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The first question asks if Smales horseshoe is a map of the unit square into itself, and this is true
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In general, Smales horseshoe is actually defined on any square in the plane, or any region of the plane that can be continually deformed to the unit square
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But regardless, this question is true
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Question 2 asks if Smales horseshoe maps some close points far apart, and vice versa
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So points that are close go far apart, and points that are far apart go close
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And this is true
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If you go back to the kneading bread analogy, which actually works very good, this is why, if youre baking bread (if anyone is familiar with baking), you can stick all the chocolate chips in one place in the dough, and then knead the bread, and the chocolate chips will evenly spread across the entire dough, or any other ingredient or spice for that matter
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Its not necessary to evenly space the spices all over the dough, because once you knead the bread, spices that were close together, or in some small cluster that you dumped on top of the dough, will spread apart evenly across the dough
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And this is effectively what Smales horseshoe does on the unit square
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It can be thought of as taking the unit square and then kneading the unit square through three topological mappings
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And the result of which is that some close points are far apart, and some far apart points are then close
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For Question 3, were given the following picture of a small ball of initial conditions around theta = pi in the undamped pendulum
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With the stable and unstable manifolds drawn in, and we want to assume that the ball is very tightly clustered around theta = pi
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We want to know what these points will look like as time evolves
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If we just think about this intuitively, we know that along the stable manifolds well have shrinking, so these points will shrink in
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And along the unstable manifolds well have stretching
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So what we should see is a long line of points along the unstable manifolds
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For this reason, a cannot be correct, because you have points spreading along the stable manifold
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This would only occur in backwards time
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Part b looks the most probable
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You have stretching along the unstable manifold, and you have no growth along the stable manifold
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The fact that you cant really see shrinking that well has more to do with the artists ability that is, my ability to use Sketchpad than anything else
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I think this one is the most probable, but lets take a look at the other ones
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c would imply that they spread out in every direction equally
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This is not the case
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We know that well have shrinking along the stable manifold, so this cannot be the case
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Part d can also not be the case, because you only have growth along the stable manifold, which is exactly the opposite of what occurs
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So part b even though the shrinking is not really very good because of my artistic ability we see this stretching along the unstable manifolds like we should see
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So b is the answer to this question
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Question 4 is a series of questions about a generalized horseshoe
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Part a asks if horseshoes only turn up in the dynamics of the Smale horseshoe map, and this is false
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For example, horseshoes show up in the dynamics of the pendulum
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That actually answers part c as well
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Part b states that horseshoes are important because they play a role in proofs of chaos, and this is true as well
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See the lecture for more details on this
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Question 5 asks whether dissipation is a necessary condition for the existence of attractors, and this is true