On Double Homoclinic Bifurcation of Limit Cycles in Near-Hamiltonian Systems on the Cylinder

We study the bifurcation problem of limit cycles in near-Hamiltonian systems near a double homoclinic loop on the cylinder. We obtain a sufficient condition to find a lower bound of the maximal number of limit cycles near the loop by the coefficients of the expansions of the three Melnikov functions corresponding to the three families of periodic orbits near the double homoclinic loop. We also provide an application of our main results to a class of cylindrical systems.