Since this is the first homework solution in this course I want to take a little bit of time to explain the symbols. The green circles. Everyone should do, and there will be solutions in these videos everyone should try all of the problems with the blue squares. Those problems will be discussed on the forum, the black diamonds are for people who know a little bit more, who won a challenge. These are completely optional and again will be discussed on the forum, the first problem here will call upon the logistic map program that you were in unit 1.2. Here's mine that led version, which I showed you in the solution to that quit the task in this problem was to generate a trajectory from Mexico also point to 200 points lower with articles 2 and then to generate a trajectory, using the same are in the same number of points, but a slightly different initial condition the next task was to plot the absolute value of the difference between those trajectories versus and here's the difference here I'm generating the end is setting up a figure doing the plot. There is the figure. Let's keep that in mind, then we were supposed to repeat this for article 3.4 and our equal 3.7-2. So I'll do that quickly here is that figure asked ask workers to do the same thing for equals 3.7. 2, there is the 3rd party the next task was to compare these parts to the ones in the homework. And then answer the following questions. So which of these clots can corresponds to Oracle's too. That was the one that fell off like a stone. That was. See which other parts and figure one corresponds to Oracle's 3.4 that was the one but isolated as it was falling. That was it. And then a 3rd question which of the plots and figure one corresponds to the plot that we generate with our equals 3.7. 2, that was the chaotic one in B now. The point of this problem goes back to my example of an Eddie in a stream as a metaphor for chaos and the notion of dropping 2 wood chips and that any very close together and watching how fast they separate if the air tractor is a fixed point and you drop those 2 wood chips those 2 initial conditions in the basement of attraction of that a tractor. Both of those initial conditions would be converging to the fix pointed tractor. So the absolute value of the difference between then, which is the distance between them would converge to zero if their tractor or a periodic orbit. Then the 2 initial conditions would rattle in from 2 different directions. So the distance between them might isolate but eventually they would end up on the same periodic orbit and so that distance would converge to a fixed value not necessarily zero because they might be on different points on the periodic orbit. Kind of like 2 cars going around a racetrack, they might be going slower in the corners and faster on the street stretches so they wouldn't just stay directly opposite each other all the time, although as you can see in this case the difference does converged to zero if their tractor is chaotic if the attractiveness chaotic the 2 initial conditions will move chaotic they through the air tractor and the distance between them, we'll also changed periodically. That's a problem, you need to generate 2 trajectories 500 points along with slightly different national conditions and look at the last number in each of those trajectories there at the 2 trajectories looks pretty small. To me the choice isn't in the problem worth 0.2-5 1 and none of the above 10 to the minus 17th. It's pretty darn close to zero. So I would select the first answer in part of the idea was to repeat that for article 3.4. There's the calculation. And again, it looks like the answer is zero. Park Ji of this problem required a little bit more programing. Here's a 5,000 point trajectory at our equals 3.7-2. Here's a 5,000 point trajectory from a slightly different initial condition at the same our value. Here's a vector containing an element was difference of those 2 trajectories and with the absolute value taken and here's the average of the values in that vector Nazi which of the answers that corresponds to looks like that one the next problem was about extending that calculation. After 500,000. The answer doesn't change a whole lot, but it is a little different, it's 2.4-4 one, the fact that that difference doesn't change very much between 5,000 and 500,000 points is pretty amazing what that says is that as the initial conditions move around the chaotic a tractor. The average distance between them is pretty much the same over 5,000 points were over 500,000 points that's a consequence of the combination of sensitive dependence on initial conditions and the bounded patterned structured nature of a chaotic attractive in problem too. We went back to using the app, the task was to generate a 50.0 trajectory from ex not equals 0.2 using this very carefully chosen our primary value let's restart the simulation this doesn't look to me like anything periodic or anything that's a fixed point, I would guess this is chaotic, but there's something very interesting going on here. Look at the sort of peace and then the sort of peace. There's some patterns going on there, but they're not quite the same. So I would guess that this is a chaotic orbit in Part B of this problem. We're gonna watch for a little bit longer and see what happens to see if it's really chaotic or if it's gonna settle down to something with this out. You can watch an ongoing process of iterations by clicking Start animation. I still see that little Arrowhead thing coming through and then things look like they've go chaotic in between the work it re-occurrence of that pattern makes me suspect were nearby objectification point. Now things are gone, periodic and I'm gonna stop the animation, so we can count and see what kind of period. It is, looks like it repeats every 123456789. That's a 9 cycle,