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Question 1 states that it is always possible to measure at least one state variable from any dynamical system
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This is false
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This is something that every time-series analyst wishes were true, but its most definitely false
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For a lot of dynamical systems in the real world, we dont even know what the state variables are, let alone being able to measure one in isolation
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Most of the time, you end up measuring several of the variables munged together in some smooth generic way, or even just the influence of some of the variables on the state space
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But its definitely not the case that youre always able to measure one state-space variable from any dynamical system, especially dynamical systems in the real world
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Question 2 states that measuring a single state-space variable from a high-dimensional dynamical system effectively projects that systems dynamics onto a plane
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This is also false
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To project the dynamics onto a plane, you would need to measure two state-space variables, not one
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Measuring a single state-space variable from a high-dimensional dynamical system effectively projects that systems dynamics onto a line, not a plane
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For Question 3, we need to figure out which of the following statements are true
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Projections of attractors of dynamical systems can cause the topology of the projected attractor to differ from that of the unprojected attractor
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And this is true
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For example, if you project an attractor and cause a loop in the dynamics to cross, you go from one hole to two holes
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For experts in topology, this change from one hole to two holes would be a change in the Betti number, signaling a change in the topology
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For non-experts in topology, this increase from one hole to two holes signals a change in the topology, and so this statement is true
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For part B, projections of attractors of dynamical systems can cause trajectories in the projected attractor to cross one another
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And this is absolutely true, as we just discussed in part A
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Imagine any periodic orbit, for example, or periodic trajectory that forms a loop in a three-dimensional state space
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There is at least one direction of projection that would cause this trajectory to cross itself
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Even more simply, think about the Lorenz butterfly
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When you project this from three dimensions to two dimensions, it causes several intersections to occur where the two wings meet
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That is, projecting from three dimensions to two dimensions causes these trajectories to cross one another in the projected attractor
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Part C states that projections of attractors of dynamical systems always increase the dimension of those attractors, and this is false
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Projections always decrease the dimension of attractors
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So part D must be true: A and B above
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Question 4 asks what spectral methods tell you
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And they tell you the frequencies that are present in a signal
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Spectral methods do not tell you the dimension of the dynamical system
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And they do not tell you whether the system has recently undergone a bifurcation