WEBVTT 00:00:00.058 --> 00:00:04.076 Here is an equation you could write on a biology exam without anybody blinking. 00:00:04.690 --> 00:00:07.196 Next year's population is this year's, 00:00:07.279 --> 00:00:11.669 multiplied by a growth rate, multiplied by however much room is left. 00:00:11.865 --> 00:00:13.944 Nothing about it looks dangerous. 00:00:14.141 --> 00:00:18.448 And yet if we turn that growth rate up slowly and watch, this little 00:00:18.448 --> 00:00:23.395 rule stops settling down and starts doing something nobody had any right to expect. 00:00:24.791 --> 00:00:26.821 So let us write it down properly. 00:00:26.904 --> 00:00:30.130 Call x sub n the population in year n, 00:00:30.213 --> 00:00:35.020 measured not in animals but as a fraction of the most the environment could support. 00:00:35.333 --> 00:00:41.905 Then next year's is r times x n, times one minus x n. 00:00:42.702 --> 00:00:45.048 Read the two factors separately. 00:00:45.244 --> 00:00:47.390 The first is plain growth: 00:00:47.473 --> 00:00:50.609 more of them this year means more of them next year. 00:00:50.805 --> 00:00:52.699 The second is crowding. 00:00:52.802 --> 00:00:57.562 While the population is small that factor sits near one and hardly bites, and as the 00:00:57.562 --> 00:01:02.486 population approaches capacity it drags the whole product down towards nothing. 00:01:03.398 --> 00:01:06.092 Two things are fixed for the rest of the lecture. 00:01:06.196 --> 00:01:09.759 The population x always lies between zero and one, 00:01:09.842 --> 00:01:13.256 nought being extinction and one being a full house. 00:01:13.359 --> 00:01:18.248 And r is a knob, which we shall turn from one up to four and no further. 00:01:18.952 --> 00:01:23.919 The equation has a name, the logistic map, and the word map is worth noticing. 00:01:24.002 --> 00:01:28.624 This is not a differential equation and there is no time flowing smoothly along. 00:01:28.728 --> 00:01:32.989 There is this year, and there is next year, and a rule joining them. 00:01:33.786 --> 00:01:37.955 Here it is drawn as a picture, with the knob set at two point six. 00:01:38.151 --> 00:01:41.159 Input along the bottom, output up the side. 00:01:41.356 --> 00:01:45.176 Feed in a population, read off the one that follows it. 00:01:45.973 --> 00:01:47.239 It is a hump. 00:01:47.436 --> 00:01:50.362 A small population gives a small answer. 00:01:50.559 --> 00:01:53.633 A population near capacity gives a small answer too, 00:01:53.716 --> 00:01:55.969 because there is nowhere to put the young. 00:01:56.282 --> 00:01:59.836 And the best year of all comes from starting somewhere near the middle. 00:02:00.748 --> 00:02:02.130 Now the only thing that ever happens 00:02:02.130 --> 00:02:06.366 in this lecture is that we apply the rule again, and again, and again. 00:02:06.449 --> 00:02:07.980 So start somewhere. 00:02:08.063 --> 00:02:10.966 Say a fifth of capacity, nought point two. 00:02:11.670 --> 00:02:15.943 One turn of the handle takes nought point two to nought point four one six. 00:02:16.256 --> 00:02:17.927 On the picture that is going straight up 00:02:17.927 --> 00:02:20.364 from nought point two until you hit the curve. 00:02:20.447 --> 00:02:23.153 There is the answer, measured up the side. 00:02:23.950 --> 00:02:27.016 And here is the move that makes the whole construction work. 00:02:27.212 --> 00:02:30.916 That answer has to become next year's input, so somehow it must get 00:02:30.916 --> 00:02:34.283 from the vertical axis back onto the horizontal one. 00:02:34.387 --> 00:02:38.521 The green line, y equals x, does exactly that. 00:02:39.573 --> 00:02:44.299 Go across to it, and you are now standing above nought point four one six. 00:02:44.495 --> 00:02:48.313 Then climb to the curve for the year after, and cross back down to the line, 00:02:48.396 --> 00:02:49.813 and carry on. 00:02:49.917 --> 00:02:52.135 Each pair of moves is one year. 00:02:52.932 --> 00:02:55.649 Let it run for ten years and watch where it goes. 00:02:57.046 --> 00:02:58.822 The staircase spirals inwards 00:02:58.822 --> 00:03:02.120 and parks itself exactly where the curve crosses the line. 00:03:02.572 --> 00:03:06.985 A little over six tenths of capacity, and it stays there for ever after. 00:03:07.182 --> 00:03:10.108 Start anywhere else and it goes to the same place. 00:03:10.905 --> 00:03:14.747 That crossing is the whole story of the settled state, because a crossing 00:03:14.747 --> 00:03:19.230 is precisely a year in which next year's population equals this year's. 00:03:19.542 --> 00:03:21.506 So there are two questions. 00:03:21.702 --> 00:03:22.829 What is it? 00:03:22.933 --> 00:03:26.103 And why did the staircase walk towards it rather than away? 00:03:27.196 --> 00:03:31.214 Start with the settled state, since that is what we just watched happen. 00:03:31.318 --> 00:03:35.684 A population that has stopped moving is one where next year equals this year. 00:03:36.136 --> 00:03:37.829 Call that value x star. 00:03:37.912 --> 00:03:43.881 Then x star must equal r x star times one minus x star. 00:03:44.793 --> 00:03:49.682 Now x star could of course be zero, and we shall come back to that in a moment. 00:03:49.878 --> 00:03:52.132 Otherwise divide both sides by it. 00:03:52.328 --> 00:03:55.580 One equals r times one minus x star. 00:03:56.376 --> 00:03:58.618 Rearrange, and there it is. 00:03:58.931 --> 00:04:02.284 The settled population is one minus one over r. 00:04:02.367 --> 00:04:06.173 With the knob at two point six that comes to nought point six one five, 00:04:06.256 --> 00:04:08.811 which is precisely where our staircase parked. 00:04:09.724 --> 00:04:12.404 But now the question this whole lecture turns on. 00:04:12.487 --> 00:04:15.135 Why did the staircase walk towards that point? 00:04:15.331 --> 00:04:18.513 Nothing in the algebra we just did says it has to. 00:04:18.617 --> 00:04:21.053 Zero satisfies the very same equation, 00:04:21.136 --> 00:04:24.551 and orbits run away from zero as fast as they can go. 00:04:25.947 --> 00:04:29.489 The answer is the steepness of the curve where it crosses the line. 00:04:29.802 --> 00:04:31.764 So differentiate the rule. 00:04:31.961 --> 00:04:36.373 F prime of x is r times one minus two x. 00:04:37.170 --> 00:04:42.895 Substitute the fixed point, one minus one over r, and the r cancels beautifully. 00:04:42.998 --> 00:04:46.110 The slope at the crossing is simply two minus r. 00:04:46.307 --> 00:04:50.940 At two point six that is nought point six, with a minus sign in front. 00:04:51.737 --> 00:04:54.024 Here is what that number does for a living. 00:04:54.221 --> 00:04:57.322 Suppose you are a small distance away from x star. 00:04:57.425 --> 00:04:58.982 Very near a crossing 00:04:59.086 --> 00:05:01.940 the curve is almost exactly its own tangent line, 00:05:02.023 --> 00:05:06.436 so one turn of the handle takes that distance and multiplies it by the slope. 00:05:07.232 --> 00:05:12.353 Multiply a distance by minus nought point six and it shrinks, and it changes sides. 00:05:12.550 --> 00:05:16.277 Multiply again and it shrinks and changes sides again. 00:05:16.381 --> 00:05:19.421 Which is exactly the inward spiral we watched: 00:05:19.504 --> 00:05:23.046 over, under, over, under, closing in. 00:05:23.958 --> 00:05:26.211 And notice there are two flavours. 00:05:26.408 --> 00:05:31.120 A positive slope smaller than one shrinks distances without changing sides, 00:05:31.203 --> 00:05:35.813 and the staircase climbs to the crossing like an ordinary flight of stairs. 00:05:36.009 --> 00:05:38.343 A negative one gives the spiral. 00:05:38.540 --> 00:05:42.209 Either way, less than one in size means the point holds. 00:05:43.006 --> 00:05:47.662 So the condition is this, and it is the only piece of theory in the whole lecture. 00:05:47.975 --> 00:05:53.000 The fixed point holds on as long as the size of two minus r is less than one, 00:05:53.083 --> 00:05:56.881 and that is the same as saying r lies between one and three. 00:05:58.277 --> 00:05:59.892 Watch it get feeble. 00:05:59.995 --> 00:06:02.283 Turn the knob up to two point nine. 00:06:02.480 --> 00:06:04.696 The crossing has shifted a little to the right, 00:06:04.779 --> 00:06:08.367 and the slope there is now minus nought point nine. 00:06:09.164 --> 00:06:11.579 Start from a fifth of capacity again. 00:06:11.776 --> 00:06:15.431 Each year the distance from the crossing shrinks by only a tenth of itself, 00:06:15.514 --> 00:06:17.477 so the staircase takes an age. 00:06:17.581 --> 00:06:21.912 But a tenth of a tenth of a tenth is nothing in the end, and it still parks. 00:06:22.709 --> 00:06:27.296 At r exactly three, the slope at the crossing is exactly minus one. 00:06:27.609 --> 00:06:29.766 Distances are no longer shrunk at all. 00:06:29.849 --> 00:06:34.691 They are simply reflected, over and back, over and back, at the same size for ever. 00:06:35.306 --> 00:06:39.451 The fixed point has stopped pulling, and it has not yet started pushing. 00:06:39.725 --> 00:06:41.351 So go past it. 00:06:41.548 --> 00:06:42.988 Three point two. 00:06:43.301 --> 00:06:47.190 The slope at the crossing is minus one point two, and each year the distance 00:06:47.190 --> 00:06:49.711 from it is multiplied by one point two. 00:06:50.623 --> 00:06:54.267 Now start almost exactly on the fixed point, a hair above it, 00:06:54.350 --> 00:06:56.429 and watch what the construction does. 00:06:57.226 --> 00:06:58.469 It leaves. 00:06:58.666 --> 00:07:00.533 The staircase spirals outwards, 00:07:00.616 --> 00:07:04.796 away from a value that satisfies the fixed point equation perfectly well. 00:07:04.993 --> 00:07:07.304 The point has not gone anywhere. 00:07:07.501 --> 00:07:11.795 It has simply lost the ability to hold on to anything, and everything near it is now 00:07:11.878 --> 00:07:13.342 on its way out. 00:07:14.254 --> 00:07:16.832 Which leaves an obvious gap in the story. 00:07:16.936 --> 00:07:19.512 The population is not going to the fixed point. 00:07:19.595 --> 00:07:23.401 It is trapped between zero and one so it cannot run off. 00:07:23.484 --> 00:07:25.226 Where on earth is it going? 00:07:26.318 --> 00:07:27.189 So here is the knob 00:07:27.189 --> 00:07:32.066 at three point two again, and here is an orbit starting from a fifth of capacity. 00:07:32.170 --> 00:07:36.313 I have drawn the first ten years in grey, because they are only the transient: 00:07:36.396 --> 00:07:38.603 the population finding its feet. 00:07:39.399 --> 00:07:41.177 And here is what it finds. 00:07:41.280 --> 00:07:42.697 Not a point at all. 00:07:42.894 --> 00:07:44.172 A square. 00:07:45.085 --> 00:07:47.489 Read that square as a timetable. 00:07:47.801 --> 00:07:52.829 Up to the curve and across to the line is one year. Then again is the next year. 00:07:53.026 --> 00:07:55.893 And the fourth move brings you back exactly where you started, 00:07:55.893 --> 00:07:58.820 so year three is a repeat of year one. 00:07:59.431 --> 00:08:03.017 So the population never settles. It alternates. 00:08:03.100 --> 00:08:08.139 A lean year, a crowded year, a lean year, a crowded year, without end. 00:08:08.452 --> 00:08:10.168 Nought point five one three, 00:08:10.251 --> 00:08:14.862 then nought point eight, then nought point five one three again. 00:08:15.565 --> 00:08:18.852 And notice how far apart those two values are. 00:08:19.048 --> 00:08:22.390 This is not the old fixed point with a small wobble on top. 00:08:22.473 --> 00:08:26.582 The population is swinging from half capacity to four fifths and back, 00:08:26.665 --> 00:08:29.765 every single year, and it will do that for ever. 00:08:30.678 --> 00:08:33.209 The two values are locked to each other. 00:08:33.406 --> 00:08:39.990 F of p is q, and f of q is p, so applying the rule twice brings each of them home. 00:08:40.786 --> 00:08:42.584 This is called a two cycle, 00:08:42.667 --> 00:08:47.242 and it appeared at exactly the value of r where the fixed point lost its grip. 00:08:47.694 --> 00:08:51.596 That is no coincidence, and there is a clean way to see why. 00:08:52.992 --> 00:08:55.510 If a population repeats every two years, 00:08:55.593 --> 00:08:59.355 then applying the rule twice must bring it back to itself. 00:08:59.552 --> 00:09:03.965 So let us look at the map that takes this year straight to the year after next. 00:09:04.390 --> 00:09:10.811 That is f of f of x, and here it is in red, sitting on top of the original hump. 00:09:10.915 --> 00:09:16.338 It has two humps of its own, and it meets the line y equals x in four places. 00:09:17.134 --> 00:09:19.596 Two of those crossings are old friends. 00:09:19.700 --> 00:09:23.985 Zero is fixed under one step, so it is certainly fixed under two. 00:09:24.088 --> 00:09:28.639 And so is x star, the fixed point we already had, which is still sitting there 00:09:28.639 --> 00:09:31.276 at nought point six eight seven five. 00:09:32.072 --> 00:09:36.346 The other two are new, and they are our lean year and our crowded year. 00:09:36.960 --> 00:09:40.351 Each is a perfectly good fixed point of this doubled map. 00:09:40.455 --> 00:09:43.623 And look at how the red curve meets the line at those two. 00:09:43.706 --> 00:09:45.390 It crosses gently. 00:09:45.494 --> 00:09:48.025 A shallow slope means attracting. 00:09:48.938 --> 00:09:51.795 Now here is the event at r equal to three. 00:09:51.991 --> 00:09:55.417 Below three the red curve looked like this: 00:09:55.521 --> 00:09:59.190 tucked up against the line, touching it only at x star. 00:09:59.387 --> 00:10:04.377 As r passed three it pushed straight through, and made two new crossings, 00:10:04.460 --> 00:10:05.959 one on each side. 00:10:06.755 --> 00:10:11.156 One solution became three, and the middle one was the one that failed. 00:10:11.353 --> 00:10:14.395 It is worth saying plainly what did not happen. 00:10:14.499 --> 00:10:18.830 Nothing was destroyed and no solution went missing. The fixed point is exactly 00:10:18.830 --> 00:10:22.696 where it always was, and no population will ever go to it again. 00:10:24.511 --> 00:10:29.084 So here is the pattern, and it is about to happen over and over. 00:10:29.167 --> 00:10:33.184 A point holds on while the slope there is shallower than one in size. 00:10:33.497 --> 00:10:37.399 When the slope reaches minus one, the point does not vanish. 00:10:37.595 --> 00:10:42.960 It stays, it turns repelling, and a cycle of twice the period is born around it. 00:10:43.872 --> 00:10:46.056 Which invites the obvious question. 00:10:46.369 --> 00:10:50.642 Our new two cycle has a slope of its own, measured on that red curve. 00:10:50.746 --> 00:10:53.927 What do you suppose happens when that one reaches minus one? 00:10:54.995 --> 00:10:56.598 Turn the knob again. 00:10:56.911 --> 00:11:01.634 At r equal to three point four four nine the two cycle fails, 00:11:01.717 --> 00:11:04.353 in exactly the way the fixed point failed. 00:11:04.550 --> 00:11:07.244 It is the same mechanism one level down. 00:11:08.041 --> 00:11:12.648 The two cycle is a fixed point of the doubled map, so it has a slope of its own, 00:11:12.731 --> 00:11:16.145 and that slope walks down through minus one as we turn. 00:11:16.342 --> 00:11:20.568 When it crosses, each of the two values sheds a pair, and we have four. 00:11:21.272 --> 00:11:23.676 Here is the knob at three point five. 00:11:23.873 --> 00:11:27.786 Grey for the transient again, and then the settled orbit in yellow. 00:11:28.582 --> 00:11:32.519 Four points, and the population visits them in a fixed order. 00:11:32.715 --> 00:11:36.348 Nought point three eight three, then nought point eight two seven, 00:11:36.431 --> 00:11:40.388 then nought point five zero one, then nought point eight seven five, 00:11:40.471 --> 00:11:42.271 and then back to the first. 00:11:43.323 --> 00:11:45.887 So the years no longer come in two kinds. 00:11:45.970 --> 00:11:49.860 They come in four, and the pattern closes only every fourth year. 00:11:50.057 --> 00:11:52.412 A good year, a terrible year, 00:11:52.495 --> 00:11:57.128 a middling year, a very good year, and only then a repeat. 00:11:58.041 --> 00:11:59.350 Push the knob a little further, 00:11:59.433 --> 00:12:03.591 to three point five five, and every one of those four splits again. 00:12:03.787 --> 00:12:06.377 The orbit now needs eight years to close. 00:12:07.081 --> 00:12:11.363 The picture is already becoming hard to read, and that is exactly the point. 00:12:11.446 --> 00:12:14.652 So look instead at how little I had to turn the knob. 00:12:14.964 --> 00:12:17.995 From three to three point four five was a long way. 00:12:18.192 --> 00:12:22.361 From three point four five to three point five four was a fifth of that. 00:12:22.813 --> 00:12:26.738 From there to three point five six, shorter still. 00:12:27.534 --> 00:12:31.923 The doublings are arriving faster and faster. Which means we should stop looking 00:12:31.923 --> 00:12:34.327 at pictures for a moment and look at the numbers. 00:12:35.840 --> 00:12:38.787 Here are the values of r at which the period doubles, 00:12:38.870 --> 00:12:43.329 the gap from each one to the next, and the ratio of one gap to the one after it. 00:12:44.033 --> 00:12:46.097 Two to four, then four to eight, 00:12:46.180 --> 00:12:49.560 then eight to sixteen, then sixteen to thirty two. 00:12:49.756 --> 00:12:51.159 The gaps collapse: 00:12:51.242 --> 00:12:55.423 nearly a half at the start, and under a thousandth four steps later. 00:12:56.475 --> 00:12:59.030 But the ratio does not collapse with them. 00:12:59.226 --> 00:13:01.061 Look down the last column. 00:13:01.258 --> 00:13:02.812 Four point seven five, 00:13:02.895 --> 00:13:07.598 four point six six, four point six seven, and then it simply stops moving. 00:13:08.739 --> 00:13:13.093 Carry the cascade on and that ratio converges on a definite number. 00:13:13.406 --> 00:13:17.424 Four point six six nine two zero one, and on it goes. 00:13:17.621 --> 00:13:21.709 It is called the Feigenbaum constant, and it is written delta. 00:13:22.505 --> 00:13:27.022 Two things about it deserve a pause, and the first is only arithmetic. 00:13:27.219 --> 00:13:30.758 Because the gaps shrink by a fixed factor every time, 00:13:30.841 --> 00:13:34.242 they behave like a geometric series, and a geometric series 00:13:34.242 --> 00:13:38.481 with ratio bigger than one in the denominator adds up to something finite. 00:13:39.277 --> 00:13:42.050 So the doublings do not go on for ever in r. 00:13:42.133 --> 00:13:46.114 They pile up at a finite value, three point five six nine nine and a bit, 00:13:46.197 --> 00:13:49.959 and by the time you arrive there the period has doubled infinitely often. 00:13:50.756 --> 00:13:52.812 The second thing is much stranger. 00:13:53.008 --> 00:13:56.318 That number has nothing whatever to do with our particular hump. 00:13:56.770 --> 00:14:02.122 Replace r x times one minus x with a sine arch, or with almost any smooth curve 00:14:02.122 --> 00:14:07.626 that has one rounded maximum, run the same cascade, and out comes the same constant. 00:14:08.841 --> 00:14:11.765 Mitchell Feigenbaum found that in nineteen seventy five, 00:14:11.848 --> 00:14:15.854 on a pocket calculator, and it took him three years to get it published. 00:14:16.166 --> 00:14:20.080 Which leaves the place where the doublings pile up, and everything past it. 00:14:21.203 --> 00:14:24.269 Let me put every one of those answers on one picture. 00:14:24.465 --> 00:14:26.533 Along the bottom, the knob. 00:14:26.729 --> 00:14:30.257 Up the side, the population, but only where it ends up: 00:14:30.340 --> 00:14:33.963 the values the orbit visits once the transient is over. 00:14:34.876 --> 00:14:40.159 For r below three there is one value, and it climbs gently as we turn. 00:14:40.472 --> 00:14:41.701 That is the fixed point, 00:14:41.784 --> 00:14:45.953 one minus one over r, drawn once for every setting of the knob. 00:14:46.749 --> 00:14:48.480 At three it splits. 00:14:48.793 --> 00:14:50.846 Above three there are two values, 00:14:50.929 --> 00:14:55.771 the lean year and the crowded year, and the gap between them widens as we go. 00:14:56.567 --> 00:15:01.189 At three point four four nine each of those splits again, and there are four. 00:15:01.501 --> 00:15:05.461 You can see the tuning fork shape repeating at a smaller size. 00:15:06.258 --> 00:15:09.590 And then the splittings come so fast that I cannot draw them. 00:15:09.787 --> 00:15:15.046 Eight, sixteen, thirty two, all crammed into this sliver, all of them piling up 00:15:15.046 --> 00:15:17.450 at three point five six nine nine. 00:15:18.247 --> 00:15:20.857 Past that line, something else entirely. 00:15:20.940 --> 00:15:25.121 The orbit no longer settles onto any finite list of values at all. 00:15:25.224 --> 00:15:28.580 It wanders over a whole band, and that band is where the rest 00:15:28.580 --> 00:15:30.101 of this lecture lives. 00:15:31.753 --> 00:15:35.434 So let us go inside it, and put the knob at three point nine. 00:15:35.538 --> 00:15:39.370 Same rule, same construction, same everything. 00:15:39.683 --> 00:15:41.739 Only r has changed. 00:15:42.117 --> 00:15:43.581 Twenty years of it. 00:15:45.184 --> 00:15:48.192 No square. No loop of any size. 00:15:48.389 --> 00:15:52.778 The staircase covers the box, and shows no sign at all of closing up. 00:15:52.975 --> 00:15:55.135 And nothing random went into this. 00:15:55.331 --> 00:15:58.197 Every one of those legs was computed from the one before it 00:15:58.280 --> 00:16:00.719 by multiplying three numbers together. 00:16:01.632 --> 00:16:04.454 Which is the thing people find hardest to accept. 00:16:04.557 --> 00:16:07.493 There is no dice roll anywhere in the logistic map. 00:16:07.576 --> 00:16:11.721 The unpredictability is manufactured out of arithmetic, and here is the property 00:16:11.721 --> 00:16:13.637 that does the manufacturing. 00:16:15.150 --> 00:16:19.400 Take two starting populations that differ in the third decimal place. 00:16:19.596 --> 00:16:23.753 Nought point four zero zero, and nought point four zero one. 00:16:23.950 --> 00:16:27.712 Run both for twenty years, and plot them against the year. 00:16:28.625 --> 00:16:32.060 For the first ten years or so you would not know there were two of them. 00:16:32.143 --> 00:16:34.440 They rise together and crash together. 00:16:34.523 --> 00:16:37.228 Then, somewhere around here, they part company. 00:16:37.607 --> 00:16:40.194 And after that they have nothing to say to each other. 00:16:40.277 --> 00:16:42.923 One is booming while the other has collapsed. 00:16:43.006 --> 00:16:44.411 The tiny difference we started 00:16:44.411 --> 00:16:47.453 with has been magnified until it is the whole picture. 00:16:48.366 --> 00:16:51.548 It grows by roughly half again every single year. 00:16:51.745 --> 00:16:56.479 So a difference of one part in a thousand takes about fifteen years to become total, 00:16:56.562 --> 00:16:59.953 and a difference of one part in a million takes about thirty. 00:17:00.150 --> 00:17:03.134 Improving your measurement buys you almost nothing. 00:17:03.676 --> 00:17:07.100 The rule is completely deterministic and completely useless for long term 00:17:07.100 --> 00:17:10.979 prediction, and those two facts sit together quite comfortably. 00:17:11.176 --> 00:17:14.169 But the diagram we drew has one more surprise in it, 00:17:14.252 --> 00:17:17.608 and it is the one that made people take all of this seriously. 00:17:18.730 --> 00:17:22.444 Here is the right hand end of that diagram, magnified. 00:17:22.527 --> 00:17:27.578 Everything from three point seven up to four, and at first glance it is all band. 00:17:28.375 --> 00:17:30.070 Except that it is not. 00:17:30.267 --> 00:17:33.565 Look here, a little past three point eight two. 00:17:33.762 --> 00:17:36.641 Right in the middle of the chaos there is a clean gap. 00:17:37.182 --> 00:17:40.237 And running through the gap, three sharp curves. 00:17:40.433 --> 00:17:45.298 That is a window: a stretch of r where the wandering stops dead and the population 00:17:45.298 --> 00:17:49.467 goes back to a strict repeating schedule, this time of length three. 00:17:50.380 --> 00:17:52.505 Which is worth a moment of disbelief. 00:17:52.702 --> 00:17:55.999 Either side of this narrow strip the very same rule produces something 00:17:55.999 --> 00:17:58.194 with no pattern in it whatsoever. 00:17:58.391 --> 00:18:00.806 Let us go and look at the orbit itself. 00:18:02.110 --> 00:18:06.741 Knob at three point eight three, and the same construction we have used all along. 00:18:06.824 --> 00:18:09.983 Grey for the transient, yellow for what it settles on. 00:18:10.524 --> 00:18:15.294 Three points. Nought point one five six, then nought point five zero five, 00:18:15.377 --> 00:18:19.136 then nought point nine five seven, and straight back to the first. 00:18:19.219 --> 00:18:23.284 A crash, a recovery, a boom, and round again, for ever. 00:18:23.709 --> 00:18:27.877 That is as orderly as anything we saw at r equal to two point six. 00:18:28.074 --> 00:18:29.883 Now turn the knob by four hundredths, 00:18:29.966 --> 00:18:33.752 to three point eight seven, and watch the schedule evaporate. 00:18:34.131 --> 00:18:36.117 There it is, wandering again. 00:18:36.221 --> 00:18:40.610 The window really is narrow, and it really is surrounded on both sides by chaos. 00:18:42.007 --> 00:18:44.050 Now go back to those three curves and follow them 00:18:44.050 --> 00:18:46.048 to the right hand end of the window. 00:18:46.844 --> 00:18:48.259 Each of them splits into two, 00:18:48.342 --> 00:18:52.102 and then into four, and it accumulates and gives way to chaos again, 00:18:52.185 --> 00:18:56.934 with the very same ratio four point six six nine governing the splittings. 00:18:57.731 --> 00:19:00.750 So the window is not merely an island of order. 00:19:00.854 --> 00:19:05.000 It contains a small copy of the entire picture we spent this lecture building. 00:19:05.452 --> 00:19:08.376 And magnify that copy and you find windows inside it, 00:19:08.459 --> 00:19:10.747 each holding a smaller copy again. 00:19:11.230 --> 00:19:16.060 The structure goes all the way down. There is no scale at which it becomes simple. 00:19:17.875 --> 00:19:21.173 So here is what one innocent equation turned out to hold. 00:19:21.276 --> 00:19:24.467 For a small growth rate, a single settled population, 00:19:24.550 --> 00:19:28.696 and a staircase that walks to it because the curve crosses the line gently. 00:19:29.609 --> 00:19:33.974 As the rate grows, that crossing steepens until it can no longer hold anything, 00:19:33.974 --> 00:19:35.900 and a two cycle takes over. 00:19:35.983 --> 00:19:41.241 Then the two cycle fails the same way, and its successor after it, faster and faster, 00:19:41.324 --> 00:19:45.516 with gaps shrinking by a factor that has nothing to do with our equation. 00:19:46.730 --> 00:19:48.459 Past the point where those pile up, 00:19:48.542 --> 00:19:52.710 orbits that never repeat and that forget where they started. And inside the chaos, 00:19:52.710 --> 00:19:57.494 windows of perfect order, each containing the whole story again in miniature. 00:19:58.197 --> 00:20:02.689 None of that was put in. All of it was sitting inside r x times one minus x, 00:20:02.772 --> 00:20:04.874 waiting for somebody to turn the knob.