Abstract

Techniques of equivariant bifurcation theory are used to study the Hopf bifurcation problem on a square lattice where the group Γ = D4 ⋉ T2 acts Γ-simply on ℂ8. This enables the analysis of the stability of solutions found in a previous analysis (Silber and Knobloch 1991 Nonlinearity 4 1063-106) of the Γ-simple representation on ℂ4 to both solutions which are spatially periodic on a rhombic lattice, and to a countably infinite number of oscillatory 'superlattice' solutions. The normal form for the bifurcation is computed, and conditions for the stability of all 17 ℂ-axial branches are given. Numerical investigations indicate that there exist open regions of coefficient space where the dynamics of the cubic order truncation of the normal form are chaotic. Chaotic dynamics have not previously been found in simpler Hopf bifurcation problems in normal form.

Original languageEnglish
Pages (from-to)491-511
Number of pages21
JournalNonlinearity
Volume14
Issue number3
DOIs
Publication statusPublished - 1 May 2001

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Hopf bifurcation
Superlattices
Hopf Bifurcation
Normal Form
Bifurcation (mathematics)
Bifurcation Theory
Stability of Solutions
nonlinearity
Chaotic Dynamics
Numerical Investigation
Square Lattice
Truncation
Equivariant
Branch
coefficients
Bifurcation
approximation
Nonlinearity
Coefficient

ASJC Scopus subject areas

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

Cite this

Hopf bifurcation on a square superlattice. / Dawes, J. H P.

In: Nonlinearity, Vol. 14, No. 3, 01.05.2001, p. 491-511.

Research output: Contribution to journalArticle

Dawes, J. H P. / Hopf bifurcation on a square superlattice. In: Nonlinearity. 2001 ; Vol. 14, No. 3. pp. 491-511.
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