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

Abstract In this paper, we investigate the maximum number of limit cycles of the reduced Abel equation 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) F = { f 0 , f 1 , f 2 } on [ 0 , T ) with f 0 ≠ 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 (2024 J. Differ. Equ. ) and improves the work of Bravo et al (2015 Discrete Contin. Dyn. Syst. ; 2024 J. Differ. Equ. ). We also obtain the same maximum number of limit cycles for the equation with trinomial coefficients.