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The first question states that sensitive dependence on initial conditions can only occur in a dynamical system that has at least one unstable manifold
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This is true
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If, for example, a dynamical system had all stable manifolds, then youd have shrinking in every direction
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To have sensitive dependence on initial conditions, you need stretching
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To achieve this stretching, you need an unstable manifold
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Question 2 asks if an attractor can only exist in a dynamical system that has at least one stable manifold
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This is true
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If you had all unstable manifolds, you would have stretching in every direction, and so nothing would be attracting
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That is, there would be no shrinking in any direction, so you wouldnt be able to attract back to anything
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So this question is true
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Question 3 asks if each stable manifold in a dynamical system has an associated Lyapunov exponent lambda whose value is negative
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This is true
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Recall that negative Lyapunov exponents correspond to stable manifolds, just like negative eigenvalues correspond to stable eigenspaces
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The next question asks the difference between local and global Lyapunov exponents
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This was discussed at the very end of lecture
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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
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Question 5 asks whether Lyapunov exponents are only associated with stable manifolds
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This is false
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Lyapunov exponents are associated with all manifolds, both stable and unstable
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To make this statement true, you could have said that negative Lyapunov exponents are only associated with stable manifolds
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While positive Lyapunov exponents, for example, are associated with unstable manifolds
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And so this statement is false
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Question 6 asks if there are n Lyapunov exponents in an n-dimensional dynamical system
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Just like eigenvalues, this statement is also true
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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
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So this statement is true
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If lambda 1, which is the largest Lyapunov exponent, is positive, then it dominates in the long term, and this is true
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Question 8 asks if Lyapunov exponents are properties of attractors
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That is, they are the same for any initial condition in the basin of attraction, and that is true
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Question 9 asks us if the stable and unstable manifolds are unrelated to the eigenvectors of the linearized system
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That is absolutely false
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The eigenvectors are locally tangent to the stable and unstable manifolds of a particular fixed point
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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
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See the lecture for more details on this algorithm
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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
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And this is true
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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