Isochronicity-induced bifurcations in systems of weakly dissipative coupled oscillators

We consider the dynamics of networks of oscillators that are weakly dissipa tive perturbations of identical Hamiltonian oscillators with weak coupling. Suppose the Hamiltonian oscillators have angular frequency omega(alpha) wh en their energy is a. We address the problem of what happens in a neighbour hood of where d omega/d alpha = 0; we refer to this as a point of isochroni city for the oscillators. If the coupling is much weaker than the dissipati on we can use averaging to reduce the system to phase equations on a torus. We consider example applications to two and three weakly diffusively coupl ed oscillators with points of isochronicity and reduce to approximating flo ws on tori. We use this to identify bifurcation of various periodic solutio ns on perturbing away from a point of isochronicity.