Normal forms, resonances, and meandering tip motions near relative equilibria of Euclidean group actions

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: g = ga(v); v = '(v) with g 2 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 '(v=0) = 0; in this general case. For the specific case of the Euclidean groups SE(N); the skew product takes the form R = Rr(v) S = Rs(v) v = '(v): with r(v) 2 SO(N); s(v) 2 IR 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 to 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 ..