Group synchrony, parameter mismatches, and intragroup connections

TL;DR

This paper investigates the stability of group synchronization in oscillator networks with parameter mismatches and intra-group connections, providing conditions for stability and quantifying synchronization errors.

Contribution

It extends existing stability analysis to include parameter mismatches and intra-group connections, offering a broader understanding of group synchronization robustness.

Findings

Stable group synchronization persists with small parameter deviations.

Derived necessary and sufficient stability conditions using master stability functions.

Extended stability analysis to networks with intra-group connections, reducing complexity.

Abstract

Group synchronization arises when two or more synchronization patterns coexist in a network formed of oscillators of different types, with the systems in each group synchronizing on the same time-evolution, but systems in different groups synchronizing on distinct time-evolutions. Group synchronization has been observed and characterized when the systems in each group are identical and the couplings between the systems satisfy specific conditions. By relaxing these constraints and allowing them to be satisfied in an approximate rather than exact way, we observe that stable group synchronization may still occur in the presence of small deviations of the parameters of the individual systems and of the couplings from their nominal values. We analyze this case and provide necessary and sufficient conditions for stability through a master stability function approach, {which also allows us to quantify the synchronization error}. We also investigate the stability of group synchronization in the presence of intra group connections and for this case, extend some of the existing results in the literature. Our analysis points out a broader class of matrices describing intra group connections for which the stability problem can be reduced in a low-dimensional form.