Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities
In this paper we study the limit cycles of the planar polynomial differential systems * x=ax-y P_n(x,y),\\ y=x ay Q_n(x,y), * where P_n and Q_n are homogeneous polynomials of degree n2, and a R. Consider the functions * &()=P_n() Q_n()\\ &()=Q_n()-P_n()\\ &_1()=a()-(),\\ &_2()=(n-1)(...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2018 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:199335 |
| Acceso en línea: | https://ddd.uab.cat/record/199335 https://dx.doi.org/urn:doi:10.1016/j.jde.2018.05.019 |
| Access Level: | acceso abierto |
| Palabra clave: | Homogeneous nonlinearities Limit cycles Non-existence and uniqueness Polynomial differential systems |
| Sumario: | In this paper we study the limit cycles of the planar polynomial differential systems * x=ax-y P_n(x,y),\\ y=x ay Q_n(x,y), * where P_n and Q_n are homogeneous polynomials of degree n2, and a R. Consider the functions * &()=P_n() Q_n()\\ &()=Q_n()-P_n()\\ &_1()=a()-(),\\ &_2()=(n-1)(2a()-()) '(). *First we prove that these differential systems have at most 1 limit cycle if there exists a linear combination of _1 and _2 with definite sign. This result improves previous knwon results. Furthermore, if _1(_1a-_2)0 for some _1,_20, we provide necessary and sufficient conditions for the non-existence, and the existence and uniqueness of the limit cycles of these differential systems. When one of these mentioned limit cycles exists it is hyperbolic and surrounds the origin. |
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