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Maintained oscillators : pendulum and equation of Van der Pol.

Now g depends on the magnitude q of the oscillations.

The most simple expression we can use is :


So we can write the Van der Pol equation:


The little oscillations increase and the big decrease.

In an adimensional form:



Because e>0, the trajectories diverge in spiral near the origin [q =dq /dt=0] (indeed, the q term is neglectable).

Far from the origin, the trajectories get closer of the singular point because g>0.

Between these two kinds of trajectories exists a closed trajectory around the origin point : it is called a limit cycle. Every trajectory tend to get closer asymptoticly. This is an example of strange attractor.

The form of the limit cycle and the kind of oscillations depend on the value of e like the picture demonstrates it: