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Question 1 asks if both projections and sections reduce the dimension of an object
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And this is true
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This is precisely what these two techniques are used for in the study of nonlinear dynamics
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And while both projections and sections reduce the dimension of an object, they do so in different and useful ways
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For example, a medical x-ray, which is a projection, tells you something very different than a CT scan, which is a section
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This leads us into Question 2: a medical x-ray performs a projection operation, and this is true
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This was described very nicely in the lecture video
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Question 3 asks if measuring the value of a single state variable of a dynamical system is like performing a section
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This is not like performing a section, this is like performing a projection
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So youre taking the entire state space, and youre projecting it onto a single axis
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So this question is false
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Question 4 states that sections can be slices in time or slices in state space, and this is true
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A section in time is called a temporal section, and this is like flashing a strobe light at a pendulum
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Alternatively, you could slice in space
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This would be like taking a plane and sticking it through one of the wings of the Lorenz butterfly
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Which, as you saw in lecture, would result in a Cantor set
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So this question is true
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Poincare sections, or sections, can be slices in both time and space
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The next four questions involve taking Poincare sections, or spatial sections, of this orbit of a dynamical system
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This is the typical method for determining the period of a trajectory from a flow
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For part a, we want to take a plane of section at x = 3, so here
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If we do this, the Poincare section will have two points on it: here and here
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So the answer to part a is Two points
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If you take a spatial Poincare section and you have two points on it, this type of section is associated with a two-cycle, so this is the answer to part b
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However, if we instead take a spatial Poincare section at y = 2 so here we will get four points: here, here, here, and here
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So the answer to part c is Four points
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And similarly, just as two points result in a two-cycle, four points on a Poincare section is defined as a four-cycle
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This should illustrate to you the difficulty of assigning a value to the period of an orbit of a flow
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Depending on where you take the plane of section, your conclusions can be different, like in this case
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One plane of section said it was a two-cycle, and another plane of section described this as a four-cycle