The dynamical impact of a shortcut in unidirectionally coupled rings of oscillators

TL;DR

This paper investigates how adding a long-range connection affects the stability and bifurcation behavior of unidirectional rings of oscillators, revealing persistent destabilization mechanisms and the emergence of distinct periodic solution groups.

Contribution

It demonstrates that the destabilization mechanism in unidirectional oscillator rings persists under small non-local perturbations and characterizes the effects of strong shortcuts on solution stability.

Findings

Destabilization persists under small non-local perturbations.

Resonance conditions modulate the Eckhaus line.

Strong shortcuts split solutions into unstable and Eckhaus-like groups.

Abstract

We study the destabilization mechanism in a unidirectional ring of identical oscillators, perturbed by the introduction of a long-range connection. It is known that for a homogeneous, unidirectional ring of identical Stuart-Landau oscillators the trivial equilibrium undergoes a sequence of Hopf bifurcations eventually leading to the coexistence of multiple stable periodic states resembling the Eckhaus scenario. We show that this destabilization scenario persists under small non-local perturbations. In this case, the Eckhaus line is modulated according to certain resonance conditions. In the case when the shortcut is strong, we show that the coexisting periodic solutions split up into two groups. The first group consists of orbits which are unstable for all parameter values, while the other one shows the classical Eckhaus behavior.