Uniqueness of the limit cycles for complex differential equations with two monomials
We prove that any complex differential equation with two monomials of the form z˙=azkz¯l+bzmz¯n, with k,l,m,n non-negative integers and a,b∈C, has one limit cycle at most. Moreover, we characterise when such a limit cycle exists and prove that then it is hyperbolic. For an arbitrary equation of the...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2023 |
| 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:2072/532584 |
| Acceso en línea: | http://hdl.handle.net/2072/532584 |
| Access Level: | acceso abierto |
| Palabra clave: | Centre-focus problem Polynomial differential equation Uniqueness of limit cycles |
| Sumario: | We prove that any complex differential equation with two monomials of the form z˙=azkz¯l+bzmz¯n, with k,l,m,n non-negative integers and a,b∈C, has one limit cycle at most. Moreover, we characterise when such a limit cycle exists and prove that then it is hyperbolic. For an arbitrary equation of the above form, we also solve the centre-focus problem and examine the number, position, and type of its critical points. In particular, we prove a Berlinskiĭ-type result regarding the geometrical distribution of the critical points stabilities. © 2022 The Author(s) |
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