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Hopf bifurcation from rotating waves and patterns in physical space.

University of Houston Research Report UH/MD-242, 1998

* J. Nonlin. Sci * ** 10 ** (2000) 69-101

** Marty Golubitsky, Victor G. LeBlanc and Ian Melbourne **

** Abstract **

Hopf bifurcations from time periodic rotating waves to two frequency tori have
been studied for a number of years by a variety of authors including Rand
and Renardy. Rotating waves are solutions to partial differential equations
where time evolution is the same as spatial rotation. Thus rotating waves
can exist mathematically only in problems that have at least **SO**(2)
symmetry. In this paper we study the effect on this Hopf bifurcation
when the problem has more than **SO**(2) symmetry. These effects manifest
themselves in physical space and not in phase space. We use as motivating
examples the experiments of Gorman * et al.* on porous plug burner
flames,
of Swinney * et al.* on the Taylor-Couette system, and of a variety of
people on meandering spiral waves in the Belousov-Zhabotinsky reaction.
In our analysis we recover and complete Rand's classification of modulated
wavy vortices in the Taylor-Couette system.

It is both curious and intriguing that the spatial manifestations of the two
frequency motions in each of these experiments is different and it is these
differences that we seek to explain. In particular, we give a mathematical
explanation of the differences between the nonuniform rotation of cellular
flames in Gorman's experiments and the meandering of spiral waves in the
Belousov-Zhabotinsky reaction.

Our approach is based on the center bundle construction of Krupa with
compact group actions and its extension to noncompact group actions by
Sanstede, Scheel, and Wulff.

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