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

J. H.P. Dawes

doi:10.1016/S0375-9601(99)00653-2

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

J. H.P. Dawes

University of Cambridge

Research output: Contribution to journal › Article › peer-review

5 Citations (SciVal)

Abstract

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.

Original language	English
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	https://doi.org/10.1016/S0375-9601(99)00653-2
Publication status	Published - 1 Nov 1999

Funding

I have benefited from useful discussions with Michael Proctor and Alastair Rucklidge. I also wish to thank two anonymous referees for a large number of helpful comments which have greatly improved the format and clarity of this paper. This work was funded by the EPSRC.

Keywords

Quasiperiodic solutions
Symmetric bifurcation theory

ASJC Scopus subject areas

General Physics and Astronomy

Access to Document

10.1016/S0375-9601(99)00653-2

Cite this

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abstract = "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.",

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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

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JF - Physics Letters, Section A: General, Atomic and Solid State Physics

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