A Chebyshev criterion for at most two non-zero limit cycles in Abel equations

TL;DR

This paper introduces a Chebyshev criterion to determine the maximum number of limit cycles in Abel equations, establishing bounds for specific coefficient classes and improving existing results.

Contribution

It provides the first Chebyshev criterion for bounding limit cycles in Abel equations with coefficients from an extended Chebyshev system, advancing the understanding of the Smale-Pugh problem.

Findings

Maximum of three limit cycles for certain coefficient classes

Reestablished and improved previous bounds on limit cycles

Applied criterion to equations with trinomial coefficients

Abstract

In this paper, we investigate the maximum number of limit cycles of the reduced Abel equation $\dot{x}=A(t)x^{3}+B(t)x^{2}$ 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) $\mathcal{F}=\{f_{0},f_{1},f_{2}\}$ on $[0,T)$ with $f_{0}\not=0$. 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.