Codimension-3 bifurcations of a class of homoclinic loop with saddle-point

In this work, bifurcations in the class that the homoclinic orbit connects the strong stable and strong unstable manifolds of a saddle are investigated for four-dimensional vector fields. The existence, numbers, coexistence and incoexistence of 1-homoclinic orbit, 1-periodic orbit, 2n2n-homoclinic orbit and 2n2n-periodic orbit are obtained, the approximate expressions of the corresponding bifurcation surfaces and the bifurcation diagrams are also presented.