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arXiv · 2608.15618

On the Number of Limit Cycles in Generalized Abel Equations with Coefficients Having the Chebyshev Property

Jianfeng Huang·Renhao Tian·Yulin Zhao
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Abstract

This paper concerns the maximum number of limit cycles of generalized Abel differential equations $dx/dt = A(t)x^p + B(t)x^q$, where $A$ and $B$ belong to the linear span of a family of functions having the Chebyshev property. Motivated by a recent open problem posed by Huang et al. (Nonlinearity, 2026), we investigate whether this maximum number can be bounded in terms of $p$, $q$, and the structure of the family. Under some natural hypotheses and by means of first- and second-order analyses using Melnikov functions, we provide lower bounds for this maximum number. In contrast to previous work, no specific form for the coefficients is assumed. We then apply these estimates to Abel equations with trigonometric polynomial, polynomial, and hyperbolic coefficients. In the trigonometric polynomial case, we reestablish the results of Álvarez et al. (J. Math. Anal. Appl., 2008) and Huang et al. (SIAM J. Appl. Dyn. Syst., 2020), while in the polynomial case, we improve the classical lower bound given by Lins-Neto.

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Jianfeng Huang, Renhao Tian, Yulin Zhao. 2026-08-16. On the Number of Limit Cycles in Generalized Abel Equations with Coefficients Having the Chebyshev Property. https://arxiv.org/abs/2608.15618

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