The first question states that sensitive dependence on initial conditions can only occur in a dynamical system that has at least one unstable manifold
This is true
If, for example, a dynamical system had all stable manifolds, then youd have shrinking in every direction
To have sensitive dependence on initial conditions, you need stretching
To achieve this stretching, you need an unstable manifold
Question 2 asks if an attractor can only exist in a dynamical system that has at least one stable manifold
This is true
If you had all unstable manifolds, you would have stretching in every direction, and so nothing would be attracting
That is, there would be no shrinking in any direction, so you wouldnt be able to attract back to anything
So this question is true
Question 3 asks if each stable manifold in a dynamical system has an associated Lyapunov exponent lambda whose value is negative
This is true
Recall that negative Lyapunov exponents correspond to stable manifolds, just like negative eigenvalues correspond to stable eigenspaces
The next question asks the difference between local and global Lyapunov exponents
This was discussed at the very end of lecture
The difference is that local Lyapunov exponents reflect the behavior of trajectories on small patches of the attractor, rather than looking at the entire attractor as a whole
Question 5 asks whether Lyapunov exponents are only associated with stable manifolds
This is false
Lyapunov exponents are associated with all manifolds, both stable and unstable
To make this statement true, you could have said that negative Lyapunov exponents are only associated with stable manifolds
While positive Lyapunov exponents, for example, are associated with unstable manifolds
And so this statement is false
Question 6 asks if there are n Lyapunov exponents in an n-dimensional dynamical system
Just like eigenvalues, this statement is also true
Just like it was the case with the eigenstuff, where you have repeated eigenvalues, you can also have repeated Lyapunov exponents, but you have one Lyapunov exponent for every dimension of the dynamical system
So this statement is true
If lambda 1, which is the largest Lyapunov exponent, is positive, then it dominates in the long term, and this is true
Question 8 asks if Lyapunov exponents are properties of attractors
That is, they are the same for any initial condition in the basin of attraction, and that is true
Question 9 asks us if the stable and unstable manifolds are unrelated to the eigenvectors of the linearized system
That is absolutely false
The eigenvectors are locally tangent to the stable and unstable manifolds of a particular fixed point
In fact, one method of finding stable and unstable manifolds is to linearize the system about a fixed point, find the eigenvectors of that linear system, and then run time forward and back along those eigenvectors
See the lecture for more details on this algorithm
And finally, Question 10 asks if a point on a stable or unstable manifold of a dynamical system stays on that manifold as time evolves
And this is true
The stable and unstable manifolds are dynamical invariants under the action of a dynamical system, so if a point is actually on a stable or unstable manifold, it will stay on that manifold for all time