Bifurcations and patterns in the Kuramoto model with inertia

TL;DR

This paper investigates how bifurcations and pattern formations occur in the Kuramoto model with inertia on various graphs, revealing the influence of frequency distribution and network structure on stability and emergent behaviors.

Contribution

It identifies specific bifurcations in the Kuramoto model with inertia on complex networks, linking pattern emergence to frequency distribution and connectivity.

Findings

Identified pitchfork and Hopf bifurcations in the model.

Showed bifurcation type depends on frequency distribution and network.

Analyzed stability using Penrose diagrams and linear stability analysis.

Abstract

In this work, we analyze the Kuramoto model (KM) with inertia on a convergent family of graphs. It is assumed that the intrinsic frequencies of the individual oscillators are sampled from a probability distribution. In addition, a given graph, which may also be random, assigns network connectivity. As in the original KM, in the model with inertia, the weak coupling regime features mixing, the state of the network when the phases (but not velocities) of all oscillators are distributed uniformly around the unit circle. We study patterns, which emerge when mixing loses stability under the variation of the strength of coupling. We identify a pitchfork (PF) and an Andronov-Hopf (AH) bifurcations in the model with multimodal intrinsic frequency distributions. To this effect, we use a combination of the linear stability analysis and Penrose diagrams, a geometric technique for studying stability of mixing. We show that the type of a bifurcation and a nascent spatiotemporal pattern depend on the interplay of the qualitative properties of the intrinsic frequency distribution and network connectivity.