Complex dynamics in the Hopf bifurcation with broken translation symmetry

doi:10.1016/0167-2789(95)00227-8

Physica D: Nonlinear Phenomena

Volume 90, Issues 1–2, 15 January 1996, Pages 56-78

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Abstract

The dynamics of the normal form for the O(2)-equivariant Hopf bifurcation perturbed by linear translation symmetry-breaking terms are investigated. A variety of stable small amplitude complex behavior is uncovered even for parameters for which the unperturbed system has no stable small amplitude solutions. In some cases this behavior can be attributed to the proximity to global bifurcations of Šil'nikov and Šil'nikov-Hopf type. In other cases cascades of gluing bifurcations involving various two-frequency states are present. Since the equations studied can be obtained by center manifold reduction of the coupled complex Ginzburg-Landau (CCGL) equations in a finite container this behavior must also be present in the CCGL equations. The theory may explain the origin of erratic reversals in traveling wave convection and in spiral vortex flow.

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Complex dynamics in the Hopf bifurcation with broken translation symmetry

Abstract

Phys. Lett. A

Physica D

Physica D

Physica D

Physica D

Physica D

Physica D

Physica D

Physica D

Flow regimes in a circular Couette system with independently rotating cylinders

J. Fluid Mech.

Symmetry and symmetry-breaking bifurcations in fluid dynamics

Ann. Rev. Fluid Mech.

Traveling and standing waves in binary-fluid convection in finite geometries

Phys. Rev. Lett.

On the Hopf bifurcation with broken O(2) symmetry

Dynamics of slowly varying wavetrains in finite geometry