8.5.1 The Van der Pol equation

Van der Pol’s equation [1] for the displacement $x(t)$ of a nonlinear oscillator is

$$\ddot{x}-\mu (1-x^2) \dot{x} + x =0 \tag{1}$$

for free motion. The term scaled by the constant coefficient $\mu$ describes a kind of nonlinear damping effect: it is intuitively clear that if $\mu > 0$, that term describes a kind of negative damping if $x < 1$, but that it switches to positive damping when $x > 1$. The full behaviour is more complicated than that simple description suggests, because of the nonlinear nature of the term $x^2 \dot{x}$, but the argument conveys the correct flavour insofar as trajectories starting with small values of $x$ tend to spiral outwards (because of the negative damping) while trajectories starting with large $x$ tend to spiral inwards.

For small values of $x$, the linearised version of the equation is simply

$$\ddot{x}-\mu \dot{x} + x \approx 0 \tag{2}$$

which describes a linear oscillator with damping coefficient $-\mu$. The only singular point of eq. (1) is at $x=0$, $\dot{x}=0$, and we can deduce immediately from the linearised equation (2) that this is a centre if $\mu=0$, a stable spiral if $ \mu < 0$ and an unstable spiral if $\mu > 0$.

You can find more information about the Van der Pol equation from the Wikipedia page. One of the things revealed there, and expanded in other pages linked from there, is that the equation played a role in the early developments leading to chaos theory. Electrical engineers testing circuits of the kind Van der Pol was originally interested noticed that they did interesting things when driven with an external sinusoidal signal. This would appear in eq. (1) as a forcing term $A \cos \omega t$ on the right-hand side, for forcing with amplitude $A$ at frequency $\omega$.

When driven close to the frequency of the limit cycle, they found that the oscillator was sometimes entrained: the oscillation frequency was “pulled” to match the drive frequency. Something a bit similar can happen if two pendulum clocks or two metronomes are sitting on the the same flexible table, or hanging from the same non-rigid wall: the two clocks or metronomes may, after a while, synchronise with each other if their separate frequencies were already fairly close. This phenomenon was first reported by the 17th-century Dutch astronomer and scientist Christiaan Huygens, and it is often know by the term “Huygens’ clocks”.

Coming back to the electrical engineers, they noticed that for some ranges of parameter values there was a kind of background noise accompanying the periodic sound of the limit cycle. Initially they blamed flaws in the equipment, but in 1945 the British mathematicians Mary Cartwright and J. E. Littlewood [2] were able to show that the effect was predicted by Van der Pol’s equation. It was an early example of “deterministic chaos”.

[1] Van der Pol, B., “On relaxation-oscillations”, The London, Edinburgh and Dublin Philosophical Magazine & Journal of Science, 2(7), 978–992 (1926).

[2] Mary Cartwright and J. E. Littlewood (1945) “On Non-linear Differential Equations of the Second Order”, Journal of the London Mathematical Society 20: 180 doi:10.1112/jlms/s1-20.3.180