Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation

• Autor(a) principal: da Silva, P. R. [UNESP]
• Data de Publicação: 2020
• Outros Autores: Meza-Sarmiento, I. S. [UNESP], Novaes, D. D.
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Institucional da UNESP
• Texto Completo: http://dx.doi.org/10.1007/s12591-018-0439-1; http://hdl.handle.net/11449/200147
• topic: Non-smooth vector fields; Regularization; Singular perturbation; Sliding vector fields; Vector fields

We consider piecewise smooth vector fields (PSVF) defined in open sets M⊆ Rn with switching manifold being a smooth surface Σ. We assume that M\ Σ contains exactly two connected regions, namely Σ + and Σ -. Then, the PSVF are given by pairs X= (X+, X-) , with X= X+ in Σ + and X= X- in Σ -. A regularization of X is a 1-parameter family of smooth vector fields Xε, ε> 0 , satisfying that Xε converges pointwise to X on M\ Σ , when ε→ 0. Inspired by the Fenichel Theory, the sliding and sewing dynamics on the discontinuity locus Σ can be defined as some sort of limit of the dynamics of a nearby smooth regularization Xε. While the linear regularization requires that for every ε> 0 the regularized field Xε is in the convex combination of X+ and X-, the nonlinear regularization requires only that Xε is in a continuous combination of X+ and X-. We prove that, for both cases, the sliding dynamics on Σ is determined by the reduced dynamics on the critical manifold of a singular perturbation problem. We apply our techniques in the description of the nonlinear regularization of normal forms of PSVF in R2 and in R3.