The Upper Bound Estimation of Abelian Integral for a Class of Quadratic Reversible System under Small Perturbations

Abstract

In this article, by using the Riccati equation method, we investigate the maximal values of the isolated zeros of the Abelian integral for a set of quadratic reversible systems (r11) that belong to genus one, while experiencing varying 3rd, 2nd, and 1st-polynomial perturbations. Specifically, we aimed to find the upper bound for the maximal zeros of the system’s limit cycle (a special dynamic behavior in a stable state, characterized by the existence of specific periodic orbits). We know that the Abelian integral is a function of h, so when studying the maximal zeros of the function related to h, we not only consider the highest degree of the relevant function but also take into account the parity of the function and the range of values of h. Then through variable substitution, a smaller upper bound can be obtained: our findings show that the maximal values of the isolated zeros count under varying 3rd, 2nd, and 1st-polynomial perturbations is 12, improving upon previous results where the upper bound was 34 for the 3rd polynomial perturbation and 22 for the 2nd and 1st polynomial perturbations. This study represents an improvement upon previous research.