C. Letellier & J.-M. Malasoma
Unimodal order in the image of the simplest equivariant jerk system,
Physical Review E, 64, 067202, 2001. Online
The simplest equivariant chaotic dynamics is investigated in terms of its image, that is, under the 2⃗1 mapping allowing one to obtain a projection of the dynamics without any residual symmetry. The inversion symmetry is therefore deleted. The bifurcation diagram can thus be predicted from the unimodal order although the first-return map computed in the original phase space exhibits three critical points. This feature is the same as the one observed on the Burke and Shaw system although this latter system has a rotation symmetry.
L. A. Aguirre, U. S. Freitas, C. Letellier & J. Maquet
Structure selection technique applied to continuous time nonlinear models,
Physica D, 158, 1-18, 2001. Online
This paper addresses the problem of choosing the multinomials that should compose a polynomial mathematical model starting from data. The mathematical representation used is a nonlinear differential equation of the polynomial type. Some approaches that have been used in the context of discrete-time models are adapted and applied to continuous-time models. Two examples are included to illustrate the main ideas. Models obtained with and without structure selection are compared using topological analysis. The main differences between structure-selected models and complete structure models are: (i) the former are more parsimonious than the latter, (ii) a predefined fixed-point configuration can be guaranteed for the former, and (iii) the former set of models produce attractors that are topologically closer to the original attractor than those produced by the complete structure models.
C. Letellier, O. Ménard, Th. Klinger, A. Piel & G. Bonhomme.
Dynamical analysis and map modelling of a thermionic diode plasma experiment,
Physica D, 156, 169-178, 2001. Online
We perform a topological analysis of a chaotic behavior in a plasma experiment: a thermionic diode experiment. The stretching and folding mechanisms are schemed by a three branch template, that is, a structure more complicated than the common horseshoe template. Moreover, a discrete model for the first-return map to a Poincaré section has been obtained from experimental data by using a global modeling technique. The template and the model give strong indications for a low dimension dynamics underlying the experimental data.
C. Lainscsek, C. Letellier & F. Schürrer
Ansatz library for global modeling with a structure selection,
Physical Review E, 64, 016206, 2001.
Abstract. The information contained in a scalar time series and its time derivatives is used to obtain a global model for the underlying dynamics. This model provides a description of the time evolution of the system studied in a space spanned by the time series and its successive time derivatives which is expected to be equivalent to the original phase space. Differential models are, in general, very complicated and do not necessarily capture all properties of the original dynamics. The possibility of choosing a form among an ansatz library for the original system which allows a structure selection for the differential model is considered. It allows for the reduction of the complexity of the model and the recovery of the right property when the differential model is transformed back into the space associated with the ansatz.
C. Letellier, A. Dinklage, H. El-Naggar, C. Wilke & G. Bonhomme
Experimental evidence for a torus breakdown through a global bifurcation in a glow discharge plasma,
Physical Review E, 63, 042702, 2001.
A global bifurcation scenario for a two-frequency torus breakdown depicted by Baptista and Caldas  is observed on a glow-discharge experiment. The torus is broken through a crisis with an unstable periodic orbit. The torus section before the bifurcation is a sided polygon that has a number of edges equal to the period of the unstable orbit. Since the discharge is an extended system the two-frequency torus breakdown is shown to be a possible way to space-time chaos.
C. Letellier & R. Gilmore
Covering dynamical systems: Two-fold covers,
Physical Review E, 63, 016206, 2001.
We study the relation between a dynamical system, which is unchanged (equivariant) under a discrete symmetry group G and another locally identical dynamical system with no residual symmetry. We also study the converse mapping: lifting a dynamical system without symmetry to a multiple cover, which is equivariant under G. This is done in R3 for the two element rotation and inversion groups. Comparisons are done for the equations of motion, the strange attractors that they generate, and the branched manifolds that classify these strange attractors. A dynamical system can have many inequivalent multiple covers, all equivariant under the same symmetry group G. These are distinguished by the value of a certain topological index. Many examples are presented. A new global bifurcation, the ‘‘peeling bifurcation,’’ is described.
 Physica D 132, 325, 1999