2025/06/17 by Jianfeng Huang, Renhao Tian, Huang, Jianfeng +3
Mathematics · Medicine · #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.2506.14091
openalex publication_date 2025/06/17 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
In this paper, we investigate the maximum number of limit cycles of the reduced Abel equation x=A(t)x3+B(t)x2 on an interval [0,T]. The Smale-Pugh problem asks whether this maximum number is bounded in terms of a given class of coefficients. We establish for the first time a Chebyshev criterion, providing a positive answer to the problem when this class spanned by an extended Chebyshev system (ET-system) F=\f0,f1,f2\ on [0,T) with f0\not=0. As an application, we prove that the equation has at most three limit cycles (including x=0) when the coefficients A and B are both linear trigonometric functions or quadratic polynomials. This reestablishes the result of Yu et al. (J. Differ. Equ., 2024) and improves the work of Bravo et al. (Disc. Cont. Dyn. Syst., 2015 & J. Differ. Equ., 2024). We also obtain the same maximum number of limit cycles for the equation with trinomial coefficients.