On the number of zeros of Abelian integrals for discontinuous quadratic differential systems

TL;DR

This paper establishes upper bounds on the number of zeros of Abelian integrals in discontinuous quadratic differential systems using Picard-Fuchs equations and Chebyshev criterion, providing sharp bounds for specific cases.

Contribution

It introduces a novel application of Picard-Fuchs equations and Chebyshev criterion to analyze zeros of Abelian integrals in discontinuous quadratic systems, including sharp bounds for degree 2 perturbations.

Findings

Upper bounds for zeros of Abelian integrals in four types of quadratic systems

Sharp bounds for degree 2 perturbations within each period annulus

Application of Picard-Fuchs and Chebyshev methods to discontinuous systems

Abstract

Applying the Picard-Fuchs equation to the discontinuous differential system, we obtain the upper bounds of the number of zeros for Abelian integrals of four kinds of quadratic differential systems when it is perturbed inside all discontinuous polynomials with degree $n$. Furthermore, by using the {\it Chebyshev criterion}, we obtain the sharp upper bounds on each period annulus for $n=2$.