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Limit cycles for a class of discontinuous generalized Liénard polynomial differential equations
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Mereu, Ana Cristina (UFSCar(Brazil). Department of Physics, Chemistry and Mathematics)

Data: 2013
Resum: 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.
Nota: Número d'acord de subvenció EC/FP7/2012/316338
Nota: Número d'acord de subvenció MICINN/2008/03437
Nota: Número d'acord de subvenció AGAUR/2009/SGR-410
Nota: Agraïments/Ajudes: The second author is partially supported by a FAPESP-BRAZIL grant 2012/20884-8. Both authors are also supported by the joint project CAPES–MECD grant PHB-2009-0025-PC.
Drets: Tots els drets reservats.
Llengua: Anglès
Document: article ; recerca ; preprint
Matèria: Limit cycles ; Non-smooth Liénard systems ; Averaging theory
Publicat a: Electronic Journal of Differential Equations, Vol. 2013 Núm. 195 (2013) , p. 1-8, ISSN 1072-6691

9 p, 763.1 KB

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