Quasi phase reduction of all-to-all strongly coupled $\lambda-\omega$ oscillators near incoherent states

TL;DR

This paper introduces quasi phase reduction (QPR), a novel method for simplifying the dynamics of strongly coupled oscillators near incoherent states, applicable to $ ext{lambda}- ext{omega}$ systems, and demonstrates its effectiveness with Stuart-Landau oscillators.

Contribution

The paper develops QPR, a new reduction technique for strongly coupled oscillators with polar symmetry, extending phase reduction methods to a broader class of systems.

Findings

QPR reduces the system to N+2 degrees of freedom near incoherent states.

Exact stability boundaries for incoherent states are obtained for Stuart-Landau oscillators.

QPR can be extended to include coupling through multiple harmonics.

Abstract

The dynamics of an ensemble of $N$ weakly coupled limit-cycle oscillators can be captured by their $N$ phases using standard phase reduction techniques. However, it is a phenomenological fact that all-to-all strongly coupled limit-cycle oscillators may behave as "quasiphase oscillators", evidencing the need of novel reduction strategies. We introduce here quasi phase reduction (QPR), a scheme suited for identical oscillators with polar symmetry ($\lambda-\omega$ systems). By applying QPR we achieve a reduction to $N+2$ degrees of freedom: $N$ phase oscillators interacting through one independent complex variable. This "quasi phase model" is asymptotically valid in the neighborhood of incoherent states, irrespective of the coupling strength. The effectiveness of QPR is illustrated in a particular case, an ensemble of Stuart-Landau oscillators, obtaining exact stability boundaries of uniform and nonuniform incoherent states for a variety of couplings. An extension of QPR beyond the neighborhood of incoherence is also explored. Finally, a general QPR model with $N+2M$ degrees of freedom is obtained for coupling through the first $M$ harmonics.