Christophe LETELLIER
20/01/2008

G. Byrne, R. Gilmore & C. Letellier,

Distinguishing between folding and tearing mechanisms in strange attractors,

Physical Review E,70, 056214, 2004. Online

Abstract

We establish conditions for distinguishing between two topologically identical strange attractors that are enclosed by identical bounding tori, one of which is generated by a flow restricted to that torus, the other of which is generated by a flow in a different bounding torus and either imaged or lifted into the first bounding torus.

J. Maquet, C. Letellier & L. A. Aguirre,

Scalar modeling and analysis of a 3D biochemical reaction model

Journal of Theoretical Biology,228(3), 421-430, 2004. Online

Abstract

For many systems it is advantageous if analysis and modeling can be accomplished from a scalar time series because this greatly facilitates the experimental setup. Moreover, in real-life systems it is hardly true that all the state variables are available for analysis and modeling. Since the late 1980s, techniques have been put forward for building mathematical models from a scalar time series. One of the objectives of this paper is to verify if it is possible to obtain global non-linear models (non-linear differential equations) from scalar time series. Such data are obtained using a model of biochemical reaction with aperiodic (chaotic) oscillations as recently observed in the case of a glycolytic reaction [1]. The main objective, however, is to investigate which state variable is more convenient for the task in practice. It is shown that observability indices seem to quantify quite well which variable should be preferred as the observable. The validity of the results are established performing rigorous topological analysis on the original system and the obtained models. The influence of noise, always present in experimental time series, on the dynamics underlying such a system is also investigated.

C. Letellier, S. Elaydi, L. A. Aguirre & Aziz-Alaoui

Difference equationsversusdifferential equations, a possible equivalence?

Physica D,195(1-2), 29-49, 2004. Online

Abstract

When a set of nonlinear differential equations is investigated, most of time there is no analytical solution and only numerical integration techniques can provide accurate numerical solutions. In a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which the time variable is discrete. In numerical investigations a fourth-order Runge–Kutta integration scheme is usually sufficient. Nevertheless, sometimes a set of difference equations may be required and, in this case, standard schemes like the forward Euler, backward Euler or central difference schemes are used. The major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small integration time steps. In some cases, it may be of interest to obtain difference equations with the same type of solutions as for the differential equations but with significantly large time steps. Nonstandard schemes as introduced by Mickens [2] allow to obtain more robust difference equations. In this paper, using such nonstandard scheme, we propose some difference equations as discrete analogues of the Rössler system for which it is shown that the dynamics is less dependent on the time step size than when a nonstandard scheme is used. In particular, it has been observed that the solutions to the discrete models are topologically equivalent to the solutions.

A. Bultel, C. Letellier & A. Bourdon

Dynamical analysis of a helium glow discharge. I A model,

Physics Letters A,323(3-4), 267-277, 2004. Online

Abstract

In this Letter, we investigate a model elaborated by Wilke et al. to explain various regimes observed in a helium glow discharge [3] for which the underlying dynamics can be chaotic, quasi-periodic, etc. We found that this model does not obey all the required physical principles. A new one is therefore proposed. It is mainly based on separated balance equations for charged species resulting from the propagation of ionization waves.

L. A. Aguirre, G. F. V. Amaral, R. A. M. Lopes & C. Letellier

Constraining the topology of neural networks to ensure dynamics with symmetry properties,

Physical Review E,69, 026701, 2004. Online]

Abstract

This paper addresses the training of network models from data produced by systems with symmetry properties. It is argued that although general networks are global approximators, in practice some properties such as symmetry are very hard to learn from data. In order to guarantee that the final network will be symmetrical, constraints are developed for two types of models, namely, the multilayer perceptron (MLP) network and the radial basis function (RBF) network. In global modeling problems it becomes crucial to impose conditions for symmetry in order to stand a chance of reproducing symmetry-related phenomena. Sufficient conditions are given for MLP and RBF networks to have a set of fixed points that are symmetrical with respect to the origin of the phase space. In the case of MLP networks such conditions reduce to the absence of bias parameters and the requirement of odd activation functions. This turns out to be important from a dynamical point of view since some phenomena are only observed in the context of symmetry, which is not a structurally stable property. The results are illustrated using bench systems that display symmetry, such as the Duffing-Ueda oscillator and the Lorenz system.

E. A. Mendes & C. Letellier

Displacement in the parameter space versus spurious solution of discretization with large time step,

Journal of Physics A,37, 1203-1218, 2004. Online

Abstract

In order to investigate a possible correspondence between differential and difference equations, it is important to possess discretization of ordinary differential equations. It is well known that when differential equations are discretized, the solution thus obtained depends on the time step used. In the majority of cases, such a solution is considered spurious when it does not resemble the expected solution of the differential equation. This often happens when the time step taken into consideration is too large. In this work, we show that, even for quite large time steps, some solutions which do not correspond to the expected ones are still topologically equivalent to solutions of the original continuous system if a displacement in the parameter space is considered. To reduce such a displacement, a judicious choice of the discretization scheme should be made. To this end, a recent discretization scheme, based on the Lie expansion of the original differential equations, proposed by Monaco and Normand-Cyrot will be analysed. Such a scheme will be shown to be sufficient for providing an adequate discretization for quite large time steps compared to the pseudo-period of the underlying dynamics.

[1] **K. Nielsen, P. G. Sorensen & F. Hynne**, Chaos in glycolysis, *Journal of Theoretical Biology*, **186**, 303-306, 1997.

[2] **R. Mickens**, *Nonstandard finite difference models of differential equations*, World Scientific, 1994.

[3] **C. Wilke, R.W. Leven & H. Deutsch**, Experimental and numerical study of prechaotic and chaotic regimes in a helium glow discharge, *Physics Letters A*, **136**, 114-120, 1989.