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

TL;DR

This paper investigates the maximum number of limit cycles in a class of piecewise smooth generalized Abel equations with two zones, revealing how the separation line affects the bifurcation of limit cycles using Melnikov theory and Chebyshev criteria.

Contribution

It provides new estimates for the maximum number of limit cycles in piecewise Abel equations, extending previous results and showing the influence of the discontinuity location.

Findings

Maximum limit cycles depend on the separation line position.

Discontinuous equations can have more limit cycles than continuous ones.

Established lower bounds for limit cycles in different 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 $\theta_1\in (0,\pi)\cup (\pi,2\pi)$.