On a conjecture on the integrability of Liénard systems
We consider the Liénard differential systems ̇x=y+F(x), ̇y=x (1), in C2 where F(x) is an analytic function satisfying F(0) = 0 and F'(0) ≠ 0. Then these systems have a strong saddle at the origin of coordinates. It has been conjecture that if such systems have an analytic first integral defined...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2020 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:221375 |
| Acceso en línea: | https://ddd.uab.cat/record/221375 https://dx.doi.org/urn:doi:10.1007/s12215-018-00398-6 |
| Access Level: | acceso abierto |
| Palabra clave: | Liénard system First integral Strong saddle |
| Sumario: | We consider the Liénard differential systems ̇x=y+F(x), ̇y=x (1), in C2 where F(x) is an analytic function satisfying F(0) = 0 and F'(0) ≠ 0. Then these systems have a strong saddle at the origin of coordinates. It has been conjecture that if such systems have an analytic first integral defined in a neighborhood of the origin, then the function F(x) is linear, i.e. F(x) = ax. Here we prove this conjecture, and show that when F(x) is linear and system (1) has an analytic first integral, this is a polynomial. |
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