On the Number of Limit Cycles in Generalized Abel Equations
Jianfeng Huang, Joan Torregrosa, Jordi Villadelprat
Abstract
Given $p,q\in\mathbb{Z}_{\geq 2}$ with $p\neq q$, we study generalized Abel differential equations $\frac{dx}{d\theta}=A(\theta)x^p+B(\theta)x^q,$ where $A$ and $B$ are trigonometric polynomials of degrees $n, m\ge 1,$ 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 $\mathcal{H}_{p,q}(n,m)$, such that the above differential equation has at most $\mathcal{H}_{p,q}(n,m)$ limit cycles. In the present paper, by means of a second order analysis using Melnikov functions, we provide lower bounds of $\mathcal{H}_{p,q}(n,m)$ 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 $\mathcal{H}_{3,2}(n,m)\geq 2(n+m)-1.$