Singular Complex Periodic Solutions of van der Pol’s Equation

Abstract

The solutions that are studied here arise when limit cycle solutions of van der Pol’s equation with parameter $\epsilon $ are continued analytically into the complex $\epsilon $-plane. These analytic continuations are described, for small $\epsilon $, by the perturbation expansions in powers of $\epsilon $ which were developed at length by Andersen, Dadfar, and Geer [SIAM J. Appl. Math., 42(1982), pp. 678–693 and 44(1984), pp. 881–895]. We show that the moving singularities that they found are caused by a family of singular complex periodic solutions. These occur along two infinitely long branches in the complex $\epsilon $-plane; half of one being plotted in Fig. 2. Numerical integration is needed to obtain these singular solutions. As Fig. 4 for the large value $\epsilon = 10 + 3.042i$ shows, the complex solution is primarily real over most of the cycle. And this primarily real part behaves much like the main part of the real relaxation oscillation, falling from near 2 at its start toward 1. But the rapid transition from 1 to -2 that then occurs prior to the start of the next half cycle now takes place via a complex singularity, rather than smoothly as in the real case. An asymptotic description of the singular complex periodic solutions for large values of $| \epsilon |$ is given using singular perturbation theory and matched asymptotic expansions. It shares many features with the corresponding theory for the real case, and, as Table 1 shows, is numerically accurate. Moving complex singularities similar to those studied here are likely to occur with other perturbation series solutions of differential equations and to hinder their effectiveness.