So let's summarize Unit 5. This was the second of two units on bifurcations and bifurcation diagrams. This unit focus exclusively on the logistic map, the iterated logistic function. I began by reminding us of the idea of final state diagrams. So for the logistic equation, for a given r value, you might make a final state diagram as follows. Iterate the function for a hundred times, then iterate another two hundred times and plot those iterates on a unit interval, and the result is a final state diagram and this is very similar to the phase line for differential equations. It shows the equilibrium or long-term behavior but it doesn't show arrows for the directions the orbits move, and that's because for the logistic equation and iterated functions in general, the orbit can bounce around and it doesn't move smoothly through all continuous values. So here are some examples, and we did a bunch early on in this unit. Here's a value of r which has period two behavior, it hits that period two behavior very quickly. The final state diagram had two dots, these two final values, so I can iterate out for a hundred and then iterate the next two hundred and it would just be bouncing back and forth between these two. It's a little bit more interesting for cases where we have aperiodic behavior. Here is a different r value that has an aperiodic orbit, and here the final state diagram would consist of many many points between these two extreme values. So it looks like the orbits never get much below 0.18 or above 0.96, but they can take on all values in between and so if I plotted two hundred orbits, two hundred iterates, I would see two hundred dots and they would essentially fill up this line. So those are final state diagrams and then to make a bifurcation diagram, we just glue all of those final state diagrams together. So each vertical slice of a bifurcation diagram is a single final state diagram. Another way I'd like to think about this is that the bifurcation diagram is like a dictionary that lets you look up the behavior for different r values. So suppose someone says, "what does a logistic equation do for r=3.6?" Say, well, I don't know but I can look that up in the bifurcation diagram. It's a dictionary or a compendium of all possible behaviors for all different r values. So I look over here at 3.6, and I would go up, and imagine taking a really thin slice out of this, and that would be my final state diagram for that value. So the bifurcation diagram summarizes a great deal of information. And we noticed that there's some interesting and intriguing, I hope, patterns in this bifurcation diagram. There are lots of transitions from periodic behavior to chaotic behavior, periodic to chaotic, periodic to chaotic, and so on. So there's a lot going on in this bifurcation diagram. To get a closer look, I mentioned or introduced to you to a web program that lets you zoom in on the bifurcation diagram many many many times, and explore, see what you see, see what it looks like. So we zoomed in and explored, and along the way as we were doing that, we needed to adjust some times the number of orbits that were skipped and we need to do that if there's long transient behavior, takes a long time to reach those final states. And sometimes we need to increase the number of orbits plotted. If we zoomed in a lot in the vertical direction, so that we're only seeing a small region, we lose resolution, so by plotting more points we can gain resolution. The plotting of points on the screen takes a long time, so don't let this be too large, unless you're zoomed in a fair amount. So let me close this summary just by reminding you that the bifurcation diagram, this amazingly complex figure all comes from this simple equation: f(x) = rx(1-x). And this is just a parabola. A parabola is almost as simple a function as you could come up with, it's not a complicated topic from algebra, but when iterated, there's an almost infinite amount of, if not complexity, then certainly an infinite amount of structure or regularity in these fractal patterns that one sees. So the function is simple and the iteration process is also very simple. The code that I used to make the bifurcation diagram is a short piece of code, in Python, the one I used to make the figure a couple of slides ago, that code is just forty-seven lines long and that includes comments on all the plotting commands as well. The basic algorithm, the basic recipe for iteration is a very very simple one. So iteration, this very simple and infinitely repetitive process, and a very simple function like a parabola, can produce this remarkably complex structure that we see in the bifurcation diagram. It's pretty amazing to me that such a complex structure such as the bifurcation diagram can arise from such a simple equation like the logistic equation. If you have experienced programming, I definitely suggest that you try coding up a bifurcation diagram program. it's fun to do, it's not a hard programming task, and it's very satisfying, and I think sort of amazing, to see the bifurcation diagram emerge as a result of the program you wrote. When I write such a program like I did for this unit, when I see the bifurcation diagram emerge I sometimes sort of wonder, have this question, where did that come from, who did that? Well, of course, I know I did it, I wrote the program, and maybe the computer did it, it's making the picture. But where does the infinite structure of the bifurcation diagram come from? Where does it arise? For that matter where does the randomness, unpredictability and butterfly effect arise out of this simple parabolic equation. I think the answer in both of these cases is that it arises from iteration, that the act of iterating, itself a very simple process and infinitely repetitious, the act of iterating a simple equation gives rise to properties or features that weren't present in the equation in the first place. We've seen randomness and unpredictability arise, and we've seen a really rich complexity arise from a very simple equation that gets iterated. This in turn calls attention to one of the things that I think is different and maybe noteworthy about the study of dynamical systems. In physics, and probably elsewhere in the physical sciences, sometimes when one has the equation in hand, the story is sort of over, at least as far as the science is concerned. That we try to understand a process, and once we have an equation for it, the process is understood, story over. One can then use that equation to make predictions build bridges, to build computers. But the understanding is encoded in the equation itself. I think the same isn't true for the iterated logistic equation. If you have the equation, it's just the beginning. The equation itself, just looking at the equation, doesn't make evident that there's a butterfly effect lurking, or that there's this infinite complexity and structure in the bifurcation diagram. Those are features of the system that only emerge via the act of iteration. And, very often, the only way to see what the system will do is to see what the system will do. And what I mean by that is one has to just start iterating and see what happens. There's not always an analytic method, a pencil and paper, a deductive way to figure out if an iterated function will be chaotic or not. But if you iterate it on a computer, you can see very quickly what's going on. So this is a feature of dynamical systems that I think is somewhat different than how equations are used elsewhere. The equation maybe isn't the end of the story, but it's the beginning. One has to iterate it and take a bit of a more experimental approach, to see what properties emerge from the act of iteration. And often the properties that emerge are properties that weren't present in the original equation. So, in any event, this brings us to the end of Unit 5. We've looked at the bifurcation diagram for the logistic equation. In the next unit, we'll focus in on this feature of repeated period doublings, those pitchforks of sideways U's that come up again and again and again in the bifurcation diagram. And amazingly, we'll see that this isn't just some mathematical or geometric curiosity, but it actually has some really important statements to make about actual physical properties of systems in the material world. See you then.