Fixed points and stability in the two-network frustrated Kuramoto model

TL;DR

This paper analyzes the fixed points and stability of a modified two-network Kuramoto model with competing populations, revealing conditions for synchronization, fragmentation, and instability through analytical and numerical methods.

Contribution

It introduces a two-network Kuramoto model with frustration parameters and provides a fixed point analysis for different synchronization regimes, advancing understanding of competing oscillator populations.

Findings

Identified conditions for phase synchronization and fragmentation.

Demonstrated stability and instability regimes through analytical and numerical methods.

Revealed how frustration parameters influence system dynamics.

Abstract

We examine a modification of the Kuramoto model for phase oscillators coupled on a network. Here, two populations of oscillators are considered, each with different network topologies, internal and cross-network couplings and frequencies. Additionally, frustration parameters for the interactions of the cross-network phases are introduced. This may be regarded as a model of competing populations: internal to any one network phase synchronisation is a target state, while externally one or both populations seek to frequency synchronise to a phase in relation to the competitor. We conduct fixed point analyses for two regimes: one, where internal phase synchronisation occurs for each population with the potential for instability in the phase of one population in relation to the other; the second where one part of a population remains fixed in phase in relation to the other population, but where instability may occur within the first population leading to `fragmentation'. We compare analytic results to numerical solutions for the system at various critical thresholds.