SYMMETRY INVARIANCE AND CENTER MANIFOLDS FOR DYNAMICAL-SYSTEMS

In this paper we analyse the role of general (possibly non-linear) tim e-independent Lie point symmetries in the study of finite-dimensional autonomous dynamical systems, and their relationship with the presence of manifolds invariant under the dynamical flow. We first show that s table and unstable manifolds are left invariant by all Lie point symme tries admitted by the dynamical system. An identical result cannot hol d for the centre manifolds, because they are in general not uniquely d efined. This non-uniqueness, and the possibility that Lie point symmet ries map a centre manifold into a different one, lead to some interest ing features which we will discuss in detail. We can conclude that-onc e the reduction of the dynamics to the centre manifold has been perfor med-the reduced problem automatically inherites a Lie point symmetry f rom the original problem: this permits to extend properties, well know n in standard equivariant bifurcation theory, to the case of general L ie point symmetries; in particular, we can extend classical results, o btained by means of the Lyapunov-Schmidt projection, to the case of bi furcation equations obtained by means of reduction to the centre manif old. We also discuss the reduction of the dynamical system into normal form (in the sense of Poincare-Birkhoff-Dulac) and respectively into the <<Shoshitaishvili form>> (in both cases one centre manifold is giv en by a <<flat>> manifold), and the relationship existing between non- uniqueness of centre manifolds, perturbative expansions, and analytici ty requirements. Finally, we present some examples which cover several aspects of the preceding discussion.