Stability of the 3-torus solution in a ring of coupled Duffing oscillators

L. Borkowski,A. Stefanski

doi:10.1140/epjst/e2020-900276-4

L. Borkowski, A. Stefanski

Open Access

https://doi.org/10.1140/epjst/e2020-900276-4

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Abstract

The dynamics of the ring of unidirectionally coupled single-well Duffing oscillators is analyzed in numerical simulation for identical nodal oscillators. The research is concentrated on the existence of the stable 3D torus attractor in this system. It is shown that 3-frequency quasi-periodicity can be robustly stable in wide range of parameters of the system under consideration. As an explanation of this stability, the conjecture on the coexistence and superposition of two independent effects characterized with irrational frequencies, i.e., the classical Newhouse, Ruelle and Takens scenario and rotating wave flow, is formulated.

Highlights

The European Physical Journal Special Topics in sets of coupled systems [12,15,16,17,18,19,23,24,25,26,27,28,29] and self or externally driven oscillators [30,31,32,33]
The existence of robustly stable 3D torus attractor in this system, which usually appears as a result of the transition from periodic (PRW) to chaotic rotating waves (CRW), is analyzed [27,28,29]
Revolutions of this circle demonstrate pictorially the PRW oscillating around the global equilibrium

Summary

Introduction

The European Physical Journal Special Topics in sets of coupled systems [12,15,16,17,18,19,23,24,25,26,27,28,29] and self or externally driven oscillators [30,31,32,33]. This article refers directly to our previous publication on the experimental observation of the 3-frequency quasiperiodic solution in a ring of seven coupled Duffing oscillators [9] and indirectly to other related works [37,38,39]. One of them refers to a property of spatio-temporal symmetry of the invariant ring of identical oscillators under the cyclic group [17,27]. Other one is connected with the rotational symmetry and the suppression of lockings in modulated rotating waves [40]. Another yet is based on some discrepancy in frequencies of the torus [29] diminishing an effect of frequency resonance which is destroying for the torus. As an explanation of this phenomenon, the conjecture on the coexistence and superposition of two independent effects characterized with irrational frequencies, i.e., the classical NRT scenario and rotating wave flow

Investigated system

System analysis

Destabilization of equilibrium and onset of rotating waves

Numerical simulations

Summary and conclusions