- 93-298 Cicogna G., Gaeta G.
- Symmetry Invariance and Center Manifolds for Dynamical Systems
Nov 16, 93
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Abstract. In this paper we analyze the role of general (possibly nonlinear)
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 center manifolds,
because they are in general not uniquely defined.
This nonuniqueness, and the possibility that Lie point
symmetries map a center 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 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-Birkhoff-Dulac) and
respectively into the "Shoshitaishvili form" (in both cases one
center manifold is given by a "flat" 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.