Normal Forms, Resonances, and Meandering Tip Motions near Relative Equilibria of Euclidean Group Actions

Abstract

The equivariant dynamics near relative equilibria to actions of noncompact, finite‐dimensional Lie groups G can be described by a skew‐product flow on a center manifold: $\dot{g} = g{\textbf a}(v), \dot{v} = \varphi (v)$ with $g\in G$ , with v in a slice transverse to the group action, and a(v) in the Lie algebra of G. We present a normal form theory near relative equilibria $\varphi(v$=$0)=0,$ in this general case. For the specific case of the Euclidean groups $SE(N),$ the skew product takes the form $$\dot{R} = R {\textbf r}(v),\qquad \dot{S} = R {\textbf s}(v),\qquad \dot{v} = \varphi (v)$$ with ${\textbf r}(v)\in SO(N),\;{\textbf s}(v)\in\mathr^N$ . We give a precise meaning to the intuitive idea of tip motion of a meandering spiral: it corresponds to the dynamics of $S(t)$ . This clarifies the notion of meander radii and drift resonance in the plane $N=2$ . For illustration, we discuss the unbounded tip motions associated with a weak focus in v, on the verge of Hopf bifurcation, in the case of resonant Hopf and rotation frequencies of the spiral, and study resonant relative Hopf bifurcation. We also encounter random Brownian tip motions for trajectories $v(t)\rightarrow \Gamma,$ which become homoclinic for $t\rightarrow +\infty$ . We conclude with some comments on the homoclinic tip shifts and drift resonance velocities in the Bogdanov‐Takens bifurcation, which turn out to be small beyond any finite order.