Geometric Singular Perturbation Theory for Systems with Symmetry

In this paper we focus on a class of symmetric vector fields in the context of singularly perturbed fast-slow dynamical systems. Our main question is to know how symmetry properties of a dynamical system are affected by singular perturbations. In addition, our approach uses tools in geometric singular perturbation theory [8], which address the persistence of normally hyperbolic compact manifolds. We analyse the persistence of such symmetry properties when the singular perturbation parameter ε is positive and small enough, and study the existing relations between symmetries of the singularly perturbed system and symmetries of the limiting systems, which are obtained from the limit ε→ 0 in the fast and slow time scales. This approach is applied to a number of examples.