On the number of limit cycles in generalized abel equations
Huang, Jianfeng (Jinan University. Department of Mathematics (China))
Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Villadelprat Yagüe, Jordi (Universitat Rovira i Virgili. Departament d'Enginyeria Informàtica i Matemàtiques)

Date:	2020
Abstract:	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.
Grants:	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
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Generalized Abel equations ; Melnikov theory ; Second order perturbation ; Limit cycles
Published in:	SIAM Journal on Applied Dynamical Systems, Vol. 19, Issue 4 (2020) , p. 2343-2370, ISSN 1536-0040

DOI: 10.1137/20M1340083

Postprint 23 p, 412.0 KB

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