Periods of solutions of periodic differential equations
Smooth non-autonomous T-periodic differential equations x'(t)=f(t,x(t)) defined in \R\K^n, where \K is \R or \C and n 2 can have periodic solutions with any arbitrary period~S. We show that this is not the case when n=1. We prove that in the real C^1-setting the period of a non-constant periodi...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2016 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:169473 |
| Acceso en línea: | https://ddd.uab.cat/record/169473 |
| Access Level: | acceso abierto |
| Palabra clave: | Holomorphic differential equations Periodic differential equations Periodic orbit |
| Sumario: | Smooth non-autonomous T-periodic differential equations x'(t)=f(t,x(t)) defined in \R\K^n, where \K is \R or \C and n 2 can have periodic solutions with any arbitrary period~S. We show that this is not the case when n=1. We prove that in the real C^1-setting the period of a non-constant periodic solution of the scalar differential equation is a divisor of the period of the equation, that is T/S\N. Moreover, we characterize the structure of the set of the periods of all the periodic solutions of a given equation. We also prove similar results in the one-dimensional holomorphic setting. In this situation the period of any non-constant periodic solution is commensurable with the period of the equation, that is T/S\Q. |
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