Interplay of the mechanisms of synchronization by common noise and global coupling for a general class of limit-cycle oscillators

TL;DR

This paper develops an analytical framework to understand how common noise and global coupling influence synchronization in limit-cycle oscillators, considering both identical and nonidentical cases with various coupling types.

Contribution

It introduces a comprehensive phase reduction theory that accounts for amplitude effects, analyzing synchronization mechanisms and frequency behaviors in diverse oscillator models.

Findings

Synchronization conditions derived for identical oscillators with intrinsic noise.

Frequency repulsion observed under negative coupling in nonidentical oscillators.

Analytical results validated through numerical simulations of Van der Pol, Duffing, and FitzHugh--Nagumo systems.

Abstract

We construct an analytical theory of interplay between synchronizing effects by common noise and by global coupling for a general class of smooth limit-cycle oscillators. Both the cases of attractive and repulsive coupling are considered. The derivation is performed within the framework of the phase reduction, which fully accounts for the amplitude degrees of freedom. Firstly, we consider the case of identical oscillators subject to intrinsic noise, obtain the synchronization condition, and find that the distribution of phase deviations always possesses lower-law heavy tails. Secondly, we consider the case of nonidentical oscillators. For the average oscillator frequency as a function of the natural frequency mismatch, limiting scaling laws are derived; these laws exhibit the nontrivial phenomenon of frequency repulsion accompanying synchronization under negative coupling. The analytical theory is illustrated with examples of Van der Pol and Van der Pol--Duffing oscillators and the neuron-like FitzHugh--Nagumo system; the results are also underpinned by the direct numerical simulation for ensembles of these oscillators.