Phase-Locked Mode Stability for Coupled van der Pol Oscillators

Abstract

The critical variational equation governing the stability of phase-locked modes for a pair of diffusively coupled van der Pol oscillators is presented in the form of a linear oscillator with a periodic damping coefficient that involves the van der Pol limit cycle. The variational equation is transformed into a Hill’s equation, and stability boundaries are obtained by analytical and numerical methods. We identify a countable set of resonances and obtain expressions for the associated stability boundaries as power series expansions of the associated Hill determinants. We establish an additional “zero mean damping” condition and express it as a Padé approximant describing a surface that combines with the Hill determinant surfaces to complete the stability boundary. The expansions obtained are evaluated to visualize the first three resonant surfaces which are compared with numerically determined slices through the stability boundaries computed over the range 0.4<ε<5. [S0739-3717(00)00502-X]