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

This paper devotes to the study of the classical Abel equation d x d t = g (t) x 3 + f (t) x 2 , where g (t) and f (t) are trigonometric polynomials of degree m ≥ 1. 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, Disc. Cont. Dyn. Syst. Ser. B, 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 Álvarez 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, which gives a positive answer to the sixth of 33 open problem proposed by Gasull in paper (Gasull, SeMA J., 2021). [ABSTRACT FROM AUTHOR]