Limit cycle bifurcations near a double 2-polycycle on the cylinder

In this paper, we consider the bifurcation problem of limit cycles near a double 2-polycycle for cylinder near-Hamiltonian system. We first study the stability of a general homoclinic loop type II on the cylinder, and then obtain certain results on the number and distribution of limit cycles for the cylinder near-Hamiltonian system near the double 2-polycycle by using the Melnikov function method and the method of stability-changing of a homoclinic loop type I or II. Furthermore, we provide a way to find alien limit cycles for cylinder near Hamiltonian system. As applications, we obtain the number of limit cycles of a class of cylinder near-Hamiltonian systems.