More limit cycles for complex differential equations with three monomials
In this paper we improve, by almost doubling, the existing lower bound for the number of limit cycles of the family of complex differential equations with three monomials, z(center dot) = Azkzl + Bzmzn + Czpzq, being k,l, m, n, p, q non-negative integers and A, B, C is an element of C. More concrete...
| Autores: | , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2072/480012 |
| Acceso en línea: | http://hdl.handle.net/2072/480012 |
| Access Level: | acceso abierto |
| Palabra clave: | Polynomial differential equation Number of limit cycles Centre-focus problem Lyapunov quantities 51 |
| Sumario: | In this paper we improve, by almost doubling, the existing lower bound for the number of limit cycles of the family of complex differential equations with three monomials, z(center dot) = Azkzl + Bzmzn + Czpzq, being k,l, m, n, p, q non-negative integers and A, B, C is an element of C. More concretely, if N = max (k + l, m + n, p + q) and H3(N) is an element of N boolean OR {infinity} denotes the maximum number of limit cycles of the above equations, we show that for N >= 4, H3(N) >= N - 3 and that for some values of N this new lower bound is N + 1. We also present examples with many limit cycles and different configurations. Finally, we show that H 3 ( 2 ) >= 2 and study in detail the quadratic case with three monomials proving in some of them non-existence, uniqueness or existence of exactly two limit cycles. |
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