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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 starts with the Duffing equation.  Your job is to modify it to \
the damped, driven pendulum, then to reproduce and extend the results from \
session 8 (which used diffeq_pendulum.cpp).\
\>", "Text"],

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

Cell["\<\
Name the equation \"diffeq\".  Note the \"==\" in defining the equation.  \
\>", "Text"],

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

Cell["\<\
Specify the range in time over which we will solve the differential equation. \
 We won't be able to use the solution outside of this range. (I.e., we'll \
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 \
initial conditions.  Setting MaxSteps to a large number is needed if tmax is \
large.\
\>", "Text"],

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NDSolve returns an \"interpolating function\", which can be evaluated later \
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\>", "Text"]
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Cell["Plot the Time Dependence and Phase Space", "Section"],

Cell["\<\
We can just use \"Plot\" with Evaluate and the \"interpolating function\" \
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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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\>", "Text"],

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Cell["\<\
Now do the phase space plot.  You may need to change AxesOrigin to get the \
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\>", "Text"],

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Cell["\<\
Here is the same plot, with the starting time t set to skip the transient \
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\>", "Text"],

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Cell["Poincare Sections", "Section"],

Cell["\<\
The idea of a Poincare section is to plot a point in phase space once every \
period of the external force, 2\[Pi]/(external frequency).  The resulting \
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 \
transients have died out.\
\>", "Text"],

Cell["\<\
Set the external period and how many periods we'll consider.  Define a \
function timeperiod[i] giving the corresponding time as a function of the \
period number.\
\>", "Text"],

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Cell["\<\
Just evaluate at the relevant points.  Flatten[expr,1] strips off a layer of \
{}'s.\
\>", "Text"],

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Cell["\<\
The power spectrum is found by taking a Fourier transform (FFT) of the \
signal.  We generate points at the same time intervals as in \
diffeq_pendulum.cpp.\
\>", "Text"],

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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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