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