On the Number of Limit Cycles in Small Perturbations of a Piecewise Linear Hamiltonian System with a Heteroclinic Loop

In this paper, the authors consider limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is $n~(n\geq1),$ it is obtained that $n$ limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most $n+\frac{n+1}{2}]$ limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in $\varepsilon.$ Especially, for $n=1,2,3$ and $4$, a precise result on the maximal number of zeros of the first Melnikov function is derived.