Bifurcations Of Heteroclinic Cycles With The Bifurcation Of Singular Points

In this paper we study one type of bifurcations of heterodimcnsional cycles with supercritical bifurcation in three-dimensional vector fields. First,we give the normal forms in small neighborhood of the two equilibria.Under the nontransversal condition of the curve of T-points,we construct a Poincare return map.Based on the bifurcation analysis of Poincare return map,we present the parametric representation of the two nontransversal T-points under the small perturbation, and the expressions of paramet-ric curves of the homoclinic orbits and heteroclinic orbits around the nontransversal T-points.What’s more,we establish the images of the parametric curves of the homo-clinic orbits and heteroclinic orbits,and the existence conditions of the closed bifurca-tion curves or the symmetrical curves closed to the two nontransversal T-points.The predictions deduced from the parametric curves agree with the numerical results in MATLAB. Lastly based on the bifurcation analysis of Poincare return map,we get the expressions of parametric curves of the periodic orbit as well as the approximate shapes of the parametric curves.