Nontrivial Twisted States in Nonlocally Coupled Stuart-Landau Oscillators

TL;DR

This paper introduces a novel nontrivial twisted state in nonlocally coupled Stuart-Landau oscillators, characterized by inhomogeneous amplitude and phase profiles, analyzed through stability, bifurcation, and robustness studies.

Contribution

It presents the discovery and detailed analysis of nontrivial twisted states, including their stability, bifurcations, and robustness in a system of coupled oscillators.

Findings

Identification of nontrivial twisted states with specific winding numbers.

Analysis of stability and bifurcations of these states.

Demonstration of robustness to heterogeneities.

Abstract

A twisted state is an important yet simple form of collective dynamics in an oscillatory medium. Here, we describe a nontrivial type of twisted state in a system of nonlocally coupled Stuart-Landau oscillators. The nontrivial twisted state (NTS) is a coherent traveling wave characterized by inhomogeneous profiles of amplitudes and phase gradients, which can be assigned a winding number. To further investigate its properties, several methods are employed. We perform a linear stability analysis in the continuum limit and compare the results with Lyapunov exponents obtained in a finite-size system. The determination of covariant Lyapunov vectors allows us to identify collective modes. Furthermore, we show that the NTS is robust to small heterogeneities in the natural frequencies and present a bifurcation analysis revealing that NTSs are born/annihilated in a saddle-node bifurcation and change their stability in Hopf bifurcations. We observe stable NTSs with winding number 1 and 2. The latter can lose stability in a supercritical Hopf bifurcation, leading to a modulated 2-NTS.