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

Haihua Liang, Jianfeng Huang

Abstract

This paper studies the number of limit cycles, known as the Smale-Pugh problem, for the generalized Abel equation \begin{align*} \frac{dx}{dθ}=A(θ)x^p+B(θ)x^q, \end{align*} where $A$ and $B$ are are piecewise trigonometrical polynomials of degree $ m $ with two zones $0\leqθ<θ_1$ and $θ_1\leqθ\leq2π$. 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 $H_{θ_1}(m)$, 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., $H_{2π}(m)\geq 4m-2$ for $θ_1=2π$, 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, $H_π(m)\geq 8m+2$ and $H_{θ_1}(m)\geq 14m-6$ for $θ_1\in (0,π)\cup (π,2π)$.

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}{dθ}=A(θ)x^p+B(θ)x^q, \end{align*} where and are are piecewise trigonometrical polynomials of degree with two zones and . 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 , is affected by the location of the separation line . 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., for , 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, and for .

Paper Structure

This paper contains 8 sections, 20 theorems, 139 equations, 2 tables.

Table of Contents

  1. Introduction and statements of main results
  2. Melnikov function for equation \ref{['equa3']}
  3. Expression of $M_{1}(\rho)$.
  4. Expression of $M_{2}(\rho)$.
  5. Expression of $M_{2}(\rho)$ for $\theta_1\neq \pi$
  6. Expression of $M_{2}(\rho)$ for $\theta_1=\pi$
  7. Chebyshev family for the first non-vanishing Melnikov function
  8. Proof of the main results

Key Result

Theorem 1.1

Assume that $M_n$ is the first non-vanishing Melnikov function of equation equa3. Let $Z_n(m)$ ($m\geq1$) be the maximum number of isolated zeros of $M_n$ on $\hat{I}$, counted with multiplicity. Then, the value of $Z_n(m)$ with respect to the value of $\theta_1$ are given in Table table1.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 28 more