## Abstract

A complete classification of the generic D_{4}*T ^{2}-equivariant Hopf bifurcation problems is presented. This bifurcation arises naturally in the study of extended systems, invariant under the Euclidean group E(2), when a spatially uniform quiescent state loses stability to waves of wavenumber k not=0 and frequency omega not=0. The D _{4}*T^{2} symmetry group applies when periodic boundary conditions are imposed in two orthogonal horizontal directions. The centre manifold theorem allows a reduction of the infinite dimensional problem to a bifurcation problem on C^{4}. In normal form, the vector field on C ^{4} commutes with an S^{1} symmetry, which is interpreted as a time translation symmetry. The spatial and spatio-temporal symmetries of all possible solutions are classified in terms of isotropy subgroups of D _{4}*T^{2}*S^{1}.

Original language | English (US) |
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Article number | 003 |

Pages (from-to) | 1063-1107 |

Number of pages | 45 |

Journal | Nonlinearity |

Volume | 4 |

Issue number | 4 |

DOIs | |

State | Published - 1991 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics