Isochronicity-induced bifurcations in systems of weakly dissipative coupled oscillators

We consider the dynamics of networks of oscillators that are weakly dissipative perturbations of identical Hamiltonian oscillators with weak coupling. Suppose the Hamiltonian oscillators have angular frequency ω(α) when their energy is α. We address the problem of what happens in a neighbourhood of where dω/dα = 0; we refer to this as a point of isochronicity for the oscillators. If the coupling is much weaker than the dissipation we can use averaging to reduce the system to phase equations on a torus. We consider example applications to two and three weakly diffusively coupled oscillators with points of isochronicity and reduce to approximating flows on tori. We use this to identify bifurcation of various periodic solutions on perturbing away from a point of isochronicity. Communicated by J. Stark