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) over dot = ga(nu), (nu) over dot = phi(nu) with g i s an element of G, with nu in a slice transverse to the group action, and a (nu) in the Lie algebra of G. We present a normal form theory near relative equilibria phi(nu=0) = 0, in this general case. For the specific case of t he Euclidean groups SE(N), the skew product takes the form (R) over dot = Rr(nu), (S) over dot = Rs(nu), (nu) over dot = phi(nu) with r (nu) is an element of SO(N), s(nu) is an element of R-N. We give a p recise 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 meande r radii and drift resonance in the plane N = 2. For illustration, we discus s the unbounded tip motions associated with a weak focus in nu, on the verg e of Hopf bifurcation, in the case of resonant Hopf and rotation frequencie s of the spiral, and study resonant relative Hopf bifurcation. We also enco unter random Brownian tip motions for trajectories nu(t) --> Gamma, which b ecome homoclinic for t --> +infinity. We conclude with some comments on the homoclinic tip shifts and drift resonance velocities in the Bogdanov-Taken s bifurcation, which turn out to be small beyond any finite order.