Normal Forms and Lorenz Attractors

Normal forms for eleven cases of bifurcations of codimension-3 are considered, basically, in systems with a symmetry, which can be reduced to one of the two three-dimensional systems. The first system is the well-known Lorenz model in a special notation, the second is the Shimizu-Morioka model. In contrast with two-dimensional normal forms which admit, in principle, a complete theoretical study, in three-dimensional systems such analysis is practically impossible, except for particular parameter values when a system is close to an integrable system. Therefore, the main method of the investigation is qualitatively-numerical. In that sense, a description of principal bifurcations which lead to the appearance of the Lorenz attractor is given for the models above, and the boundaries of the regions of the existence of this attractor are selected. We pay special attention to bifurcation points corresponding to a formation of a homoclinic figure-8 of a saddle with zero saddle value and that of a homoclinic figure-8 with zero separatrix value. In L. P. Shil'nikov 1981, it was established that these points belong to the boundary of the existence of the Lorenz attractor. In the present paper, the bifurcation diagrams near such points for the symmetric case are given and a new criterion of existence of the nonorientable Lorenz is also suggested.