SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2343-2370, January 2020.
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 $\mathcal{H}_{3,2}(n,m)\geq 2(n+m)-1.$

中文翻译：

---

关于广义Abel方程的极限环数

SIAM应用动力系统杂志，第19卷，第4期，第2343-2370页，2020年1月。
给定$ p，q \ in \ mathbb {Z} _ {\ geq 2} $与$ p \ neq q $，我们研究广义的Abel微分方程$ \ frac {dx} {d \ theta} = A（\ theta） x ^ p + B（\ theta）x ^ q，$其中$ A $和$ B $分别是度数$ n，m \ ge 1，$的三角多项式，我们对极限环的数量感兴趣（即，孤立的周期性轨道）。更具体地说，在这种情况下，一个开放的问题是证明整数的存在，该整数仅取决于$ p，q，m $和$ n $，我们用$ \ mathcal {H} _ {p，q表示}（n，m）$，因此上述微分方程最多具有$ \ mathcal {H} _ {p，q}（n，m）$个极限环。在本文中，通过使用Melnikov函数的二阶分析，我们提供了$ \ mathcal {H} _ {p，q}（n，m）$的下限，据我们所知，该下限较大比以前的文献中出现的要多。尤其是，