Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold
Bastos, Jefferson L. R (Universidade Estadual Paulista. Instituto de Biociências, Letras e Ciências Exatas)
Buzzi, Claudio (Universidade Estadual Paulista. Instituto de Biociências, Letras e Ciências Exatas)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Novaes, Douglas D. (Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica)

Date:	2019
Abstract:	We study the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. Our main result is that 7 is a lower bound for the Hilbert number of this family. In order to get our main result, we develop the Melnikov functions for a class of nonsmooth differential systems, which generalizes, up to order 2, some previous results in the literature. Whereas the first order Melnikov function for the nonsmooth case remains the same as for the smooth one (i. e. the first order averaged function) the second order Melnikov function for the nonsmooth case is different from the smooth one (i. e. the second order averaged function). We show that, in this case, a new term depending on the jump of discontinuity and on the geometry of the switching manifold is added to the second order averaged function.
Grants:	Ministerio de Economía y Competitividad MTM2016-77278-P Ministerio de Economía y Competitividad MTM2013-40998-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Melnikov theory ; Averaging theory ; Nonsmooth differential systems ; Piecewise linear differential systems ; Nonlinear switching manifold ; Limit cycles ; Hilbert number
Published in:	Journal of differential equations, Vol. 267, Issue 6 (September 2019) , p. 3748-3767, ISSN 1090-2732

DOI: 10.1016/j.jde.2019.04.019

Postprint 23 p, 372.4 KB

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