Exact dynamical solution of the Kuramoto-Sakaguchi Model for finite networks of identical oscillators

TL;DR

This paper derives an exact analytical solution for the dynamics of finite networks of identical oscillators in the Kuramoto-Sakaguchi model, revealing complex attractor structures and bifurcations, and providing insights into synchronization phenomena.

Contribution

It provides the first exact solution for finite networks of identical oscillators in the KS model, including stability analysis and bifurcation insights.

Findings

Rich attractor and repeller structures identified.

Bifurcation analysis uncovers the repulsive regime.

Numerical simulations confirm analytical results.

Abstract

We study the Kuramoto-Sakaguchi (KS) model composed by any N identical phase oscillators symmetrically coupled. Ranging from local (one-to-one, R = 1) to global (all-to-all, R = N/2) couplings, we derive the general solution that describes the network dynamics next to an equilibrium. Therewith we build stability diagrams according to N and R bringing to the light a rich scenery of attractors, repellers, saddles, and non-hyperbolic equilibriums. Our result also uncovers the obscure repulsive regime of the KS model through bifurcation analysis. Moreover, we present numerical evolutions of the network showing the great accordance with our analytical one. The exact knowledge of the behavior close to equilibriums is a fundamental step to investigate phenomena about synchronization in networks. As an example, at the end we discuss the dynamics behind chimera states from the point of view of our results.