Two-community noisy Kuramoto model with general interaction strengths: Part II

TL;DR

This paper extends the analysis of the noisy Kuramoto model on two communities by allowing different intra- and inter-community interaction strengths, providing a complete classification of phase diagrams and bifurcation points.

Contribution

It introduces a generalized model with variable interaction strengths and fully characterizes its phase diagram and bifurcations using geometric and perturbation methods.

Findings

Complete phase diagram classification in four-dimensional parameter space

Identification of all bifurcation points and solution boundaries

Illustrative phase diagrams demonstrating complex behaviors

Abstract

We generalize the study of the noisy Kuramoto model, considered on a network of two interacting communities, to the case where the interaction strengths within and across communities are taken to be different in general. Using a geometric interpretation of the self-consistency equations developed in Part I of this series as well as perturbation arguments we are able to identify all solution boundaries in the phase diagram. This allows us to completely classify the phase diagram in the four dimensional parameter space and identify all possible bifurcation points. Furthermore, we analyze the asymptotic behavior of the solution boundaries. To illustrate these results and the rich behavior of the model we present phase diagrams for selected regions of the parameter space.