In the last segment, I introduced the notion of a return map: plots of x_n+1 vs x_n. These bring out the correlations between successive points in a trajectory. Now, a hard thing about this field when you're first getting into it, we use lots of different representations to understand & explain dynamical systems, and we often switch between these representations quickly and without explanation. I'm going to try to narrate when I switch between representations and tell you what each one is good for. So what I'm gonna do right now is try to help you make the connection between the time-domain plots, which are x_n vs n, and return maps, which are x_n+1 vs x_n, and I'm going to do that by showing you lots of different side-by-side pictures in those two domains, like I started to at the end of the previous segment. The app that you used in the last unit to explore the logistic map makes that very easy, and it even puts in all those lines that I was drawing in different colors to show where the iterates are, and it makes them alot straighter than I did. The right-hand plot in this app shows what happens if you iterate the logistic map from this initial condition, using this R-parameter value, for this many iterates. And you can see from this plot that those iterates reach a fixed point. You probably recall, if we raise the R- parameter, the fixed point will move up. Here, this is R=2; that's R=2.1. You also probably recall, if we increase R further, say to 2.2, you start seeing a little bit of overshoot. Let's increase R a little bit further to make that even more pronounced. There's R=2.6 and you can really see the oscillatory convergence on the right-hand side. Now look in here: this is a cobweb plot, this is x_n+1 vs x_n, like a drew a bunch of last time, and you can start seeing that this convergence is oscillatory. Let's make it bigger. There's R=2.8, and you're starting to see the squared-off, inwards spiral that I was drawing in the last segment. You also recall, I hope, that when we raised R further, there was a bifurcation causing this fixed point to go away (actually it just became unstable, it didn't go away), and a 2-cycle to appear. Here's what that looks like on the return map. As we raise R further, the 2 cycle gets bigger and bigger. Here's 3.3, here's 3.4, and at a certain point, we get a 4-cycle. Now this gets a little bit hard to see what the asymptotic behavior is, but you can use this "Remove Iterate" button right here to remove some of the iterates, starting from the beginning. So what I'm doing here is removing the transient. And if I remove enough of the transient off the front of the trajectory, all you see is the attractor, and you can see very clearly that it's a 4-cycle, and you can see very clearly from the left-hand plot what it is about the geometry of the yellow curve and the blue line that makes that happen. For R=3.65, the iterates are bouncing all over the range of the function. This is a chaotic attractor, and you can also see from this picture on the left why this is sometimes called a "cobweb diagram." At R=3.8, the trajectory is fully chaotic, but look what happens at 3.83. A 3-cycle. So there's been another bifurcation between R=3.8 and R=3.83, from a chaotic orbit to a periodic orbit. So how can we capture all that richness of behavior - the different attractors and the bifurcations between them - on one plot? We can use another useful representation called the "bifurcation diagram", and it is plotted on the axes of x_n vs R. Now how to think about this? Each slice of this plot is one of those time-domain plots that we built before, viewed from the side, like this. That's an attempt to draw a human eye looking down the side of that plot. Now with the bifurcation diagram, we're interested in the attractor, the asymptotic behavior. So when you're doing this, you actually remove the transient from the front of the trajectory as well. Now think if you were looking at this plot from the side, that's your eye: what are you gonna see? You're gonna see a bunch of dots in one place, this high. Remember, you're not gonna see these guys, because we threw them out. So what you're gonna see at the R=2 mark of that bifurcation diagram is all those dots right on top of each other. This slice through the bifurcation diagram corresponds to this plot, and that slice of the bifurcation diagram corresponds to this plot. So what is this eye gonna see? Again, we're throwing out the transient. Viewed from the side, that right-hand plot is gonna look like 2 dots ("dot-dot") and then "dot-dot", and so on & so forth. So I've drawn the picture on either side of the bifurcation that caused the fixed point to go away, and the 2- cycle to be born. What if I look in between there more carefully? This is what I'd see. Remember, the fixed point moves up, then bifurcates at this point into a 2-cycle, and that 2- cycle gets wider and wider. And if you do that really carefully with a computer, rather than a tablet and a stylus, you see something really beautiful. And next time, we're going to dig into that structure in more detail.