Limit cycles for a piecewise polynomial potential perturbation of a symmetric 8-loop Hamiltonian
Buzzi, Claudio (Universidade Estadual Paulista "Júlio de Mesquita Filho". Instituto de Biociências, Letras e Ciências Exatas)
Tonon, Durval José (Universidade Federal de Goiás. Instituto de Matemática e Estatística)
Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2026
Abstract:	We consider the symmetric 8-loop Hamiltonian potential given by H (x,y) = y2/2 + (x-1)2 (x+1)2 and examine a piecewise polynomial potential perturbation. We begin by obtaining the first-order Melnikov functions M1, M2, M3, defined near the three period annuli located at the center points (-1,0), (1, 0), and the saddle point at (0, 0), respectively. Upper bounds for the number of simple zeros of these Melnikov functions are provided both individually and simultaneously in terms of the degree n. These simple zeros correspond to limit cycles bifurcating from the three period annuli in various configurations. For low degrees n = 2, 3, 4, 5, we explicitly present all possible simultaneous configurations of limit cycles that can bifurcate. Due to the large number of cases and computational difficulties, for degrees 6 ≤ n ≤ 21, we explore the potential configurations of limit cycles bifurcating from each period annulus, proving the existence of the configuration that we believe to be maximal. Bifurcation diagrams estimating the size of the regions in the parameter space where the best configuration of limit cycles exists are also presented.
Grants:	Generalitat de Catalunya 2021/SGR-00113 Agencia Estatal de Investigación PID2022-136613NB-I00 Agencia Estatal de Investigación CEX2020-001084-M European Commission 777911
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Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Piecewise smooth vector fields ; Hamiltonian systems ; Global dynamics ; Bifurcations ; Limit cycles ; Cyclicity of period annuli
Published in:	Nonlinear differential equations and applications, Vol. 33, Num. 1 (January 2026) , art. 7, ISSN 1420-9004

DOI: 10.1007/s00030-025-01146-3

Available from: 2027-01-31 Postprint 24 p, 601.8 KB

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Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
Articles > Research articles
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