On bifurcations of the Lorenz attractor in the Shimizu-Morioka model

is considered in which complex behavior of trajectories has been discovered [1] by means of computer simulation. These equations were put forward in [1] as a model for studying the dynamics of the Lorenz system for large Rayleigh number. Some physical applications of the model were pointed out in [2]. It was shown in papers [3,4] that there are two types of Lorenz-like attractors in this model. The first is an orientable Lorenz-like attractor and the second is nonorientable containing a countable set of saddle periodic orbits with negative multipliers. Of special interest in this model is the fact that the boundary of the region of existence of a Lorenz-like attractor includes two codimension two points (see fig. 1). The first is Q~ (or = 0.608, A = 1.044) at which the saddle value or = h I + h 3 equals zero on the bifurcation curve l 1 corresponding to the formation of the homoclinic "figure-eight-butterfly" (here h 1 < h 1 < 0 < h 3 are eigenvalues of the origin). The second point Q A (Or =0.549, A-0 .605) corresponds to the vanishing of the so-called separatrix value A (for details see [5]) on the bifurcation curve l 2 corresponding to the formation of symmetric pair of double-circuit homoclinic loops F~ and F 2. The curve 12 is divided by QA into two components l~ and l 2 which are selected by conditions A > 0 and A < 0 respectively. When crossing l~(l 2) the saddle periodic orbits born from loops F 1 and F 2 have their invariant manifolds homeomorphic to a cylinder (to a M6bius strip), respectively. It was stated by Sil'nikov [6] that under some additional restrictions on the eigenvalues of a multi-dimensional saddle, there exist regions of parameter values in neighbourhoods of such points for which systems close to the symmetric system with two homoclinic loops have a Lorenzlike attractor #1. An appropriate Poincar6 map was shown in [6] to have a one-dimensional