Geometric structure of the complex Lorenz model
V.Yu. Toronov 
, V.L. Derbov , A.G. Vladimirov 
,
Saratov State University, Saratov, Russia
Research Institute of Physics, St.Petersburg State University,
St.-Peterburg - Petrodvorets 198904 Russia
The complex generalization of Lorenz model [1,2]
where x and y are the complex variables, covers a variety of dynamical systems possessing a dispersion instability [1], such as the baroclinic instability in a heated liquid and the pulsation threshold in a laser. However, being a rather general and significant nonlinear dynamical model, Eqs.(1) were not studied as systematically as they are worth.
The issue of the present work is to reveal some general properties of CLM solutions by means of studying the geometry of the phase space. For this purpose we introduce a specific projective space, in which the states differing only by the common phase of variables x and y are considered to be equivalent. The phase space and the projective space occur to be the parts of a princile fiber bundle [3] with the Pancharatnam's connection [4] on it. We show that all the physical information about the system can be extracted from the equations of motion in this projective space. Using these equations, the existence of surfaces bounding the limit sets of trajectories in the projective space is proved. This reveals some peculiarities of phase dynamics of x and y and a restriction for bifurcations in CLM. Particularly, the equation
provides the necessary condition for the existence of the complex counterpart of the real Lorenz "butterfly" [2]. Thus, for CLM the homoclinic bifurcation is the codimension-two one.
Basing on these results, we derive a one-dimensional map, associated with CLM dynamics near the homoclinic bifurcation. Analysing this map we reveal the hierarchy of bifurcations inherent to the "complex" behavior of CLM.
This work was supported by the State Committee for High School of Russia (grant No. 95-0-2.1-59).
