Melnikov functions of arbitrary order for piecewise smooth differential systems in Rn and applications
Chen, Xingwu (Sichuan University. Department of Mathematics)
Li, Tao (Southwestern University of Finance and Economics. School of Economic Mathematics)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2022
Abstract:	In this paper we develop an arbitrary order Melnikov function to study limit cycles bifurcating from a periodic submanifold for autonomous piecewise smooth differential systems in R with two zones separated by a hyperplane. This result not only extends some of the known results on the Melnikov theory in dimension and order but also compensates for some defects of the averaging theory in studying the limit cycle bifurcation of autonomous systems from a periodic submanifold. To demonstrate the application of our theoretical result and its superiority for some systems to the existing averaging theory, we study the maximum number of limit cycles bifurcating from an n-dimensional periodic submanifold caused by non-smooth centers of the fold-fold type, providing an upper bound for any order piecewise polynomial perturbations of degree m. Concerning the planar case of the unperturbed system, a piecewise Hamiltonian system, we obtain a better upper bound for piecewise polynomial Hamiltonian perturbations up to order two. The realizability of these upper bounds is also discussed.
Grants:	Agencia Estatal de Investigación PID2019-104658GB-I00 Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1617 European Commission 777911
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 ; Hilbert's 16th problem ; Fold-fold singularity ; Limit cycle bifurcation ; Piecewise smooth differential system
Published in:	Journal of differential equations, Vol. 314 (March 2022) , p. 340-369, ISSN 1090-2732

DOI: 10.1016/j.jde.2022.01.019

Postprint 26 p, 400.8 KB

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