Bifurcation of limit cycles in a class of piecewise smooth generalized Abel equations with two asymmetric zones
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918410639638528 |
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| author | Liang, Haihua Huang, Jianfeng |
| author_facet | Liang, Haihua Huang, Jianfeng |
| contents | 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 $θ_1\in (0,π)\cup (π,2π)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_25149 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bifurcation of limit cycles in a class of piecewise smooth generalized Abel equations with two asymmetric zones Liang, Haihua Huang, Jianfeng Classical Analysis and ODEs 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 $θ_1\in (0,π)\cup (π,2π)$. |
| title | Bifurcation of limit cycles in a class of piecewise smooth generalized Abel equations with two asymmetric zones |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2603.25149 |