TY - JOUR

T1 - On the number of limit cycles in generalized Abel equations

AU - Huang, Jianfeng

AU - Torregrosa, Joan

AU - Villadelprat, Jordi

N1 - Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - Generalized Abel equations

KW - Limit cycles

KW - Melnikov theory

KW - Second order perturbation

UR - http://www.scopus.com/inward/record.url?scp=85095916710&partnerID=8YFLogxK

U2 - 10.1137/20M1340083

DO - 10.1137/20M1340083

M3 - Article

AN - SCOPUS:85095916710

VL - 19

SP - 2343

EP - 2370

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

SN - 1536-0040

IS - 4

ER -