Limit cycles in piecewise quadratic Kolmogorov systems
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 probl...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2026 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:320603 |
| Acceso en línea: | https://ddd.uab.cat/record/320603 https://dx.doi.org/urn:doi:10.1016/j.cnsns.2025.109285 |
| Access Level: | acceso abierto |
| Palabra clave: | Kolmogorov systems Poincaré map Center-focus Cyclicity Limit cycles Weak focus order Lyapunov quantities |
| Sumario: | 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. |
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