# Van der Pol Oscillator - Demo 4

## Frequency Locking

Increasing the nonlinearity tends to increase the tendency of the oscillators to lock. So now we change the driving strength to *a=0.5*

There is a long transient, which is not plotted - if you want to see it put *tran=0*.

The power spectrum....

...and the Poincare section...

...confirm that the dynamics is now periodic. From the power spectrum you can check that the frequency is the drive frequency *1.15*.

Alternatively with the drive strength as in demonstration 3 we can reduce
the frequency mismatch:

and here is the power spectrum

Reducing the frequency mismatch or increasing the driving strength increases the tendency towards frequency locking. You can now try to verify more quantitatively the different regions in the
drive-mismatch plot of figure 2, including the behavior of the transients.

We can imagine the orbit in the full 3-dimensional phase space lying on a *torus* - with one axis given by the drive phase i.e. the *Z* direction (equating 0 and
*2*), and the other given by the orbit in the *X-Y* plane. In the case of a periodic orbit, the dynamics traces out a curve that spirals around the torus, exactly repeating itself on successive passes and the section gives a point. In the case of a quasiperiodic orbit the curve eventually fills out the complete surface of the torus and the section is a circle (or distorted circle). This idea can be made precise: there exists an "invariant torus" i.e. a torus such that any point on the torus remains on the torus on the dynamics. The question of frequency locking is the question of whether the attractor is a curve on the torus (periodic, locked), or the complete torus (quasiperiodic, unlocked). On the other hand the onset of chaos is associated with the breakdown of the invariant torus.

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Last modified Sunday, January 16, 2000

Michael Cross