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Branches of stable three-tori using Hamiltonian methods in hopf bifurcation on a rhombic lattice

Kirk, Vivien and Marsden, Jerrold E. and Silber, Mary (1996) Branches of stable three-tori using Hamiltonian methods in hopf bifurcation on a rhombic lattice. Dynamics and Stability of Systems, 11 (4). pp. 267-302. ISSN 1465-3389. doi:10.1080/02681119608806228.

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This paper uses Hamiltonian methods to find and determine the stability of some new solution branches for an equivanant Hopf bifurcation on C^4. The normal form has a symmetry group given by the semi-direct product of D_2 with T^2 S^1. The Hamiltonian part of the normal form is completely integrable and may be analyzed using a system of invariants. The idea of the paper is to perturb relative equilibria in this singular Hamiltonian limit to obtain new three-frequency solutions to the full normal form for parameter values near the Hamiltonian limit. The solutions obtained have fully broken symmetry, that is, they do not lie in fixed point subspaces. The methods developed in this paper allow one to determine the stability of this new branch of solutions. An example shows that the branch of three-tori can be stable.

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Additional Information:© 1996. We have benefited from discussions with George Haller. The research of VK was supported by AURC grant A18/63090/F342908 and by DOE Contract DE-FG0395-ER25251. The research of JM was partially supported by NSF grant DMS-9302992 and by DOE Contract DE-FG0395-ER25251. The research of MS was supported by NSF grants DMS-94101l5 and DMS-9404266, and by an NSF CAREER award DMS-9502266.
Funding AgencyGrant Number
Department of EnergyDE-FG0395-ER25251
Department of EnergyDE-FG0395-ER25251
Issue or Number:4
Record Number:CaltechAUTHORS:20100819-150609400
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19532
Deposited By: Ruth Sustaita
Deposited On:19 Aug 2010 22:32
Last Modified:08 Nov 2021 23:53

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