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

Abstract

We consider a heterogenous ensemble of dynamical systems in $\mathbb{R}^4$ 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 {\em discontinuous} transition to 
global inactivity (or {\em 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.