Bifurcations of Fine 3pointloop in Higher Dimensional Space

By using the linear independent fundamental solutions of the linear variational equation along the heteroclinic loop to establish a suitable local coordinate system in some small tubular neighborhood of the heteroclinic loop, the Poincar map is constructed to study the bifurcation problems of a fine 3point loop in higher dimensional space. Under some transversal conditions and the nontwisted condition, the existence, coexistence and incoexistence of 2pointloop, 1homoclinic orbit, simple 1periodic orbit and 2fold 1periodic orbit, and the number of 1periodic orbits are studied. Moreover, the bifurcation surfaces and existence regions are given. Lastly, the above bifurcation results are applied to a planar system and an inside stability criterion is obtained.