Abstract:

This article primarily focuses on the study of the limit cycle bifurcation problem of a class of near-Hamiltonian polynomial systems using Abel integral. First, by utilizing analytical techniques, approximate expansions of Abel integral are derived around the central singularity and in the vicinity of the heteroclinic loop, along with the calculated expressions for the coefficients. These results can be utilized to analyze the Hopf bifurcation or heteroclinic bifurcation of the perturbed system. Specifically, it is shown that the discussed near-Hamiltonian polynomial system can produce $[\frac{n+1}{4}]+[\frac{n-1}{4}]+1$ limit cycles near the central singularity and branch out $2[\frac{n+1}{4}]+[\frac{n-1}{4}]$ limit cycles near the heteroclinic loop.

Key words: Near-Hamiltonian system, Limit cycle, Abel integral, Heteroclinic loop bicurcation