Analysis of the T-point-Hopf bifurcation in the Lorenz system

In this paper we show numerically the existence of a T-point-Hopf bifurcation in the Lorenz system. This codimension-three degeneracy occurs when the nontrivial equilibria involved in the T-point heteroclinic loop undergo a subcritical Hopf bifurcation. Shil'nikov-Hopf bifurcations of the heteroclinic and the homoclinic orbits of the nontrivial equilibria are also present. Moreover, we consider a theoretical model, based on the construction of a Poincaré map, that describes the global behavior close to that T-point-Hopf bifurcation. An excellent agreement between the results provided by our theoretical model and those obtained numerically for the Lorenz system is found. Specifically, the model is able to give an explanation of the complex distribution of homoclinic connections to the origin previously described in the literature.