Abstract
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 \(\varepsilon \) 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 \(\varepsilon \rightarrow 0\) in the fast and slow time scales. This approach is applied to a number of examples.
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Acknowledgements
Pedro Toniol Cardin is supported by Grant 2019/00976-4, São Paulo Research Foundation (FAPESP). Marco Antonio Teixeira is partially supported by Grant 301275/2017-3, National Council for Scientific and Technological Development (CNPq).
We are grateful to the anonymous referees for valuable comments and suggestions.
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Cardin, P.T., Teixeira, M.A. Geometric Singular Perturbation Theory for Systems with Symmetry. J Dyn Diff Equat 34, 775–787 (2022). https://doi.org/10.1007/s10884-020-09855-2
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DOI: https://doi.org/10.1007/s10884-020-09855-2