C. Letellier & J. Maquet,
Analyse de dynamiques chaotiques : une nouvelle approche de l’activité solaire
Bulletin de la Société Française de Physique, 154, 10-13, 2006. Online
Abstract: Les processus impliqués dans la physique des plasmas étant très souvent non linéaires, c’est naturellement que cette discipline peut être désormais abordée dans le cadre de la théorie des systèmes dynamiques non linéaires. Plus que de nouvelles techniques d’analyse qui viendraient simplement s’ajouter à d’autres, c’est une nouvelle manière de voir les comportements dynamiques que propose la théorie du chaos’ : les fluctuations apparemment irrégulières se révèlent ordonnées dans l’espace des phases, des variations du comportement sont associées à des bifurcations et des modèles relativement simples au regard de la physique impliquée peuvent être obtenus. Nous proposons ici un rapide survol de ces techniques dans deux contextes différents de la physique des plasmas.
Estimating the Shannon entropy: recurrence plots versus symbolic dynamics
Physical Review Letters, 96, 254102, 2006. Online
Abstract: Recurrence plots were first introduced to quantify the recurrence properties of chaotic dynamics. A few years later, the recurrence quantification analysis was introduced to transform graphical representations into statistical analysis. Among the different measures introduced, a Shannon entropy was found to be correlated with the inverse of the largest Lyapunov exponent. The discrepancy between this and the usual interpretation of a Shannon entropy is solved here by using a new definition - still based on the recurrence plots - and it is verified that this new definition is correlated with the largest Lyapunov exponent, as expected from the Pesin conjecture. A comparison with a Shannon entropy computed from symbolic dynamics is also provided.
C. Letellier, J. Maquet, L. A. Aguirre & R. Gilmore
Evidence for low dimensional chaos in the sunspot cycles,
Astronomy & Astrophysics, 449, 379-387, 2006. Online
Abstract: 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 -variable of the Rössler 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.
G. F. V. Amaral, C. Letellier & L. A. Aguirre,
Piecewise affine models of chaotic attractors: the case of the Rössler system,
Chaos, 16, 013115, 2006. Online
Abstract: This paper proposes a procedure by which it is possible to synthesize Rössler and Lorenz dynamics by means of only two affine linear systems and an abrupt switching law. Comparison of different (valid) switching laws suggests that parameters of such a law behave as co-dimension one bifurcation parameters that can be changed to produce various dynamical regimes equivalent to those observed with the original systems. Topological analysis is used to characterize the resulting attractors and to compare them with the original attractors. The paper provides guidelines that are helpful to synthesize other chaotic dynamics by means of switching affine linear systems.
C. Letellier, L. A. Aguirre & J. Maquet,
How the choice of the observable may influence the analysis of non linear dynamical systems,
Communications in Nonlinear Science and Numerical Simulation, 11 (5), 555-576, 2006. Online
Abstract: A great number of techniques developed for studying nonlinear dynamical systems start with the embedding, in a -dimensional space, of a scalar time series, lying on an -dimensional object, . In general, the main results reached at are valid regardless of the observable chosen. In a number of practical situations, however, the choice of the observable does influence our ability to extract dynamical information from the embedded attractor. This may arise in standard problems in nonlinear dynamics such as model building, control theory and synchronization. To some degree, ease of success will thus depend on the choice of observable simply because it is related to the observability of the dynamics. Investigating the Rössler system, we show that the observability matrix is related to the map between the original phase space and the differential embedding induced by the observable. This paper investigates a form for the observability matrix for nonlinear system wich is more general than the previous one used. The problem of controllability is also mentioned.
C. Letellier, E. Roulin & O. E. Rössler,
Inequivalent topologies of chaos in simple equations,
Chaos, Solitons & Fractals, 28, 337-360, 2006. Online
Abstract: In the 1970, one of us introduced a few simple sets of ordinary differential equations as examples showing different types of chaos. Most of them are now more or less forgotten with the exception of the so-called Rössler system published in Physics Letters A, 57 (5), 397-398, 1976. In the present paper, we review most of the original systems and classify them using the tools of modern topological analysis, that is, using the templates and the bouding tori recently introduced by Tsankov and Gilmore in Physical Review Letters, 91 (13), 134104, 2003. Thus, examples of inequivalent topologies of chaotic attractors are provided in modern spirit.