##
Meandering of the spiral tip: an alternative approach

* J. Nonlin. Sci.* **7** (1997) 557-586

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

** Abstract **

Meandering of a one-armed spiral tip has been noted in chemical
reactions and numerical simulations. Barkley, Kness and Tuckerman show
that meandering can begin by Hopf bifurcation from a rigidly rotating
spiral wave (a point that is verified in a B-Z reaction by Li, Ouyang,
Petrov and Swinney). At the codimension two point where (in an
appropriate sense) the frequency at Hopf bifurcation equals the
frequency of the spiral wave, Barkley notes that spiral tip meandering
can turn to linearly translating spiral tip motion.

Barkley also presents a model showing that the linear motion of the
spiral tip is a resonance phenomenon, and this point is verified
experimentally by Li * et al* and proved rigorously by Wulff. In
this paper we suggest an alternative development of Barkley's model
extending the center bundle constructions of Krupa from compact groups
to noncompact groups and from finite dimensions to function spaces. This
approach allows us to consider various bifurcations from a rotating
wave. In particular, we can analyze in a straightforward manner the
codimension two Barkley bifurcation and the codimension two
Takens-Bogdanov bifurcation from a rotating wave. We also discuss Hopf
bifurcation from a many armed spiral showing that meandering and
resonant linear motion of the spiral tip do * not* always occur.

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