Homoclinic Bifurcation to a Transitive Attractor of Lorenz Type, II

In this paper it is proven that there is a codimension two bifurcation of a double homoclinic connection of a fixed point with a resonance condition among the eigenvalues to a transitive attractor that is like that of the geometric model of the Lorenz equations. The two key parameters are the variation of the eigenvalues from resonance and the amount that the homoclinic connection is broken. Because of the need to work near resonance of two of the eigenvalues, one of the key steps in the proof is to calculate the Poincaré–Dulac map past a fixed point in this situation. Also indicated is how bifurcation is realized for a specific cubic differential equation introduced by Rychlik, which is closely related to the Lorenz equations.