SELF-SYNCHRONIZATION OF COUPLED OSCILLATORS WITH HYSTERETIC RESPONSES

We analyze a large system of nonlinear phase oscillators with sinusoid al nonlinearity, uniformly distributed natural frequen cies and global all-to-all coupling, which is an extension of Kuramoto's model to sec ond-order systems. For small coupling, the system evolves to an incohe rent state with the phases of all the oscillators distributed uniforml y. As the coupling is increased, the system exhibits a discontinuous t ransition to the coherently synchronized state at a pinning threshold of the coupling strength, or to a partially synchronized oscillation c oherent state at a certain threshold below the pinning threshold. if t he coupling is decreased from a strong coupling with all the oscillato rs synchronized coherently, this coherence can persist until the depin ning threshold which is less than the pinning threshold, resulting in hysteretic synchrony depending on the initial configuration of the osc illators. We obtain analytically both the pinning and depinning thresh old and also explain the discontinuous transition at the thresholds fo r the underdamped case in the large system size limit. Numerical explo ration shows the oscillatory partially coherent state bifurcates at th e depinning threshold and also suggests that this state persists indep endent of the system size. The system studied here provides a simple m odel for collective behaviour in damped driven high-dimensional Hamilt onian systems which can explain the synchronous firing of certain fire flies or neural oscillators with frequency adaptation and may also be applicable to interconnected power systems.