Unfolding the torus: Oscillator geometry from time delays
type: Item
instance of: scholarly article
title: Unfolding the torus: Oscillator geometry from time delays
zbMATH Open document ID: 0804.34036
DOI: 10.1007/BF02429874
published in: Journal of Nonlinear Science
publication date: 1995-01-19T00:00:00Z
review text: A method of plotting the trajectories of systems of weakly coupled oscillators is presented. The considered system of \(n\) weakly coupled oscillators has an attracting invariant \(n\)-dimensional torus \(T^ n\) and the attractor as its subset. The flow on \(T^ n\) admits a suitable global section which is an \((n-1)\)-dimensional torus \(T^{n-1}\). Thus the dynamics of \(n\) coupled oscillators is reduced to the study of the Poincaré map \(P: T^{n-1}\to T^{n-1}\), which may have a very complicated, e.g. chaotic, behavior of orbits. Even for three oscillators, a standard projection of \(T^ 2\) onto \(\mathbb{R}^ 2\) is not injective. This makes it difficult to observe the structure of the attractor. An algorithm suggested by the authors allows a direct measurement of angles on \(T^ 2\) providing a possibility of 1-to-1 projection of \(P\) onto the ``unfolded torus''. In the case where the attractor is of knot type, its knot type is easily seen in the projection. Numerical and electronic implementations of the suggested algorithm along with a numerical study of the attractor are given.
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 34A26
Mathematics Subject Classification ID: 65J99
Mathematics Subject Classification ID: 34D45
zbMATH DE Number: 490597
zbMATH Keywords: unfolded torus
zbMATH Keywords: numerical and electronic implementations
zbMATH Keywords: weakly coupled oscillators
zbMATH Keywords: attractor
zbMATH Keywords: Poincaré map
zbMATH Keywords: chaotic
zbMATH Keywords: algorithm
zbMATH Keywords: knot type
zbMATH Keywords: numerical study
zbMATH Keywords: behavior
Wikidata QID: Q57920032
author: Peter Ashwin
author: James W. Swift
reviewed by: Eugene Ershov
MaRDI profile type: Publication
cites work: Three identical oscillators with symmetric coupling
cites work: The dynamics of \(n\) weakly coupled identical oscillators
cites work: Three coupled oscillators: Mode-locking, global bifurcations and toroidal chaos
cites work: Three-frequency quasiperiodicity, phase locking, and the onset of chaos
cites work: Q3658782
description: scientific article; zbMATH DE number 490597