The dynamics of networks of oscillators that are weakly dissipative perturbations of identical Hamiltonian oscillators with weak coupling is considered. Suppose the Hamiltonian oscillators have angular frequency w(a) when their energy is a. In previous work it has been found that the stability of the in phase oscillation for diffusively coupled oscillators
depends fundamentally on the sign of dw/da, where each oscillator is close to a level curve with energy a. Hence it is natural to investigate the dynamics of the network if a is close to a point where dw/da =0. Such a point is referred to as a point of isochronicity for the oscillators, and perturbations around such a point are considered as unfolding of isochronicity.

In the lecture I will discuss the bifurcations that occur if variations of a parameter cause the network to pass through a point of isochronicity. If the coupling is much weaker than dissipation, one can apply averaging to reduce the system to phase equations on a torus and study the bifurcations of the flow on the torus. It turns out that isochronicity can be responsible for a variety of subtle and unexpected bifurcation effects that depend crucially on the couplings. For example, for linear diffusive coupling the bifurcations are vertical, for linear diffusive coupling in the derivatives they are suppressed, and only when both types of coupling are present the bifurcations are generic at second order averaging. If coupling and dissipation are of the same order of magnitude, the asymptotic dynamics in terms of phases breaks down, and variation in the energies have to be included in the stability analysis. This is joint work with Peter Ashwin.