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