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arXiv · 2605.02145

Inhomogeneous Picard-Fuchs equations of Abelian integrals in piecewise smooth near-Hamiltonian systems

Hefei Zhao·Yun Tian
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Abstract

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 $\theta$ of the separation line as a free parameter.

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Hefei Zhao, Yun Tian. 2026-05-04. Inhomogeneous Picard-Fuchs equations of Abelian integrals in piecewise smooth near-Hamiltonian systems. https://arxiv.org/abs/2605.02145

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