Transition from homogeneous to inhomogeneous steady states in oscillators under cyclic coupling

TL;DR

This paper investigates how coupled oscillators transition from homogeneous to inhomogeneous steady states under cyclic and diffusive coupling, revealing bifurcation types and effects of parameter mismatch in both limit cycle and chaotic systems.

Contribution

It uncovers the bifurcation mechanisms underlying state transitions in coupled oscillators, highlighting differences between cyclic and diffusive coupling, and extends analysis to chaotic systems.

Findings

Transition occurs via pitchfork bifurcation in limit cycle systems.

Transition follows transcritical bifurcation in chaotic systems under cyclic coupling.

Analytical and numerical verification of bifurcation types and state transitions.

Abstract

We report a transition from homogeneous steady state to inhomogeneous steady state in coupled oscillators, both limit cycle and chaotic, under cyclic coupling and diffusive coupling as well when an asymmetry is introduced in terms of a negative parameter mismatch. Such a transition appears in limit cycle systems via pitchfork bifurcation as usual. Especially, when we focus on chaotic systems, the transition follows a transcritical bifurcation for cyclic coupling while it is a pitchfork bifurcation for the conventional diffusive coupling. We use the paradigmatic Van der Pol oscillator as the limit cycle system and a Sprott system as a chaotic system. We verified our results analytically for cyclic coupling and numerically check all results including diffusive coupling for both the limit cycle and chaotic systems.