1
00:00:03,000 --> 00:00:10,000
Question 1 asks us if the shadowing lemma tells us that noise-added trajectories on chaotic attractors are shadowed by true trajectories
2
00:00:10,000 --> 00:00:13,000
And this is true; this is precisely the spirit of the shadowing lemma
3
00:00:13,000 --> 00:00:24,000
That is, if youre on a chaotic attractor and youre bumped by a small amount of noise, then youll land on an area of the attractor that you would have gotten to anyway if youd continued along that same trajectory for infinitely long
4
00:00:24,000 --> 00:00:28,000
That is, that a noise-added trajectory on a chaotic attractor is shadowed by a true trajectory
5
00:00:28,000 --> 00:00:30,000
So a little bit of noise wont be a problem
6
00:00:30,000 --> 00:00:34,000
However, question 2 really gets at the nuances of the shadowing lemma
7
00:00:34,000 --> 00:00:43,000
It asks if the shadowing lemma tells us that a chaotic attractor is immune to any level of noise; i.e., that all noise-added trajectories, regardless of the size of the noise, will remain on the attractor
8
00:00:43,000 --> 00:00:44,000
And this is false
9
00:00:44,000 --> 00:00:50,000
If the noise is large enough, for example, you could bump the trajectory off of the attractor and into the basin of attraction of a different attractor
10
00:00:50,000 --> 00:00:58,000
So you may actually reach a trajectory, or an attractor, that you wouldnt have normally reached, even if the trajectory was infinitely long, because the noise bumped you out of the basin of attraction
11
00:00:58,000 --> 00:01:02,000
The final question asks if the shadowing lemma holds for all nonlinear systems
12
00:01:02,000 --> 00:01:07,000
And this is false; the shadowing lemma really only holds for nonlinear, chaotic dynamical systems