Limit cycles for a class of discontinuous generalized Liénard polynomial differential equations
We divide R2 in l sectors S1, ..., Sl, with l > 1 even. We define in R2 a discontinuous differential system such that in each sector Sk, for k = 1, ..., l, is defined a smooth generalized Lienard polynomial differential equation ¨x + fi(x) ˙x + gi(x) = 0, i = 1, 2 alternatively, where fi and gi a...
| Autores: | , |
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| Formato: | artículo |
| Fecha de publicación: | 2013 |
| País: | España |
| Recursos: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:150599 |
| Acesso em linha: | https://ddd.uab.cat/record/150599 |
| Access Level: | acceso abierto |
| Palavra-chave: | Limit cycles Non-smooth Liénard systems Averaging theory |
| Resumo: | We divide R2 in l sectors S1, ..., Sl, with l > 1 even. We define in R2 a discontinuous differential system such that in each sector Sk, for k = 1, ..., l, is defined a smooth generalized Lienard polynomial differential equation ¨x + fi(x) ˙x + gi(x) = 0, i = 1, 2 alternatively, where fi and gi are polynomials of degree n−1 and m respectively. We apply the averaging theory of first order for discontinuous differential systems to this class of non-smooth generalized Lienard polynomial differential systems and we show that for any n and m there are such non-smooth Lienard polynomial equations having at least max{n, m} limit cycles. Note that this number is independent of l. Roughly speaking this result shows that the non-smooth classical (m = 1) Lienard polynomial differential systems can have at least the double number of limit cycles than the smooth ones, and that the non-smooth generalized Lienard polynomial differential systems can have at least one more limit cycle than the smooth ones. Of course, these comparisons are done with the present known results. |
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