box (start=(0.9183142422696933, 0.2925514593858962), end=(0.2925514593858962, 0.2925514593858962)) Actions: - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): box is hidden from the screen — left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): para39 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): diag is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_2 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_3 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_4 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_5 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_6 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_7 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_8 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_9 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_10 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_11 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_12 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_13 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_14 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_15 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_16 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_17 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_18 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_19 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_20 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_21 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_22 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_23 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_24 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_25 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_26 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_27 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_28 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_29 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_30 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_31 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_32 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_33 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_34 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_35 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_36 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_37 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_38 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_39 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): line\_40 is hidden from the screen — box left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): chaos\_note is hidden from the screen — left the board. - [16:25.014](https://academa.ai/lectures/period-doubling-logistic-map?t=985.0143333333332): head\_inside is hidden from the screen — left the board. ##### [16:26.214](https://academa.ai/lectures/period-doubling-logistic-map?t=986.2143333333332) Narration: Take two starting populations that differ in the third decimal place. Nought point four zero zero, and nought point four zero one. Run both for twenty years, and plot them against the year. Board: Empty. Actions: - [16:26.214](https://academa.ai/lectures/period-doubling-logistic-map?t=986.2143333333332): head\_sens is shown on the screen, written out. - [16:26.214](https://academa.ai/lectures/period-doubling-logistic-map?t=986.2143333333332): tl is shown on the screen, written out. - [16:31.253](https://academa.ai/lectures/period-doubling-logistic-map?t=991.2528333333332): runA is shown on the screen, drawn. - [16:33.111](https://academa.ai/lectures/period-doubling-logistic-map?t=993.1108333333332): runB is shown on the screen, drawn. ##### [16:38.703](https://academa.ai/lectures/period-doubling-logistic-map?t=998.7028333333332) Narration: For the first ten years or so you would not know there were two of them. They rise together and crash together. Then, somewhere around here, they part company. Board: tl — an Axes (x\_range=(0.0, 20.0), x\_ticks\_every=5.0, y\_ticks\_every=0.25); head\_sens — a Heading that says "Two Populations, Almost the Same"; runA — a ParametricCurve \[blue\] labelled "x\_0 = 0.400" drawn in tl (function=\, t\_range=(0.0, 20.0), breakpoints=(1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0,…); runB — a ParametricCurve \[red\] labelled "x\_0 = 0.401" drawn in tl (function=\, t\_range=(0.0, 20.0), breakpoints=(1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0,…) Actions: - [16:46.098](https://academa.ai/lectures/period-doubling-logistic-map?t=1006.0978333333333): The segment (12.0, 0.0) to (12.0, 1.0) in tl is lit up. ##### [16:48.085](https://academa.ai/lectures/period-doubling-logistic-map?t=1008.0853333333332) Narration: And after that they have nothing to say to each other. One is booming while the other has collapsed. The tiny difference we started with has been magnified until it is the whole picture. Board: Unchanged from the preceding beat in this scene. Actions: - [16:53.873](https://academa.ai/lectures/period-doubling-logistic-map?t=1013.8728333333332): sens\_note is shown on the screen, written out. - [16:55.603](https://academa.ai/lectures/period-doubling-logistic-map?t=1015.6028333333331): tl: retire a lit segment (unemphasize\_line). ##### [16:58.49](https://academa.ai/lectures/period-doubling-logistic-map?t=1018.4898333333332) Narration: It grows by roughly half again every single year. So a difference of one part in a thousand takes about fifteen years to become total, and a difference of one part in a million takes about thirty. Improving your measurement buys you almost nothing. Board: tl — an Axes (x\_range=(0.0, 20.0), x\_ticks\_every=5.0, y\_ticks\_every=0.25); sens\_note — a Text \[text\] that says "A difference of one part in four hundred, and after fifteen years the two have nothing to do with each other."; head\_sens — a Heading that says "Two Populations, Almost the Same"; runA — a ParametricCurve \[blue\] labelled "x\_0 = 0.400" drawn in tl (function=\, t\_range=(0.0, 20.0), breakpoints=(1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0,…); runB — a ParametricCurve \[red\] labelled "x\_0 = 0.401" drawn in tl (function=\, t\_range=(0.0, 20.0), breakpoints=(1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0,…) Actions: - None. ##### [17:13.788](https://academa.ai/lectures/period-doubling-logistic-map?t=1033.7883333333332) Narration: The rule is completely deterministic and completely useless for long term prediction, and those two facts sit together quite comfortably. But the diagram we drew has one more surprise in it, and it is the one that made people take all of this seriously. Board: Unchanged from the preceding beat in this scene. Actions: - [17:27.7](https://academa.ai/lectures/period-doubling-logistic-map?t=1047.6995208333333): head\_sens is hidden from the screen — left the board. - [17:27.7](https://academa.ai/lectures/period-doubling-logistic-map?t=1047.6995208333333): sens\_note is hidden from the screen — left the board. - [17:27.7](https://academa.ai/lectures/period-doubling-logistic-map?t=1047.6995208333333): tl is hidden from the screen — left the board. - [17:27.7](https://academa.ai/lectures/period-doubling-logistic-map?t=1047.6995208333333): runA is hidden from the screen — tl left the board. - [17:27.7](https://academa.ai/lectures/period-doubling-logistic-map?t=1047.6995208333333): runB is hidden from the screen — tl left the board. ### Scene 6: [Windows of Order](https://academa.ai/lectures/period-doubling-logistic-map?t=1048.7411874999998) Span: 17:28.741–20:21.773 (1048.7411874999998s–1221.773270833333s). #### Objects - band: a Polygon \[red\] drawn in zoom (vertices=((3.7, 0.925), (3.70725, 0.9268125), (3.7145, 0.928625), (3.721…, fill\_opacity=0.28) - box: an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25) - closing: a Text \[text\] that says "One settled population, then two, then four, then eight, faster and faster; then orbits that never repeat; and inside those, windows of perfect order, each holding the whole story again in miniature." - copy\_note: a Text \[text\] that says "The window doubles to six, to twelve, to twenty four, with the same ratio $4.669$, and gives way to chaos again." - diag: a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)) - dots3: a Point \[red\] drawn in box (location=(0.156149, 0.156149), marker\_radius=0.012) - dots3\_2: a Point \[red\] drawn in box (location=(0.504572, 0.504572), marker\_radius=0.012) - dots3\_3: a Point \[red\] drawn in box (location=(0.957418, 0.957418), marker\_radius=0.012) - final: a Math \[text\] that says "$x\_(n+1) = r thin x\_n (1 - x\_n)$" - function\_plot: a FunctionPlot \[green\] drawn in zoom (function=\, x\_range=(3.832, 3.847)) - function\_plot\_2: a FunctionPlot \[green\] drawn in zoom (function=\, x\_range=(3.832, 3.847)) - function\_plot\_3: a FunctionPlot \[green\] drawn in zoom (function=\, x\_range=(3.832, 3.847)) - head\_copy: a Heading that says "The Same Story, Smaller" - head\_end: a Heading that says "What One Equation Held" - head\_three: a Heading that says "Three Years, For Ever" - head\_win: a Heading that says "A Gap in the Chaos" - knob: a Math \[text\] that says "$r = 3.83$" - line: a Line \[gray\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.6128)) - line\_10: a Line \[gray\] drawn in box (start=(0.8299948860994241, 0.5404259268177138), end=(0.5404259268177138, 0.5404259268177138)) - line\_11: a Line \[gray\] drawn in box (start=(0.5404259268177138, 0.5404259268177138), end=(0.5404259268177138, 0.9512408012087574)) - line\_12: a Line \[gray\] drawn in box (start=(0.5404259268177138, 0.9512408012087574), end=(0.9512408012087574, 0.9512408012087574)) - line\_13: a Line \[gray\] drawn in box (start=(0.9512408012087574, 0.9512408012087574), end=(0.9512408012087574, 0.17764206161275328)) - line\_14: a Line \[gray\] drawn in box (start=(0.9512408012087574, 0.17764206161275328), end=(0.17764206161275328, 0.17764206161275328)) - 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para387: a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)) - point: a Point \[yellow\] drawn in zoom (location=(3.84, 0.62)) - point\_2: a Point \[yellow\] drawn in zoom (location=(3.84, 0.3)) - point\_3: a Point \[yellow\] drawn in zoom (location=(3.845, 0.16)) - point\_4: a Point \[yellow\] drawn in zoom (location=(3.845, 0.51)) - vals3: a Math \[text\] that says "$0.156 arrow.r 0.505 arrow.r 0.957$" - win\_note: a Text \[text\] that says "Between $r = 3.8284$ and about $r = 3.857$ the wandering stops and the population keeps a strict three year schedule." - zoom: an Axes (x\_range=(3.7, 3.99), x\_ticks\_every=0.05, y\_ticks\_every=0.25) #### Beats ##### [17:28.741](https://academa.ai/lectures/period-doubling-logistic-map?t=1048.7411874999998) Narration: Here is the right hand end of that diagram, magnified. Everything from three point seven up to four, and at first glance it is all band. Board: Empty. Actions: - [17:28.741](https://academa.ai/lectures/period-doubling-logistic-map?t=1048.7411874999998): head\_win is shown on the screen, written out. - [17:28.741](https://academa.ai/lectures/period-doubling-logistic-map?t=1048.7411874999998): zoom is shown on the screen, written out. - [17:36.427](https://academa.ai/lectures/period-doubling-logistic-map?t=1056.4271874999997): band is shown on the screen, written out. ##### [17:37.666](https://academa.ai/lectures/period-doubling-logistic-map?t=1057.6656874999999) Narration: Except that it is not. Look here, a little past three point eight two. Right in the middle of the chaos there is a clean gap. Board: zoom — an Axes (x\_range=(3.7, 3.99), x\_ticks\_every=0.05, y\_ticks\_every=0.25); head\_win — a Heading that says "A Gap in the Chaos"; band — a Polygon \[red\] drawn in zoom (vertices=((3.7, 0.925), (3.70725, 0.9268125), (3.7145, 0.928625), (3.721…, fill\_opacity=0.28) Actions: - [17:39.906](https://academa.ai/lectures/period-doubling-logistic-map?t=1059.9061874999998): point is shown on the screen, grown. - [17:41.906](https://academa.ai/lectures/period-doubling-logistic-map?t=1061.9061874999998): point is hidden from the screen. - [17:44.91](https://academa.ai/lectures/period-doubling-logistic-map?t=1064.9101875): point\_2 is shown on the screen, grown. ##### [17:46.103](https://academa.ai/lectures/period-doubling-logistic-map?t=1066.1026874999998) Narration: And running through the gap, three sharp curves. That is a window: a stretch of r where the wandering stops dead and the population goes back to a strict repeating schedule, this time of length three. Board: zoom — an Axes (x\_range=(3.7, 3.99), x\_ticks\_every=0.05, y\_ticks\_every=0.25); head\_win — a Heading that says "A Gap in the Chaos"; band — a Polygon \[red\] drawn in zoom (vertices=((3.7, 0.925), (3.70725, 0.9268125), (3.7145, 0.928625), (3.721…, fill\_opacity=0.28); point\_2 — a Point \[yellow\] drawn in zoom (location=(3.84, 0.3)) Actions: - [17:46.91](https://academa.ai/lectures/period-doubling-logistic-map?t=1066.9101875): point\_2 is hidden from the screen. - [17:48.215](https://academa.ai/lectures/period-doubling-logistic-map?t=1068.2151874999997): function\_plot is shown on the screen, drawn. - [17:48.515](https://academa.ai/lectures/period-doubling-logistic-map?t=1068.5151874999997): function\_plot\_2 is shown on the screen, drawn. - [17:48.815](https://academa.ai/lectures/period-doubling-logistic-map?t=1068.8151874999999): function\_plot\_3 is shown on the screen, drawn. - [17:49.55](https://academa.ai/lectures/period-doubling-logistic-map?t=1069.5501874999998): win\_note is shown on the screen, written out. ##### [17:58.347](https://academa.ai/lectures/period-doubling-logistic-map?t=1078.3466874999997) Narration: Which is worth a moment of disbelief. Either side of this narrow strip the very same rule produces something with no pattern in it whatsoever. Let us go and look at the orbit itself. Board: zoom — an Axes (x\_range=(3.7, 3.99), x\_ticks\_every=0.05, y\_ticks\_every=0.25); win\_note — a Text \[text\] that says "Between $r = 3.8284$ and about $r = 3.857$ the wandering stops and the population keeps a strict three year schedule."; head\_win — a Heading that says "A Gap in the Chaos"; band — a Polygon \[red\] drawn in zoom (vertices=((3.7, 0.925), (3.70725, 0.9268125), (3.7145, 0.928625), (3.721…, fill\_opacity=0.28); function\_plot — a FunctionPlot \[green\] drawn in zoom (function=\, x\_range=(3.832, 3.847)); function\_plot\_2 — a FunctionPlot \[green\] drawn in zoom (function=\, x\_range=(3.832, 3.847)); function\_plot\_3 — a FunctionPlot \[green\] drawn in zoom (function=\, x\_range=(3.832, 3.847)) Actions: - [18:9.725](https://academa.ai/lectures/period-doubling-logistic-map?t=1089.7246874999998): head\_win is hidden from the screen — left the board. - [18:9.725](https://academa.ai/lectures/period-doubling-logistic-map?t=1089.7246874999998): win\_note is hidden from the screen — left the board. - [18:9.725](https://academa.ai/lectures/period-doubling-logistic-map?t=1089.7246874999998): zoom is hidden from the screen — left the board. - [18:9.725](https://academa.ai/lectures/period-doubling-logistic-map?t=1089.7246874999998): band is hidden from the screen — zoom left the board. - [18:9.725](https://academa.ai/lectures/period-doubling-logistic-map?t=1089.7246874999998): function\_plot is hidden from the screen — zoom left the board. - [18:9.725](https://academa.ai/lectures/period-doubling-logistic-map?t=1089.7246874999998): function\_plot\_2 is hidden from the screen — zoom left the board. - [18:9.725](https://academa.ai/lectures/period-doubling-logistic-map?t=1089.7246874999998): function\_plot\_3 is hidden from the screen — zoom left the board. ##### [18:10.925](https://academa.ai/lectures/period-doubling-logistic-map?t=1090.9246875) Narration: Knob at three point eight three, and the same construction we have used all along. Grey for the transient, yellow for what it settles on. Board: Empty. Actions: - [18:10.925](https://academa.ai/lectures/period-doubling-logistic-map?t=1090.9246875): head\_three is shown on the screen, written out. - [18:10.925](https://academa.ai/lectures/period-doubling-logistic-map?t=1090.9246875): box is shown on the screen, written out. - [18:10.925](https://academa.ai/lectures/period-doubling-logistic-map?t=1090.9246875): para383 is shown on the screen, written out. - [18:10.925](https://academa.ai/lectures/period-doubling-logistic-map?t=1090.9246875): diag is shown on the screen, written out. - [18:11.064](https://academa.ai/lectures/period-doubling-logistic-map?t=1091.0641875): knob is shown on the screen, written out. - [18:15.987](https://academa.ai/lectures/period-doubling-logistic-map?t=1095.9871875): line is shown on the screen, written out. - [18:16.137](https://academa.ai/lectures/period-doubling-logistic-map?t=1096.1371874999998): line\_2 is shown on the screen, written out. - [18:16.287](https://academa.ai/lectures/period-doubling-logistic-map?t=1096.2871874999998): line\_3 is shown on the screen, written out. - [18:16.437](https://academa.ai/lectures/period-doubling-logistic-map?t=1096.4371874999997): line\_4 is shown on the screen, written out. - [18:16.587](https://academa.ai/lectures/period-doubling-logistic-map?t=1096.5871874999998): line\_5 is shown on the screen, written out. - [18:16.737](https://academa.ai/lectures/period-doubling-logistic-map?t=1096.7371875): line\_6 is shown on the screen, written out. - [18:16.887](https://academa.ai/lectures/period-doubling-logistic-map?t=1096.8871874999998): line\_7 is shown on the screen, written out. - [18:17.037](https://academa.ai/lectures/period-doubling-logistic-map?t=1097.0371874999998): line\_8 is shown on the screen, written out. - [18:17.187](https://academa.ai/lectures/period-doubling-logistic-map?t=1097.1871874999997): line\_9 is shown on the screen, written out. - [18:17.337](https://academa.ai/lectures/period-doubling-logistic-map?t=1097.3371874999998): line\_10 is shown on the screen, written out. - [18:17.487](https://academa.ai/lectures/period-doubling-logistic-map?t=1097.4871875): line\_11 is shown on the screen, written out. - [18:17.637](https://academa.ai/lectures/period-doubling-logistic-map?t=1097.6371874999998): line\_12 is shown on the screen, written out. - [18:17.787](https://academa.ai/lectures/period-doubling-logistic-map?t=1097.7871874999998): line\_13 is shown on the screen, written out. - [18:17.937](https://academa.ai/lectures/period-doubling-logistic-map?t=1097.9371874999997): line\_14 is shown on the screen, written out. - [18:18.018](https://academa.ai/lectures/period-doubling-logistic-map?t=1098.0181874999998): line\_25 is shown on the screen, written out. - [18:18.087](https://academa.ai/lectures/period-doubling-logistic-map?t=1098.0871874999998): line\_15 is shown on the screen, written out. - [18:18.237](https://academa.ai/lectures/period-doubling-logistic-map?t=1098.2371875): line\_16 is shown on the screen, written out. - [18:18.387](https://academa.ai/lectures/period-doubling-logistic-map?t=1098.3871874999998): line\_17 is shown on the screen, written out. - [18:18.418](https://academa.ai/lectures/period-doubling-logistic-map?t=1098.4181874999997): line\_26 is shown on the screen, written out. - [18:18.537](https://academa.ai/lectures/period-doubling-logistic-map?t=1098.5371874999998): line\_18 is shown on the screen, written out. - [18:18.687](https://academa.ai/lectures/period-doubling-logistic-map?t=1098.6871874999997): line\_19 is shown on the screen, written out. - [18:18.818](https://academa.ai/lectures/period-doubling-logistic-map?t=1098.8181874999998): line\_27 is shown on the screen, written out. - [18:18.837](https://academa.ai/lectures/period-doubling-logistic-map?t=1098.8371874999998): line\_20 is shown on the screen, written out. - [18:18.987](https://academa.ai/lectures/period-doubling-logistic-map?t=1098.9871875): line\_21 is shown on the screen, written out. - [18:19.137](https://academa.ai/lectures/period-doubling-logistic-map?t=1099.1371874999998): line\_22 is shown on the screen, written out. - [18:19.218](https://academa.ai/lectures/period-doubling-logistic-map?t=1099.2181874999999): line\_28 is shown on the screen, written out. - [18:19.287](https://academa.ai/lectures/period-doubling-logistic-map?t=1099.2871874999998): line\_23 is shown on the screen, written out. - [18:19.437](https://academa.ai/lectures/period-doubling-logistic-map?t=1099.4371874999997): line\_24 is shown on the screen, written out. - [18:19.618](https://academa.ai/lectures/period-doubling-logistic-map?t=1099.6181874999997): line\_29 is shown on the screen, written out. ##### [18:19.722](https://academa.ai/lectures/period-doubling-logistic-map?t=1099.7216874999997) Narration: Three points. Nought point one five six, then nought point five zero five, then nought point nine five seven, and straight back to the first. A crash, a recovery, a boom, and round again, for ever. Board: knob — a Math \[text\] that says "$r = 3.83$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_three — a Heading that says "Three Years, For Ever"; para383 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); line — a Line \[gray\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.6128)); line\_2 — a Line \[gray\] drawn in box (start=(0.2, 0.6128), end=(0.6128, 0.6128)); line\_3 — a Line \[gray\] drawn in box (start=(0.6128, 0.6128), end=(0.6128, 0.9087676928)); line\_4 — a Line \[gray\] drawn in box (start=(0.6128, 0.9087676928), end=(0.9087676928, 0.9087676928)); line\_5 — a Line \[gray\] drawn in box (start=(0.9087676928, 0.9087676928), end=(0.9087676928, 0.3175413678269552)); line\_6 — a Line \[gray\] drawn in box (start=(0.9087676928, 0.3175413678269552), end=(0.3175413678269552, 0.3175413678269552)); line\_7 — a Line \[gray\] drawn in box (start=(0.3175413678269552, 0.3175413678269552), end=(0.3175413678269552, 0.8299948860994241)); line\_8 — a Line \[gray\] drawn in box (start=(0.3175413678269552, 0.8299948860994241), end=(0.8299948860994241, 0.8299948860994241)); line\_9 — a Line \[gray\] drawn in box (start=(0.8299948860994241, 0.8299948860994241), end=(0.8299948860994241, 0.5404259268177138)); line\_10 — a Line \[gray\] drawn in box (start=(0.8299948860994241, 0.5404259268177138), end=(0.5404259268177138, 0.5404259268177138)); line\_11 — a Line \[gray\] drawn in box (start=(0.5404259268177138, 0.5404259268177138), end=(0.5404259268177138, 0.9512408012087574)); line\_12 — a Line \[gray\] drawn in box (start=(0.5404259268177138, 0.9512408012087574), end=(0.9512408012087574, 0.9512408012087574)); line\_13 — a Line \[gray\] drawn in box (start=(0.9512408012087574, 0.9512408012087574), end=(0.9512408012087574, 0.17764206161275328)); line\_14 — a Line \[gray\] drawn in box (start=(0.9512408012087574, 0.17764206161275328), end=(0.17764206161275328, 0.17764206161275328)); line\_15 — a Line \[gray\] drawn in box (start=(0.17764206161275328, 0.17764206161275328), end=(0.17764206161275328, 0.5595069271099131)); line\_16 — a Line \[gray\] drawn in box (start=(0.17764206161275328, 0.5595069271099131), end=(0.5595069271099131, 0.5595069271099131)); line\_17 — a Line \[gray\] drawn in box (start=(0.5595069271099131, 0.5595069271099131), end=(0.5595069271099131, 0.943937685147333)); line\_18 — a Line \[gray\] drawn in box (start=(0.5595069271099131, 0.943937685147333), end=(0.943937685147333, 0.943937685147333)); line\_19 — a Line \[gray\] drawn in box (start=(0.943937685147333, 0.943937685147333), end=(0.943937685147333, 0.20268104043408505)); line\_20 — a Line \[gray\] drawn in box (start=(0.943937685147333, 0.20268104043408505), end=(0.20268104043408505, 0.20268104043408505)); line\_21 — a Line \[gray\] drawn in box (start=(0.20268104043408505, 0.20268104043408505), end=(0.20268104043408505, 0.6189335009625182)); line\_22 — a Line \[gray\] drawn in box (start=(0.20268104043408505, 0.6189335009625182), end=(0.6189335009625182, 0.6189335009625182)); line\_23 — a Line \[gray\] drawn in box (start=(0.6189335009625182, 0.6189335009625182), end=(0.6189335009625182, 0.9033239695958989)); line\_24 — a Line \[gray\] drawn in box (start=(0.6189335009625182, 0.9033239695958989), end=(0.9033239695958989, 0.9033239695958989)); line\_25 — a Line \[yellow\] drawn in box (start=(0.156149, 0.156149), end=(0.156149, 0.50466565593017)); line\_26 — a Line \[yellow\] drawn in box (start=(0.156149, 0.50466565593017), end=(0.50466565593017, 0.50466565593017)); line\_27 — a Line \[yellow\] drawn in box (start=(0.50466565593017, 0.50466565593017), end=(0.50466565593017, 0.9574166272376591)); line\_28 — a Line \[yellow\] drawn in box (start=(0.50466565593017, 0.9574166272376591), end=(0.9574166272376591, 0.9574166272376591)); line\_29 — a Line \[yellow\] drawn in box (start=(0.9574166272376591, 0.9574166272376591), end=(0.9574166272376591, 0.1561492115545886)) Actions: - [18:20.018](https://academa.ai/lectures/period-doubling-logistic-map?t=1100.0181874999998): line\_30 is shown on the screen, written out. - [18:20.36](https://academa.ai/lectures/period-doubling-logistic-map?t=1100.3601874999997): vals3 is shown on the screen, written out. - [18:20.36](https://academa.ai/lectures/period-doubling-logistic-map?t=1100.3601874999997): dots3 is shown on the screen, written out. - [18:20.56](https://academa.ai/lectures/period-doubling-logistic-map?t=1100.5601874999998): dots3\_2 is shown on the screen, written out. - [18:20.76](https://academa.ai/lectures/period-doubling-logistic-map?t=1100.7601874999998): dots3\_3 is shown on the screen, written out. ##### [18:35.357](https://academa.ai/lectures/period-doubling-logistic-map?t=1115.3566874999997) Narration: That is as orderly as anything we saw at r equal to two point six. Now turn the knob by four hundredths, to three point eight seven, and watch the schedule evaporate. Board: knob — a Math \[text\] that says "$r = 3.83$"; vals3 — a Math \[text\] that says "$0.156 arrow.r 0.505 arrow.r 0.957$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_three — a Heading that says "Three Years, For Ever"; para383 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); line — a Line \[gray\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.6128)); line\_2 — a Line \[gray\] drawn in box (start=(0.2, 0.6128), end=(0.6128, 0.6128)); line\_3 — a Line \[gray\] drawn in box (start=(0.6128, 0.6128), end=(0.6128, 0.9087676928)); line\_4 — a Line \[gray\] drawn in box (start=(0.6128, 0.9087676928), end=(0.9087676928, 0.9087676928)); line\_5 — a Line \[gray\] drawn in box (start=(0.9087676928, 0.9087676928), end=(0.9087676928, 0.3175413678269552)); line\_6 — a Line \[gray\] drawn in box (start=(0.9087676928, 0.3175413678269552), end=(0.3175413678269552, 0.3175413678269552)); line\_7 — a Line \[gray\] drawn in box (start=(0.3175413678269552, 0.3175413678269552), end=(0.3175413678269552, 0.8299948860994241)); line\_8 — a Line \[gray\] drawn in box (start=(0.3175413678269552, 0.8299948860994241), end=(0.8299948860994241, 0.8299948860994241)); line\_9 — a Line \[gray\] drawn in box (start=(0.8299948860994241, 0.8299948860994241), end=(0.8299948860994241, 0.5404259268177138)); line\_10 — a Line \[gray\] drawn in box (start=(0.8299948860994241, 0.5404259268177138), end=(0.5404259268177138, 0.5404259268177138)); line\_11 — a Line \[gray\] drawn in box (start=(0.5404259268177138, 0.5404259268177138), end=(0.5404259268177138, 0.9512408012087574)); line\_12 — a Line \[gray\] drawn in box (start=(0.5404259268177138, 0.9512408012087574), end=(0.9512408012087574, 0.9512408012087574)); line\_13 — a Line \[gray\] drawn in box (start=(0.9512408012087574, 0.9512408012087574), end=(0.9512408012087574, 0.17764206161275328)); line\_14 — a Line \[gray\] drawn in box (start=(0.9512408012087574, 0.17764206161275328), end=(0.17764206161275328, 0.17764206161275328)); line\_15 — a Line \[gray\] drawn in box (start=(0.17764206161275328, 0.17764206161275328), end=(0.17764206161275328, 0.5595069271099131)); line\_16 — a Line \[gray\] drawn in box (start=(0.17764206161275328, 0.5595069271099131), end=(0.5595069271099131, 0.5595069271099131)); line\_17 — a Line \[gray\] drawn in box (start=(0.5595069271099131, 0.5595069271099131), end=(0.5595069271099131, 0.943937685147333)); line\_18 — a Line \[gray\] drawn in box (start=(0.5595069271099131, 0.943937685147333), end=(0.943937685147333, 0.943937685147333)); line\_19 — a Line \[gray\] drawn in box (start=(0.943937685147333, 0.943937685147333), end=(0.943937685147333, 0.20268104043408505)); line\_20 — a Line \[gray\] drawn in box (start=(0.943937685147333, 0.20268104043408505), end=(0.20268104043408505, 0.20268104043408505)); line\_21 — a Line \[gray\] drawn in box (start=(0.20268104043408505, 0.20268104043408505), end=(0.20268104043408505, 0.6189335009625182)); line\_22 — a Line \[gray\] drawn in box (start=(0.20268104043408505, 0.6189335009625182), end=(0.6189335009625182, 0.6189335009625182)); line\_23 — a Line \[gray\] drawn in box (start=(0.6189335009625182, 0.6189335009625182), end=(0.6189335009625182, 0.9033239695958989)); line\_24 — a Line \[gray\] drawn in box (start=(0.6189335009625182, 0.9033239695958989), end=(0.9033239695958989, 0.9033239695958989)); line\_25 — a Line \[yellow\] drawn in box (start=(0.156149, 0.156149), end=(0.156149, 0.50466565593017)); line\_26 — a Line \[yellow\] drawn in box (start=(0.156149, 0.50466565593017), end=(0.50466565593017, 0.50466565593017)); line\_27 — a Line \[yellow\] drawn in box (start=(0.50466565593017, 0.50466565593017), end=(0.50466565593017, 0.9574166272376591)); line\_28 — a Line \[yellow\] drawn in box (start=(0.50466565593017, 0.9574166272376591), end=(0.9574166272376591, 0.9574166272376591)); line\_29 — a Line \[yellow\] drawn in box (start=(0.9574166272376591, 0.9574166272376591), end=(0.9574166272376591, 0.1561492115545886)); line\_30 — a Line \[yellow\] drawn in box (start=(0.9574166272376591, 0.1561492115545886), end=(0.1561492115545886, 0.1561492115545886)); dots3 — a Point \[red\] drawn in box (location=(0.156149, 0.156149), marker\_radius=0.012); dots3\_2 — a Point \[red\] drawn in box (location=(0.504572, 0.504572), marker\_radius=0.012); dots3\_3 — a Point \[red\] drawn in box (location=(0.957418, 0.957418), marker\_radius=0.012) Actions: - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_2 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_3 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_4 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_5 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_6 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_7 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_8 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_9 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_10 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_11 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_12 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_13 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_14 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_15 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_16 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_17 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_18 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_19 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_20 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_21 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_22 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_23 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_24 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_25 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_26 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_27 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_28 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_29 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): line\_30 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): para383 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): dots3 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): dots3\_2 is hidden from the screen. - [18:40.302](https://academa.ai/lectures/period-doubling-logistic-map?t=1120.3021874999997): dots3\_3 is hidden from the screen. - [18:42.961](https://academa.ai/lectures/period-doubling-logistic-map?t=1122.9611874999998): knob becomes "$r = 3.87$". - [18:42.961](https://academa.ai/lectures/period-doubling-logistic-map?t=1122.9611874999998): para387 is shown on the screen, written out. - [18:42.961](https://academa.ai/lectures/period-doubling-logistic-map?t=1122.9611874999998): vals3 is hidden from the screen. - [18:43.925](https://academa.ai/lectures/period-doubling-logistic-map?t=1123.9251874999998): line\_31 is shown on the screen, written out. - [18:44.085](https://academa.ai/lectures/period-doubling-logistic-map?t=1124.0851874999998): line\_32 is shown on the screen, written out. - [18:44.245](https://academa.ai/lectures/period-doubling-logistic-map?t=1124.2451874999997): line\_33 is shown on the screen, written out. - [18:44.405](https://academa.ai/lectures/period-doubling-logistic-map?t=1124.4051874999998): line\_34 is shown on the screen, written out. - [18:44.565](https://academa.ai/lectures/period-doubling-logistic-map?t=1124.5651874999999): line\_35 is shown on the screen, written out. - [18:44.725](https://academa.ai/lectures/period-doubling-logistic-map?t=1124.7251874999997): line\_36 is shown on the screen, written out. - [18:44.885](https://academa.ai/lectures/period-doubling-logistic-map?t=1124.8851874999998): line\_37 is shown on the screen, written out. - [18:45.045](https://academa.ai/lectures/period-doubling-logistic-map?t=1125.0451874999999): line\_38 is shown on the screen, written out. - [18:45.205](https://academa.ai/lectures/period-doubling-logistic-map?t=1125.2051874999997): line\_39 is shown on the screen, written out. - [18:45.365](https://academa.ai/lectures/period-doubling-logistic-map?t=1125.3651874999998): line\_40 is shown on the screen, written out. - [18:45.525](https://academa.ai/lectures/period-doubling-logistic-map?t=1125.5251875): line\_41 is shown on the screen, written out. - [18:45.685](https://academa.ai/lectures/period-doubling-logistic-map?t=1125.6851874999998): line\_42 is shown on the screen, written out. - [18:45.845](https://academa.ai/lectures/period-doubling-logistic-map?t=1125.8451874999998): line\_43 is shown on the screen, written out. - [18:46.005](https://academa.ai/lectures/period-doubling-logistic-map?t=1126.0051874999997): line\_44 is shown on the screen, written out. - [18:46.165](https://academa.ai/lectures/period-doubling-logistic-map?t=1126.1651874999998): line\_45 is shown on the screen, written out. - [18:46.325](https://academa.ai/lectures/period-doubling-logistic-map?t=1126.3251874999999): line\_46 is shown on the screen, written out. ##### [18:46.371](https://academa.ai/lectures/period-doubling-logistic-map?t=1126.3711875) Narration: There it is, wandering again. The window really is narrow, and it really is surrounded on both sides by chaos. Board: knob — a Math \[text\] that says "$r = 3.83$"; box — an Axes (aspect=(1, 1), x\_ticks\_every=0.25, y\_ticks\_every=0.25); head\_three — a Heading that says "Three Years, For Ever"; diag — a Line \[green\] labelled "y = x" drawn in box (end=(1.0, 1.0)); para387 — a FunctionPlot \[blue\] drawn in box (function=\, x\_range=(0.0, 1.0)); line\_31 — a Line \[yellow\] drawn in box (start=(0.2, 0.0), end=(0.2, 0.6192000000000001)); line\_32 — a Line \[yellow\] drawn in box (start=(0.2, 0.6192000000000001), end=(0.6192000000000001, 0.6192000000000001)); line\_33 — a Line \[yellow\] drawn in box (start=(0.6192000000000001, 0.6192000000000001), end=(0.6192000000000001, 0.9125125631999998)); line\_34 — a Line \[yellow\] drawn in box (start=(0.6192000000000001, 0.9125125631999998), end=(0.9125125631999998, 0.9125125631999998)); line\_35 — a Line \[yellow\] drawn in box (start=(0.9125125631999998, 0.9125125631999998), end=(0.9125125631999998, 0.3089552007323829)); line\_36 — a Line \[yellow\] drawn in box (start=(0.9125125631999998, 0.3089552007323829), end=(0.3089552007323829, 0.3089552007323829)); line\_37 — a Line \[yellow\] drawn in box (start=(0.3089552007323829, 0.3089552007323829), end=(0.3089552007323829, 0.8262522936837201)); line\_38 — a Line \[yellow\] drawn in box (start=(0.3089552007323829, 0.8262522936837201), end=(0.8262522936837201, 0.8262522936837201)); line\_39 — a Line \[yellow\] drawn in box (start=(0.8262522936837201, 0.8262522936837201), end=(0.8262522936837201, 0.5555750361518521)); line\_40 — a Line \[yellow\] drawn in box (start=(0.8262522936837201, 0.5555750361518521), end=(0.5555750361518521, 0.5555750361518521)); line\_41 — a Line \[yellow\] drawn in box (start=(0.5555750361518521, 0.5555750361518521), end=(0.5555750361518521, 0.9555471774305077)); line\_42 — a Line \[yellow\] drawn in box (start=(0.5555750361518521, 0.9555471774305077), end=(0.9555471774305077, 0.9555471774305077)); line\_43 — a Line \[yellow\] drawn in box (start=(0.9555471774305077, 0.9555471774305077), end=(0.9555471774305077, 0.16438509655282763)); line\_44 — a Line \[yellow\] drawn in box (start=(0.9555471774305077, 0.16438509655282763), end=(0.16438509655282763, 0.16438509655282763)); line\_45 — a Line \[yellow\] drawn in box (start=(0.16438509655282763, 0.16438509655282763), end=(0.16438509655282763, 0.5315934035806418)); line\_46 — a Line \[yellow\] drawn in box (start=(0.16438509655282763, 0.5315934035806418), end=(0.5315934035806418, 0.5315934035806418)) Actions: - [18:46.485](https://academa.ai/lectures/period-doubling-logistic-map?t=1126.4851874999997): line\_47 is shown on the screen, written out. - [18:46.645](https://academa.ai/lectures/period-doubling-logistic-map?t=1126.6451874999998): line\_48 is shown on the screen, written out. - [18:46.805](https://academa.ai/lectures/period-doubling-logistic-map?t=1126.8051874999999): line\_49 is shown on the screen, written out. - [18:46.965](https://academa.ai/lectures/period-doubling-logistic-map?t=1126.9651874999997): line\_50 is shown on the screen, written out. - [18:47.125](https://academa.ai/lectures/period-doubling-logistic-map?t=1127.1251874999998): line\_51 is shown on the screen, written out. - [18:47.285](https://academa.ai/lectures/period-doubling-logistic-map?t=1127.2851875): line\_52 is shown on the screen, written out. - [18:47.445](https://academa.ai/lectures/period-doubling-logistic-map?t=1127.4451874999997): line\_53 is shown on the screen, written out. - [18:47.605](https://academa.ai/lectures/period-doubling-logistic-map?t=1127.6051874999998): line\_54 is shown on the screen, written out. - [18:47.765](https://academa.ai/lectures/period-doubling-logistic-map?t=1127.7651874999997): line\_55 is shown on the screen, written out. - [18:47.925](https://academa.ai/lectures/period-doubling-logistic-map?t=1127.9251874999998): line\_56 is shown on the screen, written out. - [18:48.085](https://academa.ai/lectures/period-doubling-logistic-map?t=1128.0851874999998): line\_57 is shown on the screen, written out. - [18:48.245](https://academa.ai/lectures/period-doubling-logistic-map?t=1128.2451874999997): line\_58 is shown on the screen, written out. - [18:48.405](https://academa.ai/lectures/period-doubling-logistic-map?t=1128.4051874999998): line\_59 is shown on the screen, written out. - [18:48.565](https://academa.ai/lectures/period-doubling-logistic-map?t=1128.5651874999999): line\_60 is shown on the screen, written out. - [18:48.725](https://academa.ai/lectures/period-doubling-logistic-map?t=1128.7251874999997): line\_61 is shown on the screen, written out. - [18:48.885](https://academa.ai/lectures/period-doubling-logistic-map?t=1128.8851874999998): line\_62 is shown on the screen, written out. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): box is hidden from the screen — left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): diag is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): para387 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_31 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_32 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_33 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_34 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_35 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_36 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_37 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_38 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_39 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_40 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_41 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_42 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_43 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_44 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_45 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_46 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_47 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_48 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_49 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_50 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_51 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_52 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_53 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_54 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_55 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_56 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_57 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_58 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_59 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_60 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_61 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): line\_62 is hidden from the screen — box left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): head\_three is hidden from the screen — left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): knob is hidden from the screen — left the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): zoom is shown on the screen, faded in — cast on this board again. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): band is shown on the screen, faded in — zoom came back to the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): function\_plot is shown on the screen, faded in — zoom came back to the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): function\_plot\_2 is shown on the screen, faded in — zoom came back to the board. - [18:53.825](https://academa.ai/lectures/period-doubling-logistic-map?t=1133.8246874999998): function\_plot\_3 is shown on the screen, faded in — zoom came back to the board. ##### [18:55.025](https://academa.ai/lectures/period-doubling-logistic-map?t=1135.0246874999998) Narration: Now go back to those three curves and follow them to the right hand end of the window. Board: zoom — an Axes (x\_range=(3.7, 3.99), x\_ticks\_every=0.05, y\_ticks\_every=0.25); band — a Polygon \[red\] drawn in zoom (vertices=((3.7, 0.925), (3.70725, 0.9268125), (3.7145, 0.928625), (3.721…, fill\_opacity=0.28); function\_plot — a FunctionPlot \[green\] drawn in zoom (function=\, x\_range=(3.832, 3.847)); function\_plot\_2 — a FunctionPlot \[green\] drawn in zoom (function=\, x\_range=(3.832, 3.847)); function\_plot\_3 — a FunctionPlot \[green\] drawn in zoom (function=\, x\_range=(3.832, 3.847)) Actions: - [18:55.025](https://academa.ai/lectures/period-doubling-logistic-map?t=1135.0246874999998): head\_copy is shown on the screen, written out. - [18:57.37](https://academa.ai/lectures/period-doubling-logistic-map?t=1137.3701874999997): point\_3 is shown on the screen, grown. - [18:58.066](https://academa.ai/lectures/period-doubling-logistic-map?t=1138.0661874999998): point\_4 is shown on the screen, grown. - [18:59.37](https://academa.ai/lectures/period-doubling-logistic-map?t=1139.3701874999997): point\_3 is hidden from the screen. - [19:0.066](https://academa.ai/lectures/period-doubling-logistic-map?t=1140.0661874999998): point\_4 is hidden from the screen. ##### [19:0.35](https://academa.ai/lectures/period-doubling-logistic-map?t=1140.3496874999998) Narration: Each of them splits into two, and then into four, and it accumulates and gives way to chaos again, with the very same ratio four point six six nine governing the splittings. Board: zoom — an Axes (x\_range=(3.7, 3.99), x\_ticks\_every=0.05, y\_ticks\_every=0.25); band — a Polygon \[red\] drawn in zoom (vertices=((3.7, 0.925), (3.70725, 0.9268125), (3.7145, 0.928625), (3.721…, fill\_opacity=0.28); function\_plot — a FunctionPlot \[green\] drawn in zoom (function=\, x\_range=(3.832, 3.847)); function\_plot\_2 — a FunctionPlot \[green\] drawn in zoom (function=\, x\_range=(3.832, 3.847)); function\_plot\_3 — a FunctionPlot \[green\] drawn in zoom (function=\, x\_range=(3.832, 3.847)); head\_copy — a Heading that says "The Same Story, Smaller" Actions: - [19:1.197](https://academa.ai/lectures/period-doubling-logistic-map?t=1141.1971874999997): copy\_note is shown on the screen, written out. ##### [19:11.561](https://academa.ai/lectures/period-doubling-logistic-map?t=1151.5606874999999) Narration: So the window is not merely an island of order. It contains a small copy of the entire picture we spent this lecture building. And magnify that copy and you find windows inside it, each holding a smaller copy again. Board: zoom — an Axes (x\_range=(3.7, 3.99), x\_ticks\_every=0.05, y\_ticks\_every=0.25); band — a Polygon \[red\] drawn in zoom (vertices=((3.7, 0.925), (3.70725, 0.9268125), (3.7145, 0.928625), (3.721…, fill\_opacity=0.28); function\_plot — a FunctionPlot \[green\] drawn in zoom (function=\, x\_range=(3.832, 3.847)); function\_plot\_2 — a FunctionPlot \[green\] drawn in zoom (function=\, x\_range=(3.832, 3.847)); function\_plot\_3 — a FunctionPlot \[green\] drawn in zoom (function=\, x\_range=(3.832, 3.847)); copy\_note — a Text \[text\] that says "The window doubles to six, to twelve, to twenty four, with the same ratio $4.669$, and gives way to chaos again."; head\_copy — a Heading that says "The Same Story, Smaller" Actions: - None. ##### [19:25.06](https://academa.ai/lectures/period-doubling-logistic-map?t=1165.0596874999999) Narration: The structure goes all the way down. There is no scale at which it becomes simple. Board: Unchanged from the preceding beat in this scene. Actions: - [19:29.739](https://academa.ai/lectures/period-doubling-logistic-map?t=1169.7386874999997): copy\_note is hidden from the screen — left the board. - [19:29.739](https://academa.ai/lectures/period-doubling-logistic-map?t=1169.7386874999997): head\_copy is hidden from the screen — left the board. - [19:29.739](https://academa.ai/lectures/period-doubling-logistic-map?t=1169.7386874999997): zoom is hidden from the screen — left the board. - [19:29.739](https://academa.ai/lectures/period-doubling-logistic-map?t=1169.7386874999997): band is hidden from the screen — zoom left the board. - [19:29.739](https://academa.ai/lectures/period-doubling-logistic-map?t=1169.7386874999997): function\_plot is hidden from the screen — zoom left the board. - [19:29.739](https://academa.ai/lectures/period-doubling-logistic-map?t=1169.7386874999997): function\_plot\_2 is hidden from the screen — zoom left the board. - [19:29.739](https://academa.ai/lectures/period-doubling-logistic-map?t=1169.7386874999997): function\_plot\_3 is hidden from the screen — zoom left the board. ##### [19:30.939](https://academa.ai/lectures/period-doubling-logistic-map?t=1170.9386874999998) Narration: So here is what one innocent equation turned out to hold. For a small growth rate, a single settled population, and a staircase that walks to it because the curve crosses the line gently. Board: Empty. Actions: - [19:30.939](https://academa.ai/lectures/period-doubling-logistic-map?t=1170.9386874999998): head\_end is shown on the screen, written out. - [19:32.61](https://academa.ai/lectures/period-doubling-logistic-map?t=1172.6101874999997): final is shown on the screen, written out. ##### [19:43.311](https://academa.ai/lectures/period-doubling-logistic-map?t=1183.3106874999999) Narration: As the rate grows, that crossing steepens until it can no longer hold anything, and a two cycle takes over. Then the two cycle fails the same way, and its successor after it, faster and faster, with gaps shrinking by a factor that has nothing to do with our equation. Board: final — a Math \[text\] that says "$x\_(n+1) = r thin x\_n (1 - x\_n)$"; head\_end — a Heading that says "What One Equation Held" Actions: - [19:53.377](https://academa.ai/lectures/period-doubling-logistic-map?t=1193.3771874999998): closing is shown on the screen, written out. ##### [20:0.746](https://academa.ai/lectures/period-doubling-logistic-map?t=1200.7456874999998) Narration: Past the point where those pile up, orbits that never repeat and that forget where they started. And inside the chaos, windows of perfect order, each containing the whole story again in miniature. Board: final — a Math \[text\] that says "$x\_(n+1) = r thin x\_n (1 - x\_n)$"; closing — a Text \[text\] that says "One settled population, then two, then four, then eight, faster and faster; then orbits that never repeat; and inside those, windows of perfect order, each holding the whole story again in miniature."; head\_end — a Heading that says "What One Equation Held" Actions: - [20:7.978](https://academa.ai/lectures/period-doubling-logistic-map?t=1207.9781874999999): A box is drawn around final. ##### [20:13.013](https://academa.ai/lectures/period-doubling-logistic-map?t=1213.0131874999997) Narration: None of that was put in. All of it was sitting inside r x times one minus x, waiting for somebody to turn the knob. Board: Unchanged from the preceding beat in this scene. Actions: - [20:20.732](https://academa.ai/lectures/period-doubling-logistic-map?t=1220.7316041666666): closing is hidden from the screen — left the board. - [20:20.732](https://academa.ai/lectures/period-doubling-logistic-map?t=1220.7316041666666): final is hidden from the screen — left the board. - [20:20.732](https://academa.ai/lectures/period-doubling-logistic-map?t=1220.7316041666666): head\_end is hidden from the screen — left the board.