A 9-DIMENSIONAL LORENZ SYSTEM TO STUDY HIGH-DIMENSIONAL CHAOS

We examine the dynamics of three-dimensional cells with square planfor m in dissipative Rayleigh-Benard convection. By applying a triple Four ier series ansatz up to second order, we obtain a system of nine nonli near ordinary differential equations from the governing hydrodynamic e quations. Depending on two control parameters, namely the Rayleigh num ber and the Prandtl number, the asymptotic behaviour can be stationary , periodic, quasiperiodic or chaotic. A period-doubling cascade is ide ntified as a route to chaos. Hereafter, the asymptotic behaviour progr essively evolves towards a hyperchaotic attractor. For given values of control parameters beyond the accumulation point, we observe a low-di mensional chaotic attractor as is currently done for dissipative syste ms. Although the correlation dimension strongly suggests that this att ractor could be embedded in a three-dimensional space, a topological c haracterization reveals that a higher-dimensional space must be used. Thus, we reconstruct a four-dimensional model which is found to be in agreement with the properties of the original dynamics. The nine-dimen sional Lorenz model could therefore play a significant role in develop ing tools to characterize chaotic attractors embedded in phase space w ith a dimension greater than 3.