The complex Lorenz model: Geometric structure, homoclinic bifurcation and one-dimensional map

It is shown that the phase space of the complex Lorenz model has the geometric structure associated with a fiber bundle. Using the equations of motion in the base space of the fiber bundle the surfaces bounding the attractors in this space are found. The homoclinic "butterffy" responsible for the Lorenz-like attractor appearance is shown to correspond to a codimension-two bifurcation. One-dimensional map describing bifurcation phenomena in the complex Lorenz model is constructed.