### Abstract

Techniques of equivariant bifurcation theory are used to study the Hopf bifurcation problem on a square lattice where the group Γ = D_{4} ⋉ T^{2} acts Γ-simply on ℂ^{8}. 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 ℂ^{4} 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 language | English |
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Pages (from-to) | 491-511 |

Number of pages | 21 |

Journal | Nonlinearity |

Volume | 14 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 May 2001 |

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### ASJC Scopus subject areas

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

### Cite this

*Nonlinearity*,

*14*(3), 491-511. https://doi.org/10.1088/0951-7715/14/3/304

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

Research output: Contribution to journal › Article

*Nonlinearity*, vol. 14, no. 3, pp. 491-511. https://doi.org/10.1088/0951-7715/14/3/304

}

TY - JOUR

T1 - Hopf bifurcation on a square superlattice

AU - Dawes, J. H P

PY - 2001/5/1

Y1 - 2001/5/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0035346890&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/14/3/304

DO - 10.1088/0951-7715/14/3/304

M3 - Article

VL - 14

SP - 491

EP - 511

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 3

ER -