Limit cycles in piecewise polynomial Hamiltonian systems allowing nonlinear switching boundaries
Li, Tao (School of Mathematics. Southwestern University of Finance and Economics)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2023
Abstract:	This paper aims to study the limit cycles of planar piecewise polynomial Hamiltonian systems of degree n with the switching boundary y = xm, where m and n are positive integers. We answer a version of the 16th Hilbert problem for such systems, providing an upper bound for the maximum number of limit cycles in the function of m and n. We also are devoted to giving a lower bound via perturbing piecewise linear Hamiltonian systems having at the origin a global center. For this, a complete classification of the center conditions is established. In pursuit of a better lower bound, we require that this global center is nonlinear induced by the piecewise linearity, instead of the normal linear differential center considered in most of the existing articles. This renders that the traveling time of the unperturbed periodic orbits in each smooth zone is not calculable explicitly. Thus it is difficult to use the known Melnikov functions and averaged functions to study the bifurcating limit cycles. To overcome this difficulty, we develop an arbitrary order Melnikov-like function, which does not depend on the traveling time, for general d-dimensional piecewise smooth integrable systems allowing nonlinear switching boundaries. Finally, employing the new Melnikov-like function to the considered perturbation problem, we obtain a lower bound and an upper bound for the maximum number of limit cycles bifurcating from the unperturbed periodic orbits up to any order.
Grants:	Agencia Estatal de Investigación PID2019-104658GB-I00 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:	Bifurcation theory ; Limit cycle ; Nonlinear switching boundary ; Non-smooth center ; Piecewise Hamiltonian system
Published in:	Journal of differential equations, Vol. 344 (January 2023) , p. 405-438, ISSN 1090-2732

DOI: 10.1016/j.jde.2022.11.007

Postprint 28 p, 448.7 KB

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