Bifurcations Of A Class Of3-point-loop Liyan Guo

In this paper, we are concerned with the bifurcation problems of a type of rough3-point-loop in the higher dimension for the case β1=ρ11/λ11>1,β2ρ21/λ21=1,β3=ρ31/λ31<1, and β1β2β3<1, where β1is the hyperbolic ratio of the unperturbed system at saddle point Pt, i=1,2,3. Owing to the complexity of the3-point-loop bifurcations in the higher dimension, we mainly study the existence, coexistence and incocxistcncc of3-point loop.2-point loop,1-homoclinic loop, simple1-periodic orbit and2-fold1-periodic orbit under some transversal conditions and the non-twisted condition. Lastly, the corresponding bifurcation surfaces and existence regions are given.In some small tubular neighborhood of the heteroclinic loop Γ, we obtain Poincare map and bifurcation equations by using the linear independent fundamen-tal solutions of the linear variational equation along the heteroclinic loop to establish the local coordinate and using the transformation of Silnikov coordinate. Moreover, the existence of3-point loop,2-point loop,1-homoclinic loop and1-periodic orbit is equal to the existence of positive solutions (s1, s2, s3) of bifurcation equations (s1,s2,s3≥0).Under some transversal conditions and the non-twisted condition, the follow-ing results can be derived:The3-point loop,2-point loop and1-homoclinic loop bifurcated from the heteroclinic loop F are incoexistent. but in particular situa-tions.2-point loop or1-homoclinic loop and1-periodic orbit are coexistence:Fi-nally, the corresponding bifurcation surfaces and existence regions are obtained. When Δ1=Δ2=Ε3=1, the existence of2-fold1-periodic orbit bifurcated from the heteroclinic loop F is obtained, respectively, approximate expressions of the corresponding bifurcation surfaces and existence regions are also given. But if Δi=Δj=-1,i≠j,i,j=1,2,3, and Δ=1, the2-fold1-periodic orbit bifurcated from the heteroclinic loop Γ is not obtained.