Study on limit cycles near homoclinic loops and heteroclinic loops with hyperbolic saddles

Abstract

This paper addresses the issue of limit cycles near homoclinic and heteroclinic loops with hyperbolic saddles, specifically focusing on the scenario of near-Hamiltonian systems. A key aspect of the analysis involves determining the number of zeros of the first-order Melnikov function (Abelian integral). To achieve this, it becomes necessary to calculate the coefficients of its asymptotic expansion. In this paper, we propose a new algorithm that simplifies the computation of these coefficients compared to previous approaches. An example is provided to demonstrate the advantages of our new method. Our results yield a more accurate lower bound on the number of limit cycles while significantly reducing computational workload in comparison with existing works. Additionally, we present a general theory and an application for finding the lower bounds of the maximum number of limit cycles near double homoclinic loops.