An analytical and numerical study of the bifurcations in a system of linearly-coupled oscillators

Abstract

We study a two-parameter family of ordinary differential equations in R 4 that governs the dynamics of two coupled planar oscillators. Each oscillator has a unique periodic solution that is attracting and the uncoupled product system has a unique invariant torus that is attracting. The torus persists for weak coupling and contains two periodic solutions when the coupling is linear and conservative. One of these, in which the oscillators are synchronized, persists and is stable for all coupling strengths. The other, in which the oscillators are π radiant out of phase, disappears either in a Hopf bifurcation or when fixed points appear on the orbit at a critical ratio of the coupling strength to the frequency. The out-of-phase oscillation is unstable except on an open set in the frequency-coupling-strength plane which contains moderate values of both parameters. Furthermore, there are tori bifurcating from the out-of-phase solution, which means, according to the Arnol'd theory for Hopf bifurcations in maps, that there may be periodic solutions of arbitrarily large period and chaotic solutions as well. Numerous other bifurcations occur, and there are a number of higher codimension singularities. In a large region of the frequency-coupling parameter plane stable steady states coexist with stable periodic solutions.