Abstract

The critical variational equation governing the stability of symmetric phase-locked modes for a pair of identical van der Pol oscillators with linear coupling is presented and shown to be equivalent to a Hill’s equation whose periodic coefficient involves the van der Pol limit cycle. We identify the countable set of resonances corresponding to the instability of the in-phase mode and present power series expansions for the Hill determinants which bound these resonances. An additional stability surface is associated with the transformation to Hill’s standard form and corresponds to the time-dependent damping coefficient having zero mean. We present a Padé approximant for this zero mean damping surface which is uniformly effective throughout the parameter range. Power series are also presented for the intersection of the resonance boundaries with the zero mean damping surface and the periodic solutions which occur on the intersection curves. Padé approximants computed from the power series for the intersection curves provide an improved estimate for the growth of the resonances as ε increases.

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