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

TL;DR

This paper derives inhomogeneous Picard-Fuchs equations for Abelian integrals in piecewise smooth near-Hamiltonian systems and uses them to analyze zeros of Melnikov functions.

Contribution

It explicitly obtains Picard-Fuchs equations for Abelian integrals and applies them to determine zeros of Melnikov functions in piecewise systems.

Findings

Derived inhomogeneous Picard-Fuchs equations for Abelian integrals.

Developed a recursive method for asymptotic expansions near a homoclinic loop.

Determined the maximum number of zeros of Melnikov functions near a nilpotent saddle.

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.