The goal of this segment is to explore what happens to the attractors in the dynamics of the logistic map, as the parameter R changes. By way of review, here's what happens to the logistic map if you started at the initial condition 0.2 and the parameter value 2 and you plot 50 iterates. Very rapidly the trajectory reaches a fixed point. And I showed you, changing the initial condition, doesn't really change the position of the fixed point, it just changes the transient from the initial condition to the fixed point. All the initial conditions that I just explored are in the basin of attraction of that attractor. They are all raindrops running to the same ocean. Now I'm gonna do is raise the R parameter slowly and show you what happens. I'm going back where we started with the initial condition 0.2 and the R parameter value of 2 , and raise R to 2.2, then 2.4, 2.6, you start seeing that oscillatory convergence that I mentioned the last segment, 2.8, still converging in an oscillatory fashion but the transient has got longer. It's taken longer to converge to that fixed point. 2.9 the transient is getting much longer. This oscillatory convergence, by the way, played a role in the quiz problem from the previous segment. Aha! Something different. Looks like, maybe the fixed point is no longer there. But it looks like, this might be getting closer, so let's try a few more iterates. Mmmm. Still looks like it might be converges. Let's raise R a little bit more and see what happens. Still looks like it might be converging to a fixed point, but again, this transient is really long. Let's go a little bit higher and see what happens. Now, it looks, for sure, like that fixed point is gone and something else is happening. Remember that I said that were several types of attractors, this is another one, it's called a periodic orbit or a limit cycle. These are synonyms. Here the period is two. You can imagine this, as 1 year with lots of rabbits in my backyard, and the next year with lots of foxes, and then the next year with lots of rabbits, and then the next year with lots of foxes. Which actually happens in real populations, just like the fixed point we explored for lower R values, is an attracting fixed point. This is an attracting periodic orbit. You can see that, right here, there is a convergence to the periodic orbit. If we started from a different parameter value, we would reach the same periodic orbit. Just from a different direction. By the way, an important point here, many plotting tools, like to draw lines between points. That is inappropriate when you 're working with maps. A map is a discrete time system. Time makes no sense, in between iterates. You can even think about, what it might be x of 0.5, it doesn't exist. Do not connect your dots, when you're plotting iterates of a map. Ok. What do you think will happen if we raise R further. Aha! At this value, it looks like the two cycle is no longer in existence. The fixed point is certainly gone and we've converged to something called a four-cycle. This is also a limit cycle or periodic orbit. The period is four. Here's what those two periodic orbits might sound like, if you hooked the vertical axis, of the logistic map to your piano. These changes from my fixed point to a two cycle, and from a two cycle to a four cycle, as we change the R parameter, are called bifurcations. That English word suggests forking in two, but it's mathematics use is more general. It means that there's been a change in the topology of the attractor. We 'll get back to the topology in attractors later. For now, suffice it to say that a bifurcation is a qualitative change in the attractor, not just that a fixed point moves, but that it vanishes or that the period of a periodic orbit changes. Now I sipped a word in there, parameter. R is a bifurcation parameter in the logistic map in effects that dynamics in a fundamental way, causing those qualitative changes. Let's keep increasing the value of that parameter and see what we see. There is no pattern that jumps out you here. Things don't appear to converge to anything, not to a fixed point not to a periodic orbit. Let's do a longer plot. Again, doesn't look like there is any convergence. As you do more and more iterates though, you start to see some patterns, especially for example right here, looks like something is repeating for a while. Maybe it's almost a periodic orbit, but then it's not. The patterns that you're starting to see here, are a chaotic attractor. And we gonna spend a lot of time talking about features and properties of chaotic attractors. Sometimes, by the way, they're called strange attractors. Recall the experiments that we did with the other attractors, the fixed point and the periodic orbits and varying initial conditions, when we did that the trajectory always converge to the same attractor. It does that here too, but it can be pretty hard to see in a time domain plot, like this. The issue is that the points are tracing out the same subset of states, but in a different order. It's kind of like dropping a woodchip in to an eddy, like I talked about the very first segment. If you drop a woodchip, in different parts of an eddy, the woodchip would trace out the same structure but in a different order. The structure of a chaotic attractor is a very deep and important part of non linear dynamics. Another important feature is sensitive dependence on initial conditions. What you may have heard called the butterfly effect. Again I explained this in the very first segment. Here is a slide that summarizes some of the important vocabulary and concepts that I used in this first unit of this course. You should make sure that you understand what each of these words means and how it manifests in the logistic map dynamics. Now ,the slides that I used in this lectures are lecture aids , they are not lecture notes. And that maybe a bit of a challenge, particularly in view of the fact that there is no a textbook for this course. Now, as I explained in the previous unit, this course pulls together stuff from lectures I've heard, my own thoughts, different textbooks, papers I've read so there is no single textbook, no single source, that's a map out for this course like there are some of the other complexity explorer courses. These videos are your main source of information in this course. They're designed to be short, self- contained, descriptions of individual topics. If you want to dig more deeply, or there's something that you just not getting, take a look at the links on the Supplementary Materials page, right here. This will be flashed out a lot more as the course goes on, and for each unit, and each segment of each unit, I will post links to things that you might need to understand that unit, to do the homework, and also I will post links that point to what I would suggest you look at, if there is some background you're missing, or what's the next thing to read if you wanna know more about this.