Phase-locked mode stability for coupled van der Pol oscillators

The critical variational equation governing the stability of phase-locked m odes for a pair of diffusively coupled van der Pol oscillators is presented in the form of a linear oscillator with a periodic damping coefficient tha t involves the van der Pol limit cycle. The variational equation is transfo rmed into a Hill's equation, and stability boundaries are obtained by analy tical 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 Pade approximant describi ng a surface that combines with the Hill determinant surfaces to complete t he stability boundary. The expansions obtained are evaluated to visualize t he first three resonant surfaces which are compared with numerically determ ined slices through the stability boundaries computed over the range 0.4<<e psilon><5. [S0739-3717(00)00502-X].