Symmetry Invariance and Center Manifolds for Dynamical Systems

In this paper we analyze the role of general (possibly nonlinear) time-independent Lie point symmetries in the study of nite-dimensional autonomous dynamical systems, and their relationship with the presence of manifolds invariant under the dynamical ow. We rst show that stable and unstable manifolds are left invariant by all Lie point symmetries admitted by the dynamical system. An identical result cannot hold for the center manifolds, because they are in general not uniquely de ned. This nonuniqueness, and the possibility that Lie point symmetries map a center manifold into a di erent one, lead to some interesting features which we will discuss in detail. We can conclude that once the reduction of the dynamics to the center manifold has been performed the reduced problem automatically inherites a Lie point symmetry from the original problem: this permits to extend properties, well known in standard equivariant bifurcation theory, to the case of general Lie point symmetries; in particular, we can extend classical results, obtained by means of Lyapunov-Schmidt projection, to the case of bifurcation equations obtained by means of reduction to the center manifold. We also discuss the reduction of the dynamical system into normal form (in the sense of Poincar e-Birkho -Dulac) and respectively into the "Shoshitaishvili form" (in both cases one center manifold is given by a " at" manifold), and the relationship existing between nonuniqueness of center manifolds, perturbative expansions, and analyticity requirements. Finally, we present some examples which cover several aspects of the preceding discussion.