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For the first question, were asked what the first step of calculating the Lyapunov exponent from a scalar time-series data set is
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And this is to embed the data
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That is, to perform delay-coordinate embedding on the time series
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Question 2 asks if Wolfs algorithm for calculating Lyapunov exponents uses the variational equations, and this is false
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The variational equation is a way to get at the full spectrum of Lyapunov exponents if you have the governing equations, which you usually do not
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Wolfs algorithm is a way of calculating the Lyapunov exponents given a trajectory of a dynamical system
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For Question 3, were asked why its hard to calculate the Lyapunov exponent from data
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And the answer is E, all of A through D above
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You dont get to drop points where you want them in real-world data
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The data points are almost always noisy
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The data may not be long enough to fully cover the attractor
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And the sampling rate of the data may not be high enough to capture all of the dynamics on the attractor
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These are all certainly reasons that it would be hard to calculate the Lyapunov exponent, but certainly there are more reasons as well
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Finally, for Question 4, consider this plot of the stretching factor produced by Kantzs algorithm
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Part a asks if this curve has a visible scaling region, and it does
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Region B would be considered a scaling region of this plot, so the answer to a is yes
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Part b asks what the Lyapunov exponent of this trajectory is
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The slope of a line fitted to the curve in region B
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That is, the slope of a line fitted to the scaling region of this plot