Hopf bifurcation with \(D_ 3\times D_ 3\)-symmetry
type: Item
instance of: scholarly article
title: Hopf bifurcation with \(D_ 3\times D_ 3\)-symmetry
zbMATH Open document ID: 0782.34041
DOI: 10.1007/BF00948479
author: Gerhard Dangelmayr
author: Werner Güttinger
author: Michael Wegelin
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1993-10-24T00:00:00Z
review text: The authors investigate a generic Hopf bifurcation of a hierarchical structure of three interacting sets of three oscillators. The system is assumed to be symmetric with respect to permutation of the three groups and of the individual oscillators. The Hopf bifurcation is investigated by computing isotropy subgroups and fixed point subspaces of the eight dimensional center eigenspace. The normal form admits 11 distinct periodic solutions with maximal isotropy subgroups; these are classified and their branching-types and stabilities are determined in terms of the cubic and relevant quintic coefficients of the normal form. The possibility of quasiperiodic solutions and heteroclinic cycles is discussed. The paper demonstrates that by simplifying the model to the ``macro- level'' the remaining symmetries from the ``micro-level'' may lead to hidden symmetries, which cannot be interpreted by observing only the macro-variables.
reviewed by: Alois Steindl
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C37
zbMATH DE Number: 431194
zbMATH Keywords: symmetry
zbMATH Keywords: generic Hopf bifurcation
zbMATH Keywords: hierarchical structure
zbMATH Keywords: three interacting sets of three oscillators
zbMATH Keywords: normal form
zbMATH Keywords: periodic solutions
zbMATH Keywords: isotropy subgroups
zbMATH Keywords: quasiperiodic solutions
zbMATH Keywords: heteroclinic cycles
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00948479
OpenAlex ID: W2914768962
description: scientific article; zbMATH DE number 431194