Christophe Letellier

Analysis of the dynamics of a realistic ecological model

Abstract A fairly realistic three-species food chain model based on the Leslie–Gower scheme is investigated by using tools borrowed from the nonlinear dynamical systems theory. It is observed that two co-existing attractors may be generated by this ecological model. A type-I intermittency is characterized and a homoclinic orbit is found.

Unstable periodic orbits and templates of the Rössler system: Toward a systematic topological characterization

The Rossler system has been exhaustively studied for parameter values (a in [0.33,0.557],b=2,c=4). Periodic orbits have been systematically extracted from Poincare maps and the following problems have been addressed: (i) all low order periodic orbits are extracted, (ii) encoding of periodic orbits by symbolic dynamics (from 2 letters up to 11 letters) is achieved, (iii) some rules of growth and of pruning of the periodic orbits population are obtained, and (iv) the templates of the attractors are elaborated to characterize the attractors topology. (c) 1995 American Institute of Physics.

Modeling Nonlinear Dynamics and Chaos: A Review

This paper reviews the major developments of modeling techniques applied to nonlinear dynamics and chaos. Model representations, parameter estimation techniques, data requirements, and model validation are some of the key topics that are covered in this paper, which surveys slightly over two decades since the pioneering papers on the subject appeared in the literature.

On the non-equivalence of observables in phase-space reconstructions from recorded time series

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 R ¨ ossler 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.

Investigating nonlinear dynamics from time series: The influence of symmetries and the choice of observables

When a dynamical system is investigated from a time series, one of the most challenging problems is to obtain a model that reproduces the underlying dynamics. Many papers have been devoted to this problem but very few have considered the influence of symmetries in the original system and the choice of the observable. Indeed, it is well known that there are usually some variables that provide a better representation of the underlying dynamics and, consequently, a global model can be obtained with less difficulties starting from such variables. This is connected to the problem of observing the dynamical system from a single time series. The roots of the nonequivalence between the dynamical variables will be investigated in a more systematic way using previously defined observability indices. It turns out that there are two important ingredients which are the complexity of the coupling between the dynamical variables and the symmetry properties of the original system. As will be mentioned, symmetries and the choice of observables also has important consequences in other problems such as synchronization of nonlinear oscillators. (c) 2002 American Institute of Physics.

What can be learned from a chaotic cancer model

A simple model of three competing cell populations (host, immune and tumor cells) is revisited by using a topological analysis and computing observability coefficients. Our aim is to show that a non-conventional analysis might suggest new trends in understanding the interactions of some tumor cells and their environment. The action of some parameter values on the resulting dynamics is investigated. Our results are related to some clinical features, suggesting that this model thus captures relevant phenomena to cell interactions.

Evidence for low dimensional chaos in sunspot cycles

Sunspot cycles are widely used for investigating solar activity. In 1953 Bracewell argued that it is sometimes desirable to introduce the inversion of the magnetic field polarity, and that can be done with a sign change at the beginning of each cycle. It will be shown in this paper that, for topological reasons, this so-called Bracewell index is inappropriate and that the symmetry must be introduced in a more rigorous way by a coordinate transformation. The resulting symmetric dynamics is then favourably compared with a symmetrized phase portrait reconstructed from the z -variable of the Rossler system. Such a link with this latter variable – which is known to be a poor observable of the underlying dynamics – could explain the general difficulty encountered in finding evidence of low-dimensional dynamics in sunspot data.

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

We examine the dynamics of three-dimensional cells with square planform in dissipative Rayleigh-Benard 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.

Observability of multivariate differential embeddings

The present paper extends some results recently developed for the analysis of observability in nonlinear dynamical systems. The aim of the paper is to address the problem of embedding an attractor using more than one observable. A multivariate nonlinear observability matrix is proposed which includes the monovariable nonlinear and linear observability matrices as particular cases. Using the developed framework and a number of worked examples, it is shown that the choice of embedding coordinates is critical. Moreover, in some cases, to reconstruct the dynamics using more than one observable could be worse than to reconstruct using a scalar measurement. Finally, using the developed framework it is shown that increasing the embedding dimension, observability problems diminish and can even be eliminated. This seems to be a physically meaningful interpretation of the Takens embedding theorem.

Should all the species of a food chain be counted to investigate the global dynamics

Abstract A fairly realistic three-species food-chain model based on Lotka–Volterra and Leslie–Gower schemes is investigated assuming that just a single scalar time series is available. The paper uses tools borrowed from the theory of nonlinear dynamical systems. The quality of the different phase portraits reconstructed is tested. Such a situation would arise in practice whenever only a single species is counted. It is found that the dynamical analysis can be safely performed when a single species involved in the food chain is counted if many thousands of observations are available. If not, a global model can be obtained from the available data and subsequently used to produce all the data required for a detailed analysis. In this case, however, the choice of which species to consider in order to obtain a model is crucially important.