Heteroclinic and homoclinic structures in the system of four identical globally coupled phase oscillators with nonpairwise interactions of phases

TL;DR

This paper investigates the complex dynamics of four identical globally coupled phase oscillators with nonpairwise phase interactions, focusing on heteroclinic and homoclinic structures that influence chaotic behavior.

Contribution

It identifies which heteroclinic cycles are supported or forbidden in this specific oscillator system with nonpairwise interactions.

Findings

Certain heteroclinic cycles are supported by the system.

Some heteroclinic cycles are forbidden in this configuration.

Homoclinic trajectories to saddle-foci are analyzed.

Abstract

Systems of $N$ identical globally coupled phase oscillators can demonstrate a multitude of complex behaviours. Such systems can have chaotic dynamics for $N>4$ when a coupling function is biharmonic. The case $N = 4$ does not possess chaotic attractors when the coupling is biharmonic, but has them when the coupling includes nonpairwise interactions of phases. Previous studies showed that some of chaotic attractors in this system are organized by heteroclinic networks. In present paper we discuss which heteroclinic cycles are forbidden and which are supported by this particular system. We also discuss some of the cases regarding homoclinic trajectories to saddle-foci equilibria.