Limit cycles in piecewise quadratic Kolmogorov systems
Da Cruz, Leonardo Pereira Costa (Universidade de São Paulo. Instituto de Ciências Matemáticas e de Computação)
Oliveira, Regilene (Universidade de São Paulo. Instituto de Ciências Matemáticas e de Computação)
Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2026
Abstract:	We study planar piecewise quadratic differential systems of Kolmogorov type. Specifically, we consider systems with both coordinate axes invariant and with a separation line that is straight and distinct from the invariant axes. The main results concern two different aspects. First, the center problem is solved for certain subclasses. Second, using this classification, the bifurcation of limit cycles of crossing type is investigated. We contrast the nature of Hopf-type bifurcations in smooth and piecewise smooth settings, particularly regarding the bifurcation of limit cycles of small amplitude. The classical pseudo-Hopf bifurcation is analyzed in the Kolmogorov systems class. It is worth highlighting that, in contrast to the smooth Kolmogorov quadratic systems, which have no limit cycles, the piecewise case exhibits at least six. Furthermore, we show that the maximal weak focus order, eight, does not necessarily yield the maximal number of small-amplitude limit cycles.
Grants:	Agencia Estatal de Investigación CEX2020-001084-M Agencia Estatal de Investigación PID2022-136613NB-I00 Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-00113
Note:	Altres ajuts: acords transformatius de la UAB
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Kolmogorov systems ; Poincaré map ; Center-focus ; Cyclicity ; Limit cycles ; Weak focus order ; Lyapunov quantities
Published in:	Communications in nonlinear science and numerical simulation, Vol. 152, Part C (January 2026) , art. 109285, ISSN 1878-7274

DOI: 10.1016/j.cnsns.2025.109285

16 p, 2.6 MB

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