# Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms

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1 UNESP - Departamento de matemàtica-IBILCE 2 UAB - Universitat Autònoma de Barcelona Barcelona 3 UNICAMP - Universidade Estadual de Campinas

Abstract : We consider an $n$-dimensional piecewise smooth vector field with two zones separated by a hyperplane $\Sigma$ which admits an invariant hyperplane $\Omega$ transversal to $\Sigma$ containing a period annulus A fulfilled by crossing periodic solutions. For small discontinuous perturbations of these systems we develop a Melnikov-like function to control the persistence of periodic solutions contained in $\mathcal{A}$. When $n = 3$, we provide normal forms for the piecewise linear case. Finally we apply the Melnikov-like function to study discontinuous perturbations of the given normal forms.

Keywords : Crossing periodic solutions Piecewise differential system Normal forms Lyapunov–Schmidt reduction Limit cycles

Autor: Marcio Gouveia - Jaume Llibre - Douglas Novaes - Claudio Pessoa -

Fuente: https://hal.archives-ouvertes.fr/

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