Maximum number of limit cycles for Abel equation having coefficients with linear trigonometric functions

TL;DR

This paper investigates the maximum number of limit cycles in Abel equations with linear trigonometric coefficients, providing a new criterion and solving a special case of the Smale-Pugh problem with a maximum of three limit cycles.

Contribution

It introduces a new criterion for estimating limit cycle multiplicity and completely solves the case with linear trigonometric coefficients, establishing an upper bound of three.

Findings

Maximum of three limit cycles for Abel equations with linear trigonometric coefficients.

New criterion for estimating limit cycle multiplicity.

Complete solution for the simplest case of the Smale-Pugh problem.

Abstract

This paper devotes to the study of the classical Abel equation $\frac{dx}{dt}=g(t)x^{3}+f(t)x^{2}$, where $g(t)$ and $f(t)$ are trigonometric polynomials of degree $m\geq1$. We are interested in the problem that whether there is a uniform upper bound for the number of limit cycles of the equation with respect to $m$, which is known as the famous Smale-Pugh problem. In this work we generalize an idea from the recent paper (Yu, Chen and Liu, arXiv:$2304.13528$, $2023$) and give a new criterion to estimate the maximum multiplicity of limit cycles of the above Abel equations. By virtue of this criterion and the previous results given by {\'A}lvarez et al. and Bravo et al., we completely solve the simplest case of the Smale-Pugh problem, i.e., the case when $g(t)$ and $f(t)$ are linear trigonometric, and obtain that the maximum number of limit cycles, is three.