Hopf bifurcation on a square lattice

M. Silber*, E. Knobloch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

68 Scopus citations

Abstract

A complete classification of the generic D4*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*T2 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 C4. In normal form, the vector field on C 4 commutes with an S1 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*T2*S1.

Original languageEnglish (US)
Article number003
Pages (from-to)1063-1107
Number of pages45
JournalNonlinearity
Volume4
Issue number4
DOIs
StatePublished - 1991

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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