On the number of limit cycles in generalized abel equations

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(Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Villadelprat Yagüe, Jordi

(Universitat Rovira i Virgili. Departament d'Enginyeria Informàtica i Matemàtiques)

Data:	2020
Resum:	Given p, q ∊ Z ≥ 2 with p ≠ q, we study generalized Abel differential equations (Equation presented), where A and B are trigonometric polynomials of degrees n, m ≥ 1, respectively, and we are interested in the number of limit cycles (i. e. , isolated periodic orbits) that they can have. More concretely, in this context, an open problem is to prove the existence of an integer, depending only on p, q, m, and n and that we denote by H p,q(n, m), such that the above differential equation has at most H p,q(n, m) limit cycles. In the present paper, by means of a second order analysis using Melnikov functions, we provide lower bounds of H p,q(n, m) that, to the best of our knowledge, are larger than the previous ones appearing in the literature. In particular, for classical Abel differential equations (i. e. , p = 3 and q = 2), we prove that H 3,2(n, m) ≥ 2(n + m) - 1.
Ajuts:	Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1617 Ministerio de Ciencia e Innovación MTM2016-77278-P Ministerio de Ciencia e Innovación MTM2017-86795-C3-2-P European Commission 777911
Drets:	Tots els drets reservats.
Llengua:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Matèria:	Generalized Abel equations ; Melnikov theory ; Second order perturbation ; Limit cycles
Publicat a:	SIAM Journal on Applied Dynamical Systems, Vol. 19, Issue 4 (2020) , p. 2343-2370, ISSN 1536-0040

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Registre creat el 2021-02-12, darrera modificació el 2023-10-08