On the number of zeros of Abelian integrals arising from perturbed quadratic reversible centers

Hilbert's 16th problem has been a significant topic in mathematics and its applications, with Arnold proposing a weakened version focusing on differential equations. Although considerable progress has been made in studying Hamiltonian systems, integrable non-Hamiltonian systems have received comparatively less attention. Recently, there has been a growing interest in quadratic reversible systems within this framework, leading to notable advancements. This study, which is grounded in qualitative analysis theory, investigates the upper bound on the number of zeros of Abelian integrals for a specific class of quadratic reversible systems under polynomial perturbations of degree $n$. By employing the Picard–Fuchs and the Riccati equation methods, we establish that for $n\ge 4$, the upper bound for the number of zeros of the Abelian integrals is $3n-4$. To achieve this result, we first transform the first integral of the quadratic reversible system into a standard form using numerical methods. Then, by integrating the Picard–Fuchs and the Riccati equation approaches, we derive explicit representations of the Abelian integrals and estimate their maximum number of zeros using relevant theoretical results. These findings provide an upper bound for the number of limit cycles in the system, demonstrating that when the degree of the polynomial perturbation is sufficiently large (specifically $n\ge 4$), these analytical techniques effectively determine the maximum number of zeros of the Abelian integrals.