C. Letellier, J. Maquet, H. Labro, L. Le Sceller, G. Gouesbet, F. Argoul & A. Arneodo
Analyzing chaotic behaviour in a Belousov-Zhabotinskii reaction by using a global vector field reconstruction,
Journal of Physical Chemistry A, 102, 10265-10273, 1998. Online
The Belousov-Zhabotinskyi reaction is governed by very complex chemical kinetics involving at least 20 species. Nevertheless, there are many results that suggest that such a reaction could be described by a 3D model. For instance, in the special case when the asymptotic behavior is close to a homoclinic orbit, a 3D normal-form model has been previously proposed. Rather than deriving such a model, which requiring prior knowledge about the dynamics, a 3D model is here obtained by using a global vector field reconstruction technique starting from the measured time dependence of [CeIV]. This reconstructed model is hereafter validated by comparing the topological properties of the associated attractor to the ones directly reconstructed from the time series by using derivative coordinates. Indeed, the template characterizing the reconstructed model is compatible with the one extracted from the data. Nevertheless, it is found to be not compatible with the template associated with the normal-form model, which does not generate trajectories close enough to the experimental ones. Consequently, the topological properties of the underlying dynamics are not well captured by the normal-form model
C. Letellier, L. Le Sceller & G. Gouesbet
Nonlinear dynamics: what for ?
Journal of High Temperature Material Processes, 2, 83-101, 1998. Online
This paper presents an introduction to the theory of nonlinear dynamical systems, emphasizing some basic facts and recent advances. The point of view is focused on the analysis of signals. The described methods can be applied to many fields (fluid mechanics, chemistry, astrophysics, electronics,...) and should also be useful to the community of plasma workers.
C. Letellier, J. Maquet, L. Le Sceller, G. Gouesbet & L. A. Aguirre
On the non-equivalence of observables in phase space reconstructions from recorded time series,
Journal of Physics A, 31, 7913-7927, 1998. 0nline
Abstract. In practical problems of phase-space reconstruction, it is usually the case that the reconstruction is much easier using a particular recorded scalar variable. This seems to contradict the general belief that all variables of a dynamical system are equivalent in phase-space reconstruction problems. This paper will argue that, in many cases, the choice of a particular scalar time series from which to reconstruct the original dynamics could be critical. It is argued that different dynamical variables do not provide the same level of information (observability) of the underlying dynamics and, as a consequence, the quality of a global reconstruction critically depends on the recorded variable. Examples in which the choice of observables is critical are discussed and the level of information contained in a given variable is quantified in the case where the original system is known. A clear example of such a situation arises in the Rössler system for which the performance of a global vector field reconstruction technique is investigated using time series of variables x, y or z, taken one at a time.
P. Reiterer, C. Lainscsek, F. Schürrer, C. Letellier & J. Maquet
A Nine-Dimensional Lorenz System to study high-dimensional chaos,
Journal of Physics A, 31, 7121-7139, 1998. Online
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.