Bifurcation of limit cycles in a class of piecewise smooth generalized Abel equations with two asymmetric zones

Abstract This paper studies the number of limit cycles, known as the Smale-Pugh problem, for the generalized Abel equation d x d θ = A (θ) x p + B (θ) x q, dxd =A () x^p+B () x^q, where A and B are piecewise trigonometrical polynomials of degree m with two zones 0 ≤ θ H θ 1 (m) H ₁ (m), is affected by the location of the separation line θ = θ 1. For the equation of classical Abel type, our result not only includes the estimates provided in the recent paper (Huang et al. , SIAM J. Appl. Dyn. Syst. , 2020), i. e. , H 2 π (m) ≥ 4 m − 2 for θ 1 = 2 π, but also shows that the equation in the discontinuous case can possess more than two times as many limit cycles as in the continuous case. More accurately, H π (m) ≥ 8 m + 2 and H θ 1 (m) ≥ 14 m − 6 H ₁ (m) 14m-6 for θ 1 ∈ (0, π) ∪ (π, 2 π).