In this last segment about maps, I'm going to talk about the universality of chaos, beginning with the Feigenbaum number. Recall this picture of the bifurcation diagram of the logistic map. It's pretty clear that the bifurcations in this cascade aren't evenly spaced, rather, they are getting smaller as r increases and as the period increases with r. If you look closely, you'll see that the width and height of those little pitchforks are decreasing at a constant ratio; that is, that the ratio of the width delta 3 to the width delta 2, is about the same size as the ratio of the width delta 2 to delta 1. In the high period limit, that ratio approaches the value of about 4.66. Mitchell Feigenbaum is responsible for the proof about that ratio, and that number bears his name. Now, you'll notice the limit in this equation, what that means is that delta 1 over delta 2 is close to 4.66, delta 2 over delta 3 is closer to 4.66, and so on and so forth. And in the limit of very small bifurcations, the number actually limits to the Feigenbaum number. Note that the Feigenbaum number is about the widths of the pitchforks. There is a similar result, but with a different ratio for the heights. Now, what's really amazing about this result is that it holds for any 1-D function with a quadratic maximum; that is, it goes well beyond the logistic map. It's also true for the sine map and it's also true for any other map that looks like a parabola near its maximum. In making the point about how amazing this is, Steve Strogatz says the Feigenbaum number is a new physical constant as fundamental to 1-D maps as pi is to circles. This is one manifestation of the universality of chaos. Systems as diverse as hurricanes, orbiting moons, pendulums, and the human heart all act the same in the throws of chaos. This is a large part of what fascinates me and I think many other people about this field; this universality. And the Feigenbaum number is your first taste of that. Don't take this number too far. Note: 1-D. Not 2-D or more. Also, maps. Not flows. There is some evidence that this result holds for some higher dimensions and for some flows, but the proofs don't extend to them, so it may well be true, but we don't know for sure yet. Speaking of higher dimensional maps, here is an example of a 2-dimensional map. It's called Smale's Horseshoe. Smale's Horseshoe operates not on the unit interval like the logistic map, but on the unit square. And here's the action of the map. The first thing it does is stretch out the square to be a long, thin rectangle with the same volume. The next action of the map is to take that rectangle and fold it and then chop off the extra bits. So it's a map of the unit square onto itself and it has the interesting property of taking points that are far apart and bringing them close together. At the same time, it takes some points that are close together and maps them very far apart. The stretching and folding that creates this is a paradigm in chaos. It is the cause of sensitive dependence on initial conditions. It happens in the logistic map too because that quadratic transformation kneads the unit interval. And when I say 'kneads' I don't mean that kind of 'needs', I mean this kind of 'kneads', like bread. And speaking of kneading and bread, here is an interesting 3-dimensional map. This is a lump of bread dough with a little bit of dye put on it. As in the case of Smale's Horseshoe, the kneading of the bread causes points that are close to end up far apart and far away points to end up close together. Again, that's the source of sensitive dependence on initial conditions. It's also why bakers knead dough. This progression of images shows a ball of dough, again with a tiny bit of red dye in it, after a succession of applications of the kneading map. The kneading map is taking that ball of dough, squishing it flat, folding it over and gathering it into a ball again. As you can see, working forward from the top left image, which is the initial image before any iterations of the kneading map were applied, to the second image, the top middle image which is after one application of the kneading map, the dye gets distributed pretty quickly through the bulk of the dough. Of course, thinking of kneading bread as a map, a single discrete operation, isn't really right. The action is proceeding continuously in space and time as our hands move the dough around continuously. We can only think of this as a map, a discrete time operation, if we kind of look away during the kneading operation. If we only look at the dough when it's back in a ball after each operation, that's kind of like shining a strobe light at a physical system. This is the driven damped pendulum that I built a while ago. At this particular force and frequency, the dynamics are chaotic - never quite repeating, sensitively dependent on initial conditions, and yet, patterned. If I turn off the lights and shine a strobe light at the pendulum, you only observe it every second or so. You have no idea what it did in between. You're only information is at those discrete intervals. And that's the difference between a map and a flow - discrete time vs. continuous time. We've gone through a lot of the important concepts in nonlinear dynamics in the context of maps. In the next unit of this course, we're going to circle around through a lot of those same concepts in the context of flows. So you can think of this like looking at the full dynamics of the kneading operation as you squish the dough, fold it, and gather it, rather than just looking at it every time it's a ball and then looking away while you do this, and looking back.