Limit cycle bifurcations near a 2-polycycle or double 2-polycycle of planar systems

In this paper, we study the bifurcation problem of limit cycles for a near-Hamiltonian system near a 2-polycycle or double 2-polycycle. We obtain new results on the number of limit cycles by using the method of stability-changing of a homoclinic loop. This method also provides a new way to find alien limit cycles. Especially, we found some alien limit cycles of new type.