Symmetry invariance and centre manifolds for dynamical systems

Summary

In this paper we analyse the role of general (possibly non-linear) time-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 stable and unstable manifolds are left invariant by all Lie point symmetries admitted by the dynamical system. An identical result cannot hold for the centre manifolds, because they are in general not uniquely defined. This non-uniqueness and the possibility that Lie point symmetries map a centre manifold into a different one, lead to some interesting features which we will discuss in detail. We can conclude that—once the reduction of the dynamics to the centre 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 the Lyapunov-Schmidt projection, to the case of bifurcation equations obtained by means of reduction to the centre manifold. We also discuss the reduction of the dynamical system into normal form (in the sense of Poincaré-Birkhoff-Dulac) and respectively into the «Shoshitaishvili form» (in both cases one centre manifold is given by a «flat» manifold), and the relationship existing between non-uniqueness of centre manifolds, perturbative expansions, and analyticity requirements. Finally, we present some examples which cover several aspects of the preceding discussion.