Meandering of the Spiral Tip: An Alternative Approach
J. Nonlin. Sci. 7 No. 6 (1997) 557-586.
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.
The difficulty with our methods is the inability to give computable criteria to verify that the abstract bifurcations analyzed in this paper actually occur.