Bifurcations of twisted heteroclinic loop with resonant eigenvalues

In a small tubular neighborhood of the heteroclinic orbits, we establish a local coordinate system by using the foundational solutions of the linear variational equation of the unperturbed system along the heteroclinic orbits. We study the bifurcation problems of twisted heteroclinic loop with resonant eigenvalues. Under the twisted conditions and some transversal conditions, we obtain the existence, the number, the coexistence and non-coexistence problem of 1-heteroclinic loop, 1-homoclinic loop, 1-periodic orbit, double 1-periodic orbit, and 2-heteroclinic loop, 2-homoclinic loop, 2-periodic orbit. Moreover, the relative bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graphs are drawn.