Codimension-two Bifurcations Induce Hysteresis Behavior and Multistabilities in Delay-coupled Kuramoto Oscillators

TL;DR

This paper analyzes how codimension-two bifurcations cause hysteresis and multistability in delay-coupled Kuramoto oscillators, providing a comprehensive bifurcation analysis and identifying critical points affecting system dynamics.

Contribution

It rigorously derives bifurcation sets and identifies Bautin bifurcations in the delay-coupled Kuramoto model, revealing complex multistability phenomena.

Findings

Existence of saddle-node bifurcations inducing hysteresis

Coexistence of multiple stable and unstable states near critical points

Detailed bifurcation scenarios for synchronization transitions

Abstract

Hysteresis phenomena and multistability play crucial roles in the dynamics of coupled oscillators, which are now interpreted from the point of view of codimension-two bifurcations. On the Ott-Antonsen's manifold, complete bifurcation sets of delay-coupled Kuramoto model are derived regarding coupling strength and delay as bifurcation parameters. It is rigorously proved that the system must undergo Bautin bifurcations for some critical values, thus there always exists saddle-node bifurcation of periodic solutions inducing hysteresis loop. With the aid of center manifold reduction method and the Matlab Package DDE-Biftool, the location of Bautin and double Hopf points and detailed dynamics are theoretically determined. We find that, near these critical points, at most four coherent states (two of which are stable) and a stable incoherent state may coexist, and that the system undergoes Neimark-Sacker bifurcation of periodic solutions. Finally, the clear scenarios about the synchronous transition in delayed Kuramoto model are depicted.