Symmetry breaking and symmetry locking in equivariant circle maps
type: Item
instance of: scholarly article
title: Symmetry breaking and symmetry locking in equivariant circle maps
zbMATH Open document ID: 0761.58050
DOI: 10.1016/0167-2789(92)90011-B
author: M. Krupa
author: R. M. Roberts
published in: Physica D
publication date: 1993-01-16T00:00:00Z
review text: The families of \(Z_ k\)-equivariant circle maps, typically given by \[ f_ k(x,\omega,\mu)=x+\omega-(\mu/2\pi k)\sin 2\pi kx\bmod 1, \] which commute with actions of finite cyclic or dihedral groups \(G\), are considered as models for symmetric dynamical systems which undergo symmetry breaking and symmetry increasing bifurcations. A classification of generic bifurcations of periodic orbits for maps with \(D_ k\) and \(Z_ k\) symmetry is given. The authors analyze the phenomena of symmetry locking and symmetry restoration. So, two notions of symmetry locking are defined --- full and partial. Full symmetry locking occurs when the symmetry groups of the closure of all trajectories of a map are contained in some proper subgroup of \(G\). Partial symmetry locking occurs when the symmetry groups of all trajectories in some invariant open subset of the circle are contained in a proper subgroup of \(G\). It is proved that for a ``nice class'' of maps, the full symmetry locking is essentially equivalent to frequency locking, and that partial symmetry locking exists within periodic windows, that is regions of the parameter space where an iterate of the map has an invariant subinterval on which it has the structure of a unimodal map. Also, some results that suggest that symmetry restoration occurs at the end of periodic windows are given.
Mathematics Subject Classification ID: 37C80
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 94098
zbMATH Keywords: equivariant circle maps
zbMATH Keywords: symmetric dynamical systems
zbMATH Keywords: symmetry breaking
zbMATH Keywords: symmetry increasing bifurcations
zbMATH Keywords: symmetry locking
zbMATH Keywords: symmetry restoration
reviewed by: Liliana Răileanu
MaRDI profile type: Publication
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full work available at URL: 0167-2789(92)90011-b
OpenAlex ID: W2026224077
description: scientific article; zbMATH DE number 94098