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Cell["\<\
In this notebook, we solve nonlinear differential equations, looking at the \
time dependence, phase space plots, Poincare sections, and the power \
spectrum.  

The notebook has specialized to a pendulum equation, with parameters used in \
session 8 of 780.20. \
\>", "Text"],

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Cell["Define the Differential Equation", "Section"],

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Name the equation \"diffeq\".  Note the \"==\" in defining the equation.  \
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Cell["\<\
Specify the range in time over which we will solve the differential equation. \
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have to extend this range if we get an \"outside of range\" error later.)\
\>", "Text"],

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Numerically solve the differential equaiton using NDSolve, specifying the \
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\>", "Text"],

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NDSolve returns an \"interpolating function\", which can be evaluated later \
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\>", "Text"]
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Or we can do a parametric plot (\"ParametricPlot\") with the same solution.  \
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Now do the phase space plot.  You may need to change AxesOrigin to get the \
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Here is the same plot, with the starting time t set to skip the transient \
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\>", "Text"],

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The idea of a Poincare section is to plot a point in phase space once every \
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pattern gives information about the periodicity of the signal (or indicates \
chaos).  Start the plot at a large enough time t (\"tstart\") so that the \
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\>", "Text"],

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Set the external period and how many periods we'll consider.  Define a \
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\>", "Text"],

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The power spectrum is found by taking a Fourier transform (FFT) of the \
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The basic command is Fourier.  FourierParameters chooses a particular \
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Transpose to make pairs of points from two separate lists of points.\
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