Dynamical Analysis Of The Nontransversal T-point-Hopf Bifurcation

The study had found that heteroclinic loop plays an important role in a system of autonomical ordinary differential equations which contains parameters. Nontransversal T-point is a point in the parameter plane where there is a special heteroclinic loop. If the parameter space is three-dimensional, then the branch usually appears on a curve. When the curve arrives at the Hopf bifurcation surface of a balance point contained in the heteroclinc loop, there will be a degraded situation. We are interested in this bifurcation, where it is called nontransversal T-point-Hopf bifurcation. In this article, we model by constructing Poincare mapping to discuss the global nature of the nontransveral T-point-Hopf bifurcation nearby. In the model, the intersection of the two-dimensional manifolds between the two saddle points is nontransversal. Such assumpution will make the nature of the bifurcation be different from that between the two-dimensional manifolds whose intesection is transversal. we construct a Poincare map in small neibohood of unperturbed heterodimensional cycle and further obtain the bifurcation equations. By research the bifurcation equation, we get the existence of heteroclinic loop, homoclinic orbits and periodic orbits under small perturbations and win the approximate expression and coexistence surface area of the corresponding bifurcation.