Diversity of dynamical behaviors due to initial conditions: Extension of the Ott-Antonsen ansatz for identical Kuramoto-Sakaguchi phase oscillators

Exact finite-dimensional reduction for a population of noisy oscillators and its link to Ott-Antonsen and Watanabe-Strogatz theories

Populations of globally coupled phase oscillators are described in the thermodynamic limit by kinetic equations for the distribution densities or, equivalently, by infinite hierarchies of equations for the order parameters. Ott and Antonsen [Chaos 18, 037113 (2008)] have found an invariant finite-dimensional subspace on which the dynamics is described by one complex variable per population. For oscillators with Cauchy distributed frequencies or for those driven by Cauchy white noise, this subspace is weakly stable and, thus, describes the asymptotic dynamics. Here, we report on an exact finite-dimensional reduction of the dynamics outside of the Ott-Antonsen subspace. We show that the evolution from generic initial states can be reduced to that of three complex variables, plus a constant function. For identical noise-free oscillators, this reduction corresponds to the Watanabe-Strogatz system of equations [Watanabe and Strogatz, Phys. Rev. Lett. 70, 2391 (1993)]. We discuss how the reduced system can be used to explore the transient dynamics of perturbed ensembles.

Dynamics of the generalized Kuramoto model with nonlinear coupling: Bifurcation and stability

We systematically study dynamics of a generalized Kuramoto model of globally coupled phase oscillators. The coupling of modified model depends on the fraction of phase-locked oscillators via a power-law function of the Kuramoto order parameter r through an exponent α, such that α=1 corresponds to the standard Kuramoto model, α<1 strengthens the global coupling, and the global coupling is weakened if α>1. With a self-consistency approach, we demonstrate that bifurcation diagrams of synchronization for different values of α are thoroughly constructed from two parametric equations. In contrast to the case of α=1 with a typical second-order phase transition to synchronization, no phase transition to synchronization is predicted for α<1, as the onset of partial locking takes place once the coupling strength K>0. For α>1, we establish an abrupt desynchronization transition from the partially (fully) locked state to the incoherent state, whereas there is no counterpart of abrupt synchronization transition from incoherence to coherence due to that the incoherent state remains linearly neutrally stable for all K>0. For each case of α, by performing a standard linear stability analysis for the reduced system with Ott-Antonsen ansatz, we analytically derive the continuous and discrete spectra of both the incoherent state and the partially (fully) locked states. All our theoretical results are obtained in the thermodynamic limit, which have been well validated by extensive numerical simulations of the phase-model with a sufficiently large number of oscillators.