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˙ = azk ¯zl + bzm¯zn, 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 exists and prove that then it is hyperbolic. For an arbitrary equation of...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:267124 |
| Acceso en línea: | https://ddd.uab.cat/record/267124 https://dx.doi.org/urn:doi:10.1016/j.jmaa.2022.126663 |
| Access Level: | acceso abierto |
| Palabra clave: | Polynomial differential equation Uniqueness of limit cycles Centre-focus problem |
| Sumario: | We prove that any complex differential equation with two monomials of the form z˙ = azk ¯zl + bzm¯zn, 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 exists and prove that then it is hyperbolic. For an arbitrary equation of the above form, we also solve the centrefocus problem and examine the number, position, and type of its critical points. In particular, we prove a Berlinski˘ı-type result regarding the geometrical distribution of the critical points stabilities. |
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