J. Phys. A: Math. Gen. 31 (1998) 7121–7139. Printed in the UK PII: S0305-4470(98)92905-5 A nine-dimensional Lorenz system to study high-dimensional chaos Peter Reiterer†, Claudia Lainscsek†, Ferdinand Sch¨ urrer†, Christophe Letellier‡ and Jean Maquet‡ † Institute for Theoretical Physics, Technical University of Graz, Petersgasse 16, A-8010 Graz, Austria ‡ LESP/UMR 6614—CORIA, Universit´ e de Rouen, Place Emile Blondel, 76131 Mont Saint- Aignan Cedex, France Received 30 March 1998, in final form 18 June 1998 Abstract. We examine the dynamics of three-dimensional cells with square planform in dissipative Rayleigh–B´ enard 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. 1. Introduction It is now well known that chaotic dynamics may be generated by many kinds of experiments from various fields of science such as hydrodynamics, astrophysics, chemistry, biology, electronics, etc. It is therefore of particular interest to possess an extended tool box to characterize the dynamics of the studied physical systems. From the pioneering paper by Lorenz [1], many analyses have been developed which may be separated into two classes. First, the geometrical properties of attractors on which the asymptotic motion settles down have been investigated. Such methods are based on a notion of distance in the reconstructed phase space according to Packard et al [2] and the very mathematical paper by Takens [3]. Geometrical properties present the attractive feature of being usable in a n- dimensional space. For instance, the correlation dimension [4] and Lyapunov exponents [5] have been commonly used to characterize attractors. The knowledge of Lyapunov exponents gives an indication about the predictability of a system and also about the number of active dynamical degrees of freedom involved in it [6, 7]. Nevertheless, the computation of such geometrical quantities is very sensitive to noise perturbations and not very useful in identifying different classes of systems. 0305-4470/98/347121+19$19.50 c  1998 IOP Publishing Ltd 7121