On the number of zeros of Abelian integrals for a kind of quadratic reversible centers

Hilbert ${}^{\prime }$s 16th problem is extensively studied in mathematics and its applications. Arnold proposed a weakened version focusing on differential equations. While significant progress has been made for Hamiltonian systems, less attention has been given to integrable non-Hamiltonian systems. In recent years, investigating quadratic reversible systems in integrable non-Hamiltonian systems has gained widespread attention and shown promising advancements. In this academic context, our study is based on qualitative analysis theory. It explores the upper bound of the number of zeros of Abelian integrals for a specific class of quadratic reversible systems under perturbations with polynomial degrees of n. The Picard-Fuchs equation method and the Riccati equation method are employed in our investigation. The research findings indicate that when the degree of the perturbing polynomial is n ( $n\ge 5$), the upper bound for the number of zeros of Abelian integrals is determined to be $7n-12$. To achieve this, we first numerically transform the Hamiltonian function of the quadratic reversible system into a standard form. By applying a combination of the Picard-Fuchs equation method and the Riccati equation method, we derive the representation of the Abelian integrals. Using relevant theorems, we estimate the upper bound for the number of zeros of the Abelian integrals, which consequently provides an upper bound for the number of limit cycles in the system. The research results demonstrate that when the perturbation polynomial degree is high or equal to n, the Picard-Fuchs equation method and the Riccati equation method can be applied to estimate the upper bound of the number of zeros of the Abelian integrals.