A nine-dimensional Lorenz system to study high-dimensional chaos

Abstract

We examine the dynamics of three-dimensional cells with square planform in dissipative Rayleigh-Bénard convection. By applying a triple Fourier series ansatz up to second order, we obtain a system of nine nonlinear ordinary differential equations from the governing hydrodynamic equations. Depending on two control parameters, namely the Rayleigh number and the Prandtl number, the asymptotic behaviour can be stationary, periodic, quasiperiodic or chaotic. A period-doubling cascade is identified as a route to chaos. Hereafter, the asymptotic behaviour progressively evolves towards a hyperchaotic attractor. For given values of control parameters beyond the accumulation point, we observe a low-dimensional chaotic attractor as is currently done for dissipative systems. Although the correlation dimension strongly suggests that this attractor could be embedded in a three-dimensional space, a topological characterization 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-dimensional Lorenz model could therefore play a significant role in developing tools to characterize chaotic attractors embedded in phase space with a dimension greater than 3.