Inhomogeneous Picard-Fuchs equations of Abelian integrals in piecewise smooth near-Hamiltonian systems
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918479971483648 |
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| author | Zhao, Hefei Tian, Yun |
| author_facet | Zhao, Hefei Tian, Yun |
| contents | In this paper, we explicitly obtain inhomogeneous Picard-Fuchs equations for Abelian integrals $I_{i,j}^+(h)$, where $I_{i,j}^+(h)$ is an integral along orbital arcs defined by polynomials $\frac{1}{2}y^2 + F(x)=h$. Moreover, we discuss the method of using Picard-Fuchs equations to recursively compute the asymptotic expansions of genearating functions of Abelian integrals near a homoclinic loop. As an application, we derive the maximum number of isolated zeros of Melnikov functions near a nilpotent saddle homoclinic loop for piecewise polynomials perturbations with the inclination $θ$ of the separation line as a free parameter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02145 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inhomogeneous Picard-Fuchs equations of Abelian integrals in piecewise smooth near-Hamiltonian systems Zhao, Hefei Tian, Yun Classical Analysis and ODEs In this paper, we explicitly obtain inhomogeneous Picard-Fuchs equations for Abelian integrals $I_{i,j}^+(h)$, where $I_{i,j}^+(h)$ is an integral along orbital arcs defined by polynomials $\frac{1}{2}y^2 + F(x)=h$. Moreover, we discuss the method of using Picard-Fuchs equations to recursively compute the asymptotic expansions of genearating functions of Abelian integrals near a homoclinic loop. As an application, we derive the maximum number of isolated zeros of Melnikov functions near a nilpotent saddle homoclinic loop for piecewise polynomials perturbations with the inclination $θ$ of the separation line as a free parameter. |
| title | Inhomogeneous Picard-Fuchs equations of Abelian integrals in piecewise smooth near-Hamiltonian systems |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2605.02145 |