Bifurcation from relative equilibria of noncompact group actions: Skew products, meanders, and drifts

  • Bernold Fiedler

  • Björn Sandstede

  • Arnd Scheel

  • Claudia Wulff

Bifurcation from relative equilibria of noncompact group actions: Skew products, meanders, and drifts cover
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Abstract

We consider a finite-dimensional, typically noncompact Riemannian manifold MM with a differentiable proper action of a possibly non-compact Lie group G.G. We describe GG-equivariant flows in a tubular neighborhood UU of a relative equilibrium Gu0,u0MG\cdot u_0,u_0\in M, with compact isotropy HH of u0,u_0, by a skew product flow g˙=ga(v),v˙=φ(v).\dot{g} = g {\bf a}(v),\dot{v} = \varphi(v). Here gG,aalg(G).g\in G, {\bf a}\in {\rm alg}(G). The vector vv is in a linear slice VV to the group action. The induced local flow on G×VG\times V is equivariant under the action of (g0,h)G×H(g_0,h)\in G\times H on (g,v)G×V,(g,v)\in G\times V, given by (g0,h)(g,v)=(g0gh1,hv).(g_0,h)(g,v) = (g_0 gh^{-1},hv). The original flow on UU is equivalent to the induced flow on {id}×H\{id\}\times H-orbits in G×V.G\times V.

Applications to relative equivariant Hopf bifurcation in V are presented, clarifying phenomena like periodicity, meandering, and drifting. Specific illustrations involving Euclidean groups GG are meandering spirals, in the plane, and drifting twisted scroll rings, in three-dimensional Belousov-Zhabotinsky media.

Cite this article

Bernold Fiedler, Björn Sandstede, Arnd Scheel, Claudia Wulff, Bifurcation from relative equilibria of noncompact group actions: Skew products, meanders, and drifts. Doc. Math. 1 (1996), pp. 479–505

DOI 10.4171/DM/20