Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms - Gouveia, Márcio R.A. [UNESP]; Llibre, Jaume; Novaes, Douglas D.; Pessoa, Claudio [UNESP] (Elsevier, 2016)
We consider an n-dimensional piecewise smooth vector field with two zones separated by a hyperplane σ which admits an invariant hyperplane Ω transversal to σ 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 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; Limit cycles; Lyapunov-Schmidt reduction; Normal forms; Piecewise differential system