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

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.