Jean Maquet

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.

Construction of phenomenological models from numerical scalar time series

Abstract The problem of reconstruction of ordinary differential equations from numerical scalar time series is discussed. Techniques are exemplified for Rossler and Lorenz chaotic attractors, with emphasis on improved algorithms with respect to those previously published. The steps which are still required in order to investigate experimental noisy data are discussed.

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.

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.

Structure-selection techniques applied to continous-time nonlinear models

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.

INDUCED ONE-PARAMETER BIFURCATIONS IN IDENTIFIED NONLINEAR DYNAMICAL MODELS

It is shown that nonlinear global models identified from a single time series can be used to reproduce the same sequence of bifurcations of the original system. This has been observed for simulated...

ANALYSIS OF A NONSYNCHRONIZED SINUSOIDALLY DRIVEN DYNAMICAL SYSTEM

A nonautonomous system, i.e. a system driven by an external force, is usually considered as being phase synchronized with this force. In such a case, the dynamical behavior is conveniently studied in an extended phase space which is the product of the phase space ℝm of the undriven system by an extra dimension associated with the external force. The analysis is then performed by taking advantage of the known period of the external force to define a Poincare section relying on a stroboscopic sampling. Nevertheless, it may so happen that the phase synchronization does not occur. It is then more convenient to consider the nonautonomous system as an autonomous system incorporating the subsystem generating the driving force. In the case of a sinusoidal driving force, the phase space is ℝm+2 instead of the usual extended phase space ℝm × S1. It is also demonstrated that a global model may then be obtained by using m + 2 dynamical variables with two variables associated with the driving force. The obtained model characterizes an autonomous system in contrast with a classical input/output model obtained when the driving force is considered as an input.

An Equivariant 3D model for the long‐term behavior of the solar activity

Modeling dynamics underlying the sunspot numbers is an important problem because such data indicate the relative activity of the Sun. A key point in modeling sunspot data, which follows an 11‐year cycle, is the need to take into account the reversal of the Sun’s magnetic field, which follows a 22‐year cycle. This can be done using an appropriate coordinate transformation applied to the phase portrait reconstructed from the sunspot numbers. Such a transformation introduces symmetry in the phase portrait and has the advantage of unfolding the structure of the dynamics. Global models have been obtained from such data. It is shown that the models capture the basic dynamical structure underlying the data which appears to be the structure of a Rossler attractor with an additional half twist.

Reconstruction of a set of differential equations modelling an experimental homoclinic chaos in the Belousov-Zhabotinskii reaction

We analyze chemical experimental time series by means of recent tools from nonlinear dynamics. More specifically, experiments on the Belousov-Zhabotinskii reaction [1] in a continuous flow reactor reveal a spiralling strange attractor which arises from a (global) homoclinic bifurcation. By using a global vector field reconstruction method, a set of ordinary differential equations is obtained from the measurements of the time dependence of [CeIV]. We show that a tridimensional space is sufficient to embed the behavior of the BZ reaction as suggested by previous works and that the reconstructed model allows us to exhibit topological properties which are not clearly evidenced from the experimental data. We investigate topological properties from the reconstructed set of ODEs and compare them with properties of the phase portrait directly reconstructed from the data and of the phase portrait associated with a 3D model proposed by Richetti et al.

Global Modeling and Differential Embedding

In order to reproduce the evolution of real economy over long period, a global model may be attempted to give a description of the dynamics with a small set of model coefficients. Then, the problem is to obtain a global model which is able to reproduce all the dynamical behavior of the data set studied starting from a set of initial conditions. Such a global model may be built on derivatives coordinates, i.e. the recorded time series and its successive derivatives. In this chapter, the mathematical background of a gobal modeling technique based on such a differential embedding will be exemplified on test cases of the real world (electrochemical and chemical experiments). Difficulties encountered in global modeling related to the nature of economic data records will be discussed. Properties of the time series required for a successful differential model will be defined.