Bifurcations of twisted double homoclinic loops with resonant condition

In this paper, the bifurcation problems of twisted double homoclinic loops with resonant condition are studied for (m + n)-dimensional nonlinear dynamic systems. In the small tubular neighborhoods of the homoclinic orbits, the foundational solutions of the linear variational systems are selected as the local coordinate systems. The Poincaré maps are constructed by using the composition of two maps, one is in the small tubular neighborhood of the homoclinic orbit, and another is in the small neighborhood of the equilibrium point of system. By the analysis of bifurcation equations, the existence, uniqueness and existence regions of the large homoclinic loops, large periodic orbits are obtained, respectively. Moreover, the corresponding bifurcation diagrams are given.