Asymptotic Analysis of the Biased van der Pol Equation

Abstract

We perform an asymptotic analysis of the biased van der Pol equation $\ddot{x} - ν\,(1 - x^{2})\,\dot{x} + x = a$, where the dissipation parameter is large $(ν\gg 1)$ and the bias parameter $a$ is a real number between $-1$ and $1$, corresponding to large-amplitude relaxation oscillations along a stable limit cycle. Using an abridged singular perturbation analysis, the period of relaxation oscillations $T(a,ν)$ is obtained that is accurate within $0.1\%$ of the numerical results for $ν\gg 1$.