1 99:59:59,999 --> 99:59:59,999 Since this is the first homework solution in this course I want to 2 99:59:59,999 --> 99:59:59,999 take a little bit of time to explain the symbols. 3 99:59:59,999 --> 99:59:59,999 The green circles. 4 99:59:59,999 --> 99:59:59,999 Everyone should do, 5 99:59:59,999 --> 99:59:59,999 and there will be solutions in these videos everyone should try 6 99:59:59,999 --> 99:59:59,999 all of the problems with the blue squares. 7 99:59:59,999 --> 99:59:59,999 Those problems will be discussed on the forum, 8 99:59:59,999 --> 99:59:59,999 the black diamonds are for people who know a little bit more, 9 99:59:59,999 --> 99:59:59,999 who won a challenge. 10 99:59:59,999 --> 99:59:59,999 These are completely optional and again will be discussed on the 11 99:59:59,999 --> 99:59:59,999 forum, 12 99:59:59,999 --> 99:59:59,999 the first problem here will call upon the logistic map program 13 99:59:59,999 --> 99:59:59,999 that you were in unit 1.2. 14 99:59:59,999 --> 99:59:59,999 Here's mine that led version, 15 99:59:59,999 --> 99:59:59,999 which I showed you in the solution to that quit the task in this 16 99:59:59,999 --> 99:59:59,999 problem was to generate a trajectory from Mexico also point to 200 17 99:59:59,999 --> 99:59:59,999 points lower with articles 2 and then to generate a trajectory, 18 99:59:59,999 --> 99:59:59,999 using the same are in the same number of points, 19 99:59:59,999 --> 99:59:59,999 but a slightly different initial condition the next task was to 20 99:59:59,999 --> 99:59:59,999 plot the absolute value of the difference between those 21 99:59:59,999 --> 99:59:59,999 trajectories versus and here's the difference here I'm generating 22 99:59:59,999 --> 99:59:59,999 the end is setting up a figure doing the plot. 23 99:59:59,999 --> 99:59:59,999 There is the figure. 24 99:59:59,999 --> 99:59:59,999 Let's keep that in mind, 25 99:59:59,999 --> 99:59:59,999 then we were supposed to repeat this for article 3.4 and our equal 26 99:59:59,999 --> 99:59:59,999 3.7-2. 27 99:59:59,999 --> 99:59:59,999 So I'll do that quickly here is that figure asked ask workers to 28 99:59:59,999 --> 99:59:59,999 do the same thing for equals 3.7. 29 99:59:59,999 --> 99:59:59,999 2, 30 99:59:59,999 --> 99:59:59,999 there is the 3rd party the next task was to compare these parts to 31 99:59:59,999 --> 99:59:59,999 the ones in the homework. 32 99:59:59,999 --> 99:59:59,999 And then answer the following questions. 33 99:59:59,999 --> 99:59:59,999 So which of these clots can corresponds to Oracle's too. 34 99:59:59,999 --> 99:59:59,999 That was the one that fell off like a stone. 35 99:59:59,999 --> 99:59:59,999 That was. 36 99:59:59,999 --> 99:59:59,999 See which other parts and figure one corresponds to Oracle's 3.4 37 99:59:59,999 --> 99:59:59,999 that was the one but isolated as it was falling. 38 99:59:59,999 --> 99:59:59,999 That was it. 39 99:59:59,999 --> 99:59:59,999 And then a 3rd question which of the plots and figure one 40 99:59:59,999 --> 99:59:59,999 corresponds to the plot that we generate with our equals 3.7. 41 99:59:59,999 --> 99:59:59,999 2, 42 99:59:59,999 --> 99:59:59,999 that was the chaotic one in B now. 43 99:59:59,999 --> 99:59:59,999 The point of this problem goes back to my example of an Eddie in a 44 99:59:59,999 --> 99:59:59,999 stream as a metaphor for chaos and the notion of dropping 2 wood 45 99:59:59,999 --> 99:59:59,999 chips and that any very close together and watching how fast they 46 99:59:59,999 --> 99:59:59,999 separate if the air tractor is a fixed point and you drop those 2 47 99:59:59,999 --> 99:59:59,999 wood chips those 2 initial conditions in the basement of 48 99:59:59,999 --> 99:59:59,999 attraction of that a tractor. 49 99:59:59,999 --> 99:59:59,999 Both of those initial conditions would be converging to the fix 50 99:59:59,999 --> 99:59:59,999 pointed tractor. 51 99:59:59,999 --> 99:59:59,999 So the absolute value of the difference between then, 52 99:59:59,999 --> 99:59:59,999 which is the distance between them would converge to zero if their 53 99:59:59,999 --> 99:59:59,999 tractor or a periodic orbit. 54 99:59:59,999 --> 99:59:59,999 Then the 2 initial conditions would rattle in from 2 different 55 99:59:59,999 --> 99:59:59,999 directions. 56 99:59:59,999 --> 99:59:59,999 So the distance between them might isolate but eventually they 57 99:59:59,999 --> 99:59:59,999 would end up on the same periodic orbit and so that distance would 58 99:59:59,999 --> 99:59:59,999 converge to a fixed value not necessarily zero because they might 59 99:59:59,999 --> 99:59:59,999 be on different points on the periodic orbit. 60 99:59:59,999 --> 99:59:59,999 Kind of like 2 cars going around a racetrack, 61 99:59:59,999 --> 99:59:59,999 they might be going slower in the corners and faster on the street 62 99:59:59,999 --> 99:59:59,999 stretches so they wouldn't just stay directly opposite each other 63 99:59:59,999 --> 99:59:59,999 all the time, 64 99:59:59,999 --> 99:59:59,999 although as you can see in this case the difference does converged 65 99:59:59,999 --> 99:59:59,999 to zero if their tractor is chaotic if the attractiveness chaotic 66 99:59:59,999 --> 99:59:59,999 the 2 initial conditions will move chaotic they through the air 67 99:59:59,999 --> 99:59:59,999 tractor and the distance between them, 68 99:59:59,999 --> 99:59:59,999 we'll also changed periodically. 69 99:59:59,999 --> 99:59:59,999 That's a problem, 70 99:59:59,999 --> 99:59:59,999 you need to generate 2 trajectories 500 points along with slightly 71 99:59:59,999 --> 99:59:59,999 different national conditions and look at the last number in each 72 99:59:59,999 --> 99:59:59,999 of those trajectories there at the 2 trajectories looks pretty 73 99:59:59,999 --> 99:59:59,999 small. 74 99:59:59,999 --> 99:59:59,999 To me the choice isn't in the problem worth 0.2-5 1 and none of 75 99:59:59,999 --> 99:59:59,999 the above 10 to the minus 17th. 76 99:59:59,999 --> 99:59:59,999 It's pretty darn close to zero. 77 99:59:59,999 --> 99:59:59,999 So I would select the first answer in part of the idea was to 78 99:59:59,999 --> 99:59:59,999 repeat that for article 3.4. 79 99:59:59,999 --> 99:59:59,999 There's the calculation. 80 99:59:59,999 --> 99:59:59,999 And again, 81 99:59:59,999 --> 99:59:59,999 it looks like the answer is zero. 82 99:59:59,999 --> 99:59:59,999 Park Ji of this problem required a little bit more programing. 83 99:59:59,999 --> 99:59:59,999 Here's a 5,000 point trajectory at our equals 3.7-2. 84 99:59:59,999 --> 99:59:59,999 Here's a 5,000 point trajectory from a slightly different initial 85 99:59:59,999 --> 99:59:59,999 condition at the same our value. 86 99:59:59,999 --> 99:59:59,999 Here's a vector containing an element was difference of those 2 87 99:59:59,999 --> 99:59:59,999 trajectories and with the absolute value taken and here's the 88 99:59:59,999 --> 99:59:59,999 average of the values in that vector Nazi which of the answers 89 99:59:59,999 --> 99:59:59,999 that corresponds to looks like that one the next problem was about 90 99:59:59,999 --> 99:59:59,999 extending that calculation. 91 99:59:59,999 --> 99:59:59,999 After 500,000. 92 99:59:59,999 --> 99:59:59,999 The answer doesn't change a whole lot, 93 99:59:59,999 --> 99:59:59,999 but it is a little different, 94 99:59:59,999 --> 99:59:59,999 it's 2.4-4 one, 95 99:59:59,999 --> 99:59:59,999 the fact that that difference doesn't change very much between 96 99:59:59,999 --> 99:59:59,999 5,000 and 500,000 points is pretty amazing what that says is that 97 99:59:59,999 --> 99:59:59,999 as the initial conditions move around the chaotic a tractor. 98 99:59:59,999 --> 99:59:59,999 The average distance between them is pretty much the same over 99 99:59:59,999 --> 99:59:59,999 5,000 points were over 500,000 points that's a consequence of the 100 99:59:59,999 --> 99:59:59,999 combination of sensitive dependence on initial conditions and the 101 99:59:59,999 --> 99:59:59,999 bounded patterned structured nature of a chaotic attractive in 102 99:59:59,999 --> 99:59:59,999 problem too. 103 99:59:59,999 --> 99:59:59,999 We went back to using the app, 104 99:59:59,999 --> 99:59:59,999 the task was to generate a 50.0 trajectory from ex not equals 0.2 105 99:59:59,999 --> 99:59:59,999 using this very carefully chosen our primary value let's restart 106 99:59:59,999 --> 99:59:59,999 the simulation this doesn't look to me like anything periodic or 107 99:59:59,999 --> 99:59:59,999 anything that's a fixed point, 108 99:59:59,999 --> 99:59:59,999 I would guess this is chaotic, 109 99:59:59,999 --> 99:59:59,999 but there's something very interesting going on here. 110 99:59:59,999 --> 99:59:59,999 Look at the sort of peace and then the sort of peace. 111 99:59:59,999 --> 99:59:59,999 There's some patterns going on there, 112 99:59:59,999 --> 99:59:59,999 but they're not quite the same. 113 99:59:59,999 --> 99:59:59,999 So I would guess that this is a chaotic orbit in Part B of this 114 99:59:59,999 --> 99:59:59,999 problem. 115 99:59:59,999 --> 99:59:59,999 We're gonna watch for a little bit longer and see what happens to 116 99:59:59,999 --> 99:59:59,999 see if it's really chaotic or if it's gonna settle down to 117 99:59:59,999 --> 99:59:59,999 something with this out. 118 99:59:59,999 --> 99:59:59,999 You can watch an ongoing process of iterations by clicking Start 119 99:59:59,999 --> 99:59:59,999 animation. 120 99:59:59,999 --> 99:59:59,999 I still see that little Arrowhead thing coming through and then 121 99:59:59,999 --> 99:59:59,999 things look like they've go chaotic in between the work it 122 99:59:59,999 --> 99:59:59,999 re-occurrence of that pattern makes me suspect were nearby 123 99:59:59,999 --> 99:59:59,999 objectification point. 124 99:59:59,999 --> 99:59:59,999 Now things are gone, 125 99:59:59,999 --> 99:59:59,999 periodic and I'm gonna stop the animation, 126 99:59:59,999 --> 99:59:59,999 so we can count and see what kind of period. 127 99:59:59,999 --> 99:59:59,999 It is, 128 99:59:59,999 --> 99:59:59,999 looks like it repeats every 123456789. 129 99:59:59,999 --> 99:59:59,999 That's a 9 cycle,