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 \begin{align*} \frac{dx}{d\theta}=A(\theta)x^p+B(\theta)x^q, \end{align*} where $A$ and $B$ are are piecewise trigonometrical polynomials of degree $ m $ with two zones $0\leq\theta<\theta_1$ and $\theta_1\leq\theta\leq2\pi$. By means of the first and second order analysis using the Melnikov theory and applying the new Chebyshev criterion that established by \cite{HLZ2023}, we estimate the maximum number of positive and negative limit cycles that such equations can have, and reveal how this maximum number, denoted by $H_{\theta_1}(m)$, is affected by the location of the separation line $\theta=\theta_1$. 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., $H_{2\pi}(m)\geq 4m-2$ for $\theta_1=2\pi$, 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, $H_{\pi}(m)\geq 8m+2$ and $H_{\theta_1}(m)\geq 14m-6$ for $\theta_1\in (0,\pi)\cup (\pi,2\pi)$.