Van der Pol Oscillator - Demo 3

Quasiperiodic Motion

Suppose now the Van der Pol oscillator is driven at frequency 1.15 with a strength a=0.32

The phase space orbit appears quite complicated, eventually densely filling out an area of the projection. Is this chaos?

One clue to the answer is given by the power spectrum:

The spectrum continues to have sharp peaks as in the periodic case, but now the "main" peak around frequency 1 has sub-bands. You can check that the two largest peaks are at frequencies of 1 and 1.15 by clicking on these peaks in the running applet. Thus the internal oscillation (frequency 1) and driven oscillation (frequency 1.15) appear independently in the spectrum.

It is hard from the power spectrum to tell whether the sub-bands are rationally or irrationally related to the main peak. In the former case the orbit would be periodic, with a higher period corresponding to the inverse of the smallest difference frequency in the spectrum. In the latter case the orbit would consist of two incommensurate frequencies, would never precisely revisit a point, and is called quasiperiodic.

This question is addressed better by looking at the Poincare section at some Z=const plane:

What is going on here? Remember that the third phase space coordinate Z is the phase of the drive oscillation. The Poincare section here is take at some constant Z, which corresponds to looking at the oscillation once every period of the drive. If the dynamics is periodic the Poincare section will consist of discrete points (one point if the periodic is the drive period, two if the period is twice the drive period etc.). If however the dynamics is quasiperiodic, containing an additional oscillation at a frequency incommensurate with the drive frequency, the points on the Poincare section will eventually fill out a dense curve.

Of course there are limitations in any numerical implementation to the resolution at which we can distinguish a dense curve, corresponding to a quasiperiodic orbit, and a very large number of discrete points, corresponding to a periodic orbit of very high period. But the Poincare section makes the task of discrimination easier than either the time series or the power spectrum.

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Last modified Saturday, November 27, 1999
Michael Cross