Bifurcation from Relative Equilibria of Noncompact Group Actions: Skew Products, Meanders, and Drift

We consider a nite-dimensional, typically noncompact Riemannian manifold M with a di erentiable proper action of a possibly noncompact Lie group G: We describe G-equivariant ows in a tubular neighborhood U of a relative equilibrium G u 0 ; u 0 2M , with compact isotropy H of u 0 ; by a skew product ow _ g = ga(v); _ v = '(v): Here g 2 G; a 2 alg(G): The vector v is in a linear slice V to the group action. The induced local ow on G V is equivariant under the action of (g 0 ; h) 2 G H on (g; v) 2 G V; given by (g 0 ; h)(g; v) = (g 0 gh 1 ; hv): The original ow on U is equivalent to the induced ow on fidg H-orbits in G V: Applications to relative equivariant Hopf bifurcation in V are presented, clarifying phenomena like periodicity, meandering, and drifting. Speci c illustrations involving Euclidean groups G are meandering spirals, in the plane, and drifting twisted scroll rings, in three-dimensional Belousov-Zhabotinsky media. 1991 Mathematics Subject Classi cation: 58F35, 57S20, 55R91

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