### Abstract

The Hopf bifurcation with D_{4} ⋉ T^{2} symmetry generically has open regions of the normal form coefficient space where all branches of periodic solutions bifurcate supercritically but none is stable [1]. In such regions we prove the existence of an attracting set near the origin. A new possibility for the attractor is a quasiperiodic solution branch related to Standing Cross Rolls (SCR). The new solution physically represents a planform we call Drifting Standing Cross Rolls. Unlike Standing Cross Rolls, this solution can be stable, as can a further triply-periodic solution. This explains the behaviour in the regions of coefficient space omitted by Silber and Knobloch [1] and completes their analysis.

Original language | English |
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Pages (from-to) | 158-165 |

Number of pages | 8 |

Journal | Physics Letters, Section A: General, Atomic and Solid State Physics |

Volume | 262 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - 1 Nov 1999 |

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### Keywords

- Quasiperiodic solutions
- Symmetric bifurcation theory

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

**Stable quasiperiodic solutions in the Hopf bifurcation with D _{4} ⋉ T^{2} symmetry.** / Dawes, J. H.P.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Stable quasiperiodic solutions in the Hopf bifurcation with D4 ⋉ T2 symmetry

AU - Dawes, J. H.P.

PY - 1999/11/1

Y1 - 1999/11/1

N2 - The Hopf bifurcation with D4 ⋉ T2 symmetry generically has open regions of the normal form coefficient space where all branches of periodic solutions bifurcate supercritically but none is stable [1]. In such regions we prove the existence of an attracting set near the origin. A new possibility for the attractor is a quasiperiodic solution branch related to Standing Cross Rolls (SCR). The new solution physically represents a planform we call Drifting Standing Cross Rolls. Unlike Standing Cross Rolls, this solution can be stable, as can a further triply-periodic solution. This explains the behaviour in the regions of coefficient space omitted by Silber and Knobloch [1] and completes their analysis.

AB - The Hopf bifurcation with D4 ⋉ T2 symmetry generically has open regions of the normal form coefficient space where all branches of periodic solutions bifurcate supercritically but none is stable [1]. In such regions we prove the existence of an attracting set near the origin. A new possibility for the attractor is a quasiperiodic solution branch related to Standing Cross Rolls (SCR). The new solution physically represents a planform we call Drifting Standing Cross Rolls. Unlike Standing Cross Rolls, this solution can be stable, as can a further triply-periodic solution. This explains the behaviour in the regions of coefficient space omitted by Silber and Knobloch [1] and completes their analysis.

KW - Quasiperiodic solutions

KW - Symmetric bifurcation theory

UR - http://www.scopus.com/inward/record.url?scp=0007348977&partnerID=8YFLogxK

U2 - 10.1016/S0375-9601(99)00653-2

DO - 10.1016/S0375-9601(99)00653-2

M3 - Article

AN - SCOPUS:0007348977

VL - 262

SP - 158

EP - 165

JO - Physics Letters A

JF - Physics Letters A

SN - 0375-9601

IS - 2-3

ER -