Stasis in heterogeneous networks of coupled oscillators: discontinuous transition with hysteresis

We consider a heterogeneous ensemble of dynamical systems in R4 that individually are either attracted to fixed points (and are termed inactive) or to limit cycles (in which case they are termed active). These distinct states are separated by bifurcations that are controlled by a single parameter. Upon coupling them globally, we find a discontinuous transition to global inactivity (or stasis) when the proportion of inactive components in the ensemble exceeds a threshold: there is a first–order phase transition from a globally oscillatory state to global oscillation death. There is hysteresis associated with these phase transitions. Numerical results for a representative system are supported by analysis using a system-reduction technique and different dynamical regimes can be rationalised through the corresponding bifurcation diagrams of the reduced set of equations.