The transition of a multi-dimensional lorenz system

A multi-dimensional Lorenz system, which includes thirty-three amplitude equations describing time evolu-tion of convection, is derived from two-dimensional Boussinesq equations by using the Galerkin method. Its transition is studied by numerical solution. It is found that, with Rayleigh number increasing from zero to one hundred, five different types of motion appear one after another as follows: stationary motion, periodic motion, quasiperiodic motion with two-fundamental frequencies, quasi-periodic motion with three fundamental frequencies, and chaotic motion. By comparing with the Lorenz model and Curry's fourteen-dimensional model, it is shown that as retained modes increase, the critical values of transition become larger and the types of bi-furcation change. The results of dynamic behavior happen to be in agreement with the IIIa route of the Gollub and Benson experiments.