Q:

Differential Equation power spectra

ε = .1; A = 0; ω0 = 1; ωf = 0; 
data = 
  NDSolve[
    {x''[t] == ε (1 - x[t]^2) x'[t] - ω0^2*x[t] + A*Cos[ωf*t], x[0] == 1, x'[0] == 0}, 
    {t, 0, 100}];
Periodogram[data, ScalingFunctions -> "Absolute", PlotRange -> All]

How would I create a suitable Power spectrum of the frequencies that this system has.  For these set of values (at the top), there should ideally be only one unique frequency (ω0) since the other one is not in play.  
Mathematica's example of power spectra utilizes the periodogram funcition, although I don't really understand what it does mathematically, so there is room for using another type of graph. 

A:

ε = 1/10; A = 0; ω0 = 1; ωf = 0;
sol = NDSolve[{x''[
      t] == ε*(1 - x[t]^2) x'[t] - ω0^2*x[t] + 
      A*Cos[ωf*t], x[0] == 1, x'[0] == 0}, 
   x[t], {t, -100, 100}];
Plot[x[t] /. sol, {t, -100, 100}]

Periodogram[Flatten[Table[x[t] /. sol, {t, -100, 100, 0.5}]],PlotRange -> All]

A:

Here's my shot at it:
ε = .1; A = 0; ω0 = 1; ωf = 0;

sol = First[NDSolve[
    {x''[t] == ε (1 - x[t]^2) x'[t] - ω0^2*x[t] + A*Cos[ωf*t], x[0] == 1, x'[0] == 0}, 
    x, {t, 0, 100}]];
f = x /. sol;
data = Table[f[t], {t, 0, 100}];

Periodogram[data, ScalingFunctions -> "Absolute"]

A:

Consider the equation: $$\ddot{x} = (x,\dot{x},t)$$
The numerical solution generates a sequence of values of x at discrete values of t. This sequence is known as time series.  Plotting x pk t might seem random or noisy, but often it contains harmonics at certain frequencies.  Dominant frequencies in time series data can be investigated using the Discrete Fourier transform.  For example we take N equally spaced values from the time series - $$x = x_{0},x_{1}, ..., x_{N}$$
The Discrete Fourier transform is defined as follows: $$X_{k}=\frac{1}{\sqrt{N}}\sum_{m=0}^{N}x_{m}e^{\frac{-2\pi ikm}{N}},   (k=0,1,2,...,N-1)$$
Generally $X_{k}$ is a complex number.  To analyse the frequency structure we analyse the power spectrum $P(\omega _{k})$ defined by: $$P(\omega _{k})=X_{k}\overline{X_{k}}=\left | X_{k} \right |^{2}$$
ε = .1; A = 0; ω0 = 1; ωf = 0;

ff = x[t] /. 
 First[NDSolve[{x''[
    t] == ε (1 - x[t]^2) x'[t] - ω0^2*x[t] + 
    A*Cos[ωf*t], x[0] == 1, x'[0] == 0}, 
 x, {t, 0, 300}]][[1]]

Table[ff, {t, 0, 100}] // Fourier // Abs // 
 ListLogPlot[#, PlotRange -> All] &

