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

Abstract

We consider a finite-dimensional, typically noncompact Riemannian manifold $M$ with a differentiable proper action of a possibly non-compact Lie group $G.$ We describe $G$-equivariant flows in a tubular neighborhood $U$ of a relative equilibrium $G\cdot u_0,u_0\in M$, with compact isotropy $H$ of $u_0,$ by a skew product flow $\dot{g} = g {\bf a}(v),\dot{v} = \varphi(v).$ Here $g\in G, {\bf a}\in {\rm alg}(G).$ The vector $v$ is in a linear slice $V$ to the group action. The induced local flow on $G\times V$ is equivariant under the action of $(g_0,h)\in G\times H$ on $(g,v)\in G\times V,$ given by $(g_0,h)(g,v) = (g_0 gh^{-1},hv).$ The original flow on $U$ is equivalent to the induced flow on $\{id\}\times H$-orbits in $G\times V.$

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