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