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

Kirk, V., J. E. Marsden and M. Silber

Dynamics and Stability of Systems, 11, (1996), 267-302

Abstract:

This Paper uses Hamiltonian methods to find and determine the stability of some new solution branches for an equivariant Hopf bifurcation on $\mathbb {C}$⁴ . The normal form has a symmetry group given by the semi-direct product of D₂ with T² x S¹ . 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 f this new branch of solutions. An example shows that the branch of three-tori can be stable.