Hopf-steady-state mode interactions with 0(2) symmetry
Title:
Hopf-steady-state mode interactions with 0(2) symmetry
Author:
Hill, Andrew Stewart, Ian
Appeared in:
Dynamical systems
Paging:
Volume 6 (1991) nr. 2 pages 149-171
Year:
1991
Contents:
Bifurcations in systems of ordinary or partial differential equations often involve nonlinear interactions between two or more modes, and such mode interactions have been widely studied. Here we fill a gap in the literature by considering the interaction of a steady-state bifurcation and a Hopf bifurcation in a system with 0(2) (circular) symmetry. One such interaction has been analysed previously, m the context of Taylor-Couette flow of a fluid confined between coaxial rotating cylinders, but only when both mode numbers are 1. Here we consider arbitrary mode numbers (l, m) under the assumption that both modes correspond to double eigenvalues of the jacobian. We describe the isotropy lattice, which governs the possible patterns of symmetry breaking, and find the general form of the reduced bifurcation equations by computing the 0(2)-equivariant mappings. We use this information to determine the local branching equations and stability coefficients, and solve the branching equations for non-degenerate systems. We observe that the case when l is even and m = 1 is exceptional, and analyse the simplest example: (l, m) = (2, 1). In this exceptional case the standing-wave branch associated with the Hopf bifurcation fails to exist at the mode-interaction point
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