1
00:00:03,000 --> 00:00:07,000
The first question is Stable and unstable manifolds play no role in the shape of a chaotic attractor.
2
00:00:07,000 --> 00:00:13,000
This is false; the stable and unstable manifolds are actually exactly what cause the shape of a chaotic attractor
3
00:00:13,000 --> 00:00:17,000
Question 2 is that Stable manifolds have the effect of compressing state space.
4
00:00:17,000 --> 00:00:20,000
This is absolutely true
5
00:00:20,000 --> 00:00:26,000
In contrast, the unstable manifolds have the effect of spreading out state space, not compressing it
6
00:00:26,000 --> 00:00:29,000
So question 3 is false
7
00:00:29,000 --> 00:00:37,000
For question 4: in general, an attracting fixed point in an n-dimensional nonlinear dynamical system does live at the intersection of n stable manifolds
8
00:00:37,000 --> 00:00:39,000
So this question is true
9
00:00:39,000 --> 00:00:46,000
As an example not as a proof, merely an example consider the fixed points in the dampened pendulum
10
00:00:46,000 --> 00:00:50,000
The attracting fixed points lie at the intersection of two stable manifolds
11
00:00:50,000 --> 00:00:56,000
And the non-attracting, unstable fixed points lie at the intersection of a stable and an unstable manifold
12
00:00:56,000 --> 00:01:01,000
Again, this is not a proof that this holds in general, merely an example to help you understand the question better
13
00:01:01,000 --> 00:01:03,000
However, this does hold in general
14
00:01:03,000 --> 00:01:09,000
For question 5, were given the following periodic orbit of a nonlinear dynamical system, and asked what we can say about the dimension of that system
15
00:01:09,000 --> 00:01:12,000
We can actually say something
16
00:01:12,000 --> 00:01:17,000
For example, we can say that the state space has to be more than one dimension, because its moving in two directions
17
00:01:17,000 --> 00:01:22,000
We can also say that the state space cannot be in two dimensions, because of this intersection
18
00:01:22,000 --> 00:01:27,000
For the deterministic dynamical systems we are studying, an intersection like this cannot occur
19
00:01:27,000 --> 00:01:34,000
If it could, then at this intersection point in phase space, a trajectory would have two directions to go, which would violate determinism
20
00:01:34,000 --> 00:01:39,000
So we know, because this intersection cannot occur, the state space must have at least three dimensions
21
00:01:39,000 --> 00:01:42,000
It could, however, have many more
22
00:01:42,000 --> 00:01:45,000
We can also not say that the periodic orbit is an attractor
23
00:01:45,000 --> 00:01:49,000
It could be that this is an unstable periodic orbit, for example, that does not attract anything
24
00:01:49,000 --> 00:01:55,000
To know that this is an attractor or not, wed have to find a set of initial conditions that are attracted to this periodic orbit
25
00:01:55,000 --> 00:01:57,000
The amount of information given is not enough to conclude this
26
00:01:57,000 --> 00:02:01,000
The only thing we can conclude is that the state space has at least three dimensions
27
00:02:01,000 --> 00:02:08,000
Question 6 asks if there is at least one stable manifold associated with every point on an attracting periodic orbit in a nonlinear dynamical system
28
00:02:08,000 --> 00:02:10,000
This is definitely true
29
00:02:10,000 --> 00:02:18,000
One way of seeing this is, because its an attracting periodic orbit, then for every point along the periodic orbit there needs to be a direction of shrinking
30
00:02:18,000 --> 00:02:28,000
Said differently, if at some point along the periodic orbit all manifolds were unstable, then at this point the trajectories would be ejected from the periodic orbit, and this would not be an attracting periodic orbit of the dynamical system
31
00:02:28,000 --> 00:02:31,000
For question 7, recall the Lorenz system
32
00:02:31,000 --> 00:02:37,000
This system has three state variables and three parameters, not two, so this question is false
33
00:02:37,000 --> 00:02:48,000
For question 8, the Lorenz system does not model a spring-loaded pendulum; its a truncated version of the Navier-Stokes equation, and models a chunk of fluid heated from below
34
00:02:48,000 --> 00:02:51,000
So this question is false
35
00:02:51,000 --> 00:02:57,000
Question 9 asks if there are two stable fixed points in the dynamics of the Lorenz system for some values of the systems parameters
36
00:02:57,000 --> 00:02:58,000
This question is true
37
00:02:58,000 --> 00:03:05,000
For example, consider this picture, which has r less than approximately 25, a = 16, and b = 4
38
00:03:05,000 --> 00:03:10,000
Question 10 asks if theres a chaotic attractor in the dynamics of the Lorenz system for some values of the systems parameters
39
00:03:10,000 --> 00:03:14,000
This is absolutely true, and why the Lorenz system is so famous
40
00:03:14,000 --> 00:03:17,000
For example, see this attractor, which is a chaotic attractor of the Lorenz equations
41
00:03:17,000 --> 00:03:23,000
The parameters used were rho = 28, sigma = 10, and beta = 8/3
42
00:03:23,000 --> 00:03:27,000
Note that sometimes rho, sigma, and beta are called r, a, and b
43
00:03:27,000 --> 00:03:32,000
This can be somewhat confusing, but its good to start getting in the practice of switching between the names of parameters
44
00:03:32,000 --> 00:03:39,000
Question 11 says that Nonlinear dynamical systems can have lots of attractors in different regions of their state space, but only one type at a time
45
00:03:39,000 --> 00:03:45,000
So for example, all the attractors for a particular parameter set are fixed points, or periodic orbits, or chaotic attractors
46
00:03:45,000 --> 00:03:47,000
This is not the case; this question is false
47
00:03:47,000 --> 00:03:56,000
In fact, nonlinear dynamical systems can have an arbitrary number of all types all over the state space, and theres no way to know a priori how many or what types or where they are
48
00:03:56,000 --> 00:04:01,000
This is one of the things that makes studying nonlinear dynamical systems so interesting and rich
49
00:04:01,000 --> 00:04:06,000
Number 12 states that The basins of attraction of different attractors in a dynamical system can overlap
50
00:04:06,000 --> 00:04:12,000
This is false: if this were the case, then initial conditions lying in the intersection of these basins of attraction could go to two different places
51
00:04:12,000 --> 00:04:14,000
This is not the kind of dynamical system were studying
52
00:04:14,000 --> 00:04:18,000
Question 13 asks if Lorenz was the first person to recognize chaos
53
00:04:18,000 --> 00:04:27,000
While some people argue it was actually Poincare, Lorenz is generally recognized as the first person to recognize chaos, in his seminal paper Deterministic Nonperiodic Flow
54
00:04:27,000 --> 00:04:31,000
So this question is true
55
00:04:31,000 --> 00:04:36,000
Question 14 states that Lorenz was the first person to use the term chaos for this kind of behavior
56
00:04:36,000 --> 00:04:38,000
This is false
57
00:04:38,000 --> 00:04:44,000
Yorke and Li were actually the first people to use the term chaos in their paper Period 3 Implies Chaos