Limit cycles near a homoclinic loop in two classes of piecewise smooth near-Hamiltonian systems

Abstract

For two classes of piecewise smooth near-Hamiltonian systems, by studying some properties of the expansions of two Melnikov functions near a homoclinic loop, we give a simple relation between the coefficients of $h^j(j\ge 0,j\in Z)$ appearing in the two expansions. Based on this, we further give a general condition for each of the two systems to have as many as possible limit cycles near the homoclinic loop. Hence, by using the above main results and some techniques we obtain a lower bound of the maximum number of limit cycles near a homoclinic loop for each of two concrete systems with polynomial perturbations of degree $n$  ($n\ge 1$).