Amplitude response of coupled oscillators

Abstract

We investigate the interaction of a pair of weakly nonlinear oscillators (eg. each near a Hopf bifurcation) when the coupling strength is comparable to the attraction of the limit cycles. Changes in amplitude cannot then be ignored, and there are new phenomena. We show that even for simple forms of these equations, there are parameter regimes in which the interaction causes the system to stop oscillating, and the rest state at zero, stabilized by the interaction, is the only stable solution. When the uncoupled oscillators have a local frequency that is independent of amplitude (zero shear), and the coupling is scalar, we give a complete description of the behavior. We then analyze more complicated equations to show the effects of amplitude-dependent frequency in the uncoupled equations (nonzero shear) and nonscalar coupling. For example, when there is shear, there can be bistability between a phase-locked solution and an unlocked “drift” solution, in which the oscillators go at different average frequencies. There can also be bistability between a phase locked solution and an equilibrium, nonoscillatory, solution. The equations for the zero shear case have a pair of degenerate bifurcation points, and the new phenomena in the nonzero shear case arise from a partial unfolding of these singularities. When the coupling is nonscalar, there are a variety of new behaviors, including bistability of an in-phase and an anti-phase solution for two identical oscillators, parameters for which only the anti-phase solution is stable, and “phase-trapping,” i.e. nonlocked solutions in which the oscillators still have the same average frequency. It is shown that whether the coupling is diffusive or direct has many effects on the behavior of the system.