The Periodic Solution of Van Der Pol's Equation

We give the periodic solution of the van der Pol equation $\ddot{x} + x = \varepsilon(1 - x^2)\dot{x}$ in the form of a series converging for all values of the damping parameter $\varepsilon$. For $\varepsilon$ small the series solution reduces to a perturbation series in powers of $\varepsilon$ and is obtained from this series, essentially by the analytical continuation method, using a suitable transformation of the parameter. The coefficients of the perturbation series and, by virtue of the simplicity of the transformation proposed, those of the transformed series as well are given in explicit form by recurrent analytical formulae, so that both series can be developed in principle up to any order in $\varepsilon$. Actually, the coefficients in the expansions can be obtained by symbolic computation in exact rational number form up to a very high order, the only limitation being the available computing resources.