The Logistic Map: From Fixed Points to Chaos via Period-Doubling

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?

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?

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?

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.

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.

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.