Ten limit cycles near a cubic homoclinic loop with a nilpotent cusp

TL;DR

This paper investigates the bifurcation of limit cycles near a cubic homoclinic loop in a planar system, introducing a novel method combining algebraic and differential equations to compute Melnikov functions, revealing up to ten bifurcating limit cycles.

Contribution

It presents a new approach using Abelian integrals and Picard-Fuchs equations to analyze limit cycle bifurcations near a cubic homoclinic loop.

Findings

Planar cubic systems can have up to ten limit cycles near a homoclinic loop.

The method combines algebraic structure of Abelian integrals with Picard-Fuchs equations.

Asymptotic expansion of Melnikov functions is computed near the cuspidal loop.

Abstract

In this paper, we study the bifurcation of limit cycles near a homoclinic cuspidal loop in a planar cubic near-Hamiltonian system by high-order Melnikov functions. We present a method combining the algebraic structure of Abelian integrals and Picard-Fuchs equation for computing the corresponding asymptotic expansion of Melnikov functions near the cuspidal loop. Using this system as an example, we show that planar cubic systems can have ten limit cycles bifurcating near a cubic homoclinic loop.