Weak coupling of strongly nonlinear, weakly dissipative identical oscillators

Abstract

This paper considers a network of coupled oscillators with (a) weak coupling and (b) weak dissipation (but strongly nonlinear response). By changing to energy- angle coordinates, equations for the slow evolution of the amplitudes and phases of the individual oscillators are obtained. Restricting to problems with identical oscillators and coupling weaker than dissipation, an approximation to an attracting invariant torus for two oscillators is constructed and averaging is performed on this to reduce to a single equation for the phase difference. For two van der Pol-Duffing type oscillators coupled by a cubic function this method is used to predict bifurcation from in-phase stability (through a region where neither in-phase nor antiphase solutions are stable) to antiphase stability as a coupling parameter is changed. Furthermore, the method is used to investigate analytically the destabilization of the in-phase solution when two oscillators are dissipatively coupled through one state variable. For a large class of systems the stability of the in-phase solution depends on the sign of the shear or dependence of frequency on amplitude of the individual oscillators.