This first question is simply a definition: in a two-dimensional linear dynamical system, if you have a positive real eigenvalue and a negative real eigenvalue, this is by definition a saddle point
So this question is true
The second question pretty much comes down to a definition as well
If we assume that the nonlinear system can be linearized about the fixed point, and that linearization has one positive and one negative real eigenvalue, then by definition this is a saddle point
The only thing thats a little bit weird here is making sure that the nonlinear system can be linearized at a fixed point
In this case youre basically assuming that a linearization is a good local approximation of the fixed points behavior
This is a very standard assumption made in the nonlinear dynamics community when doing this kind of analysis
The answer to this question is true
The next two questions involve the pendulum at theta = pi, so this inverted position, and no angular velocity
The first question asks if this is a saddle point
Remember that, in quiz solution 4.2, we showed in-depth that this point, and actually any odd multiple of pi, is a saddle point of this dynamical system
Note that, while we showed it for the undamped case in quiz 4.2, the same analysis holds for a damped pendulum
And if youre interested, I encourage you to do this analysis by using the damped pendulum instead of the undamped pendulum
Ill warn you, however, that the math gets a little bit more hairy, as theres more parameters involved
So, this first question is true
The second question asks if this point is the only saddle point
As we showed in the solution video for quiz 4.2, actually every odd multiple of pi is a saddle point of this dynamical system
For the damped or the undamped pendulum, this is true
So the answer to this question is false
For question 5, in the nonlinear case, stable and unstable manifolds generally are not the same thing as stable and unstable eigenvectors
This is only true very locally to a fixed point
So this question is false
To see this, consider the stable and unstable manifolds of this pendulum
As you can see, very locally the stable and unstable manifolds are the same as the eigenvectors
Or a good local approximation of the eigenvectors, at least
However, if you look globally, these two eigenvectors are not a good approximation of the stable and unstable manifolds
Question 6 is the nonlinear equivalent to a question we tackled in an earlier quiz problem
Namely, does a point along an eigenvector stay along an eigenvector under the action of a matrix
This question is the same thing, but in a nonlinear case
Just as the eigenvectors are invariant under the action of a matrix, stable and unstable manifolds are invariant under the action of a dynamical system
So this question is true
For question 7, were asked if the distance between a fixed point and a point near that fixed point that lies on its stable manifold will grow with time
This is of course false
The key word here is grow with time
We know, if a point lies on the stable manifold of a fixed point, then, as time goes forward, the distance between that point and the fixed point will shrink exponentially in fact
Question 8 is very similar to question 7, except in this case, were looking at a point on the unstable manifold of a fixed point
If a point is on the unstable manifold, then the distance between that point and the fixed point will grow exponentially over time
So this question is definitely true
For the last question, if a fixed points stable manifold connects back around and becomes its own unstable manifold, that is actually called a homoclinic orbit, not a heteroclinic orbit
So this question is false
A good way to remember this is to look at the prefix hetero
A word with the prefix hetero means other or different
So in this case, this is a path that connects two different fixed points
In the case of homoclinic, you have the of homo, which means same
In a homoclinic orbit, youre attaching the stable manifold back to its own unstable manifold
Keeping in mind the meaning of hetero versus homo in this circumstance can be quite helpful