Partial synchronization phenomena in networks of identical oscillators with non-linear coupling

TL;DR

This paper investigates a nonlinear Kuramoto-like model of identical oscillators on networks, revealing conditions for synchronization and exploring diverse partially synchronous states through analytical and numerical methods.

Contribution

It introduces a nonlinear coupling mechanism that balances attractive and repulsive interactions, analyzing stability and dynamical regimes beyond traditional synchronization.

Findings

Full synchronization is stable when repulsion is weak compared to the maximum hub degree.

Partial synchronization regimes include stationary, multistable, periodic, and chaotic states.

A measure is proposed to predict the prevalence of stationary versus time-dependent states across network types.

Abstract

We study a Kuramoto-like model of coupled identical phase oscillators on a network, where attractive and repulsive couplings are balanced dynamically due to nonlinearity in interaction. Under a week force, an oscillator tends to follow the phase of its neighbors, but if an oscillator is compelled to follow its peers by a sufficient large number of cohesive neighbors, then it actually starts to act in the opposite manner, i.e. in anti-phase with the majority. Analytic results yield that if the repulsion parameter is small enough in comparison with the degree of the maximum hub, then the full synchronization state is locally stable. Numerical experiments are performed to explore the model beyond this threshold, where the overall cohesion is lost. We report in detail partially synchronous dynamical regimes, like stationary phase-locking, multistability, periodic and chaotic states. Via statistical analysis of different network organizations like tree, scale-free, and random ones, and different network sizes, we found a measure allowing one to predict relative abundance of partially synchronous stationary states in comparison to time-dependent ones.