A new kind of T-point in the Lorenz system with a different bifurcation set

In this work we find a new kind of T-point (or Bykov point) in the Lorenz system. At this codimension-two degeneracy, a heteroclinic cycle connects the origin (when it is a real saddle) and non-trivial equilibria (when they are saddle-focus). We observe that it presents a noteworthy geometric difference from the “classical” T-point, known since the 1980s in the Lorenz system. Because the dominant eigenvalue of the two-dimensional manifold at the origin changes, a variation in the direction of the corresponding heteroclinic orbit occurs near this equilibrium. Simultaneously, there is an important change in the bifurcation set, not previously found in the literature. While at the classical T-point the homoclinic and heteroclinic curves of non-trivial equilibria arise as half-lines in the same direction (as predicted by the well-known model of Glendinning and Sparrow), now these global bifurcation curves emerge in opposite directions. To justify this change we build a theoretical model with suitable Poincaré sections in a tubular environment of the heteroclinic cycle. Finally, by introducing a fourth parameter into the Lorenz system (a new quadratic term in its third equation), we show how the classical T-point can also lead to the new bifurcation set. This transition through a nongeneric situation (which occurs when the Jacobian matrix at the origin has a double eigenvalue) implies the existence of a codimension-three degenerate T-point. We find this bifurcation in the Lorenz-like system considered and illustrate how the bifurcation sets evolve by analyzing parallel parameter planes on both sides of the degeneracy.