For small values of b (and no driving) the perturbation theory predicts an orbit x=2cost, y=-2sint i.e. a simple harmonic motion with a circular orbit in (x,v) phase space of radius 2. This can be checked by numerical simulation (here for b=0.05.
If you change the x-axis to show time (x-ax=0) you can check the sinusoidal time dependence.
The power spectrum shows a large peak at frequency 1, and odd harmonics down by several orders of magnitude. (Changing the number of points used to 512 or 1024 will give better resolution.)
For larger values of b the orbit becomes distorted from a circle, and the harmonics in the power spectrum are stronger, indicating a more non-linear oscillator.
Phase space orbit:
Power spectrum:
Return to discussion.