A first tour through examples - Chapter 6

Van der Pol’s periodic attractor

The equation of Van der Pol was designed as a model for an electronic oscillator in the 1920s. We do not go into the derivation of this model. In any case it is a mathematical description of obsolete technology based on radio tubes, the predecessors of our present transistors.
The equation is

$$ \ddot{u}+\epsilon(1-u^2)\dot{u}+u=0 $$

where $u$ represent the voltage.

As we’ve done for the harmonic oscillator and the pendulum, we pass to the phase plane representation by setting $x=u$ and $y=\dot{u}$ to get

$$ \begin{cases} \dot{x}= y\\ \dot{y}=-x-\epsilon(1-x^2) y \end{cases} $$

where $\epsilon\geq 0$. When $\epsilon=0$ we recognize the equations for the harmonic oscillator we’ve seen before.

As we can observe, the Van der Pol equation describes a system that, independent of the initial state different from $(0,0)$ eventually will end up in the same periodic behaviour. The trajectory of such a periodic solution is called a limit cycle. Here we have an attractive limit cycle. It can even be shown that this property is robust, in the sense that small changes of the right-hand side of the differential equation will not change this property.