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From nonlinear dynamics to biological data

by  Christophe Letellier,

University of Normandie 

CORIA, Avenue de l’Université, F-76800 Saint-Etienne du Rouvray (FRANCE)

 

 

Summary:

I. DYNAMICAL SYSTEMS AND STATE SPACE

The concepts of dynamical and its associated state space will be introduced. The role of singular points and their stability will be briefly introduced, and their different types (in two-dimensional space) discussed. The concept of bifurcation will be illustrated (yet in twodimensional space). When the attractor becomes recurrent (periodic, quasi-periodic, chaotic, ...), it is useful to investigate its properties through the concept of suface of section (commonly designed as Poincar´e section). For instance, the limit cycle becomes a periodic point, and a stability analysis similar to those developed for singular points can be performed, although with a different stability condition: for this latter reason, it is highly recommended to distinguish these two types of points (singular points and periodic points) which are too often designed in the literature as “fixed point”. Two markers will be introduced for a first characterization of chaotic behaviors: a Shannon entropy (more or less equivalent to the largest Lyapunov exponent) computed from a Poincar´e section and an embedding dimension.

II. ROUTE TO CHAOS AND PROPERTIES OF CHAOS

From a Poincar´e section, a bifurcation diagram can be constructed. Some examples will be provided, and some guidelines for reading them. A few main routes to chaos will be discussed: period-doubling cascade, intermittency (with its equivalent in fluid mechanics), Ruelle-Takens scenario, Curry-Yorke scenario... When the dynamics are strongly dissipative and low-dimensional, it is useful to compute a first-return to a Poincar´e section which often can offer a natural partition of the dynamics and thus allow the construction of a generating dynamics symbol. Basic elements of symbolic dynamics will be introduced for labelling periodic orbits. It will be explained how a Shannon entropy can be easily computed when a symbolic dynamics is introduced. A link between each domain and the strips of paper-sheet model (template) will be briefly illustrated.

III. A TAXONOMY OF CHAOS

Various examples of chaotic dynamics will be introduced using a taxonomy of chaos. It is based on the bounding tori, first-return map, the number of positive and negative Lypunov exponents and, ultimately the concept of template. Chaos, hyperchaos, toroidal chaos, conservative chaos, and their variant will be discussed. The challenging points of some of them will be exhibited. Some examples from the real world will be presented when known.

IV. OBSERVABILITY AND CONTROLABILITY

Commonly, once a time series is recorded in a system (numerical or experimental), no attention is paid to the variable measured. This is mainly due to the Takens theorem which states that an object of dimension d is diffeomorphically reconstructed in a space of dimension 2d + 1 using delay or derivative coordinates [3]. Nevertheless, one of the condition of the Takens theorem is that the measurement function ξ = h(x) must be generic, meaning that applying a perturbation to it must not change the result. Unfortunately, often, the measured variables are the result of a measurement function that is not generic, and the Takens theorem no longer holds. In control theory, there is a useful concept to know whether a state of the dynamical system under study can be retrieved from the measured data: this is called observability. For short, this is related to an observability matrix O d ξ which is the matrix of the d successive Lie derivatives of the measured variable (here we consider that the first Lie derivative is the measured variable iteself). It will be shown how a first (approximated) assessment of observability can be performed using a fluence graph. Using a variable providing a global observability of the state space is relevant for determining the state of the system, modeling, parameter identification, computing the embedding dimension, etc.). The dual of observability is controlability, to assess how it is possible to control the system under study. According to Kalman [1] and Lin [2], observability is the dual of controlability (established for linear system). Yet, using the fluence graph, it will be shown how the dual — a derivative of the system — of a measured variable can be identified. Then, when a global model is obtained, one of its outputs can be applied to control the system.

V. ATRIAL FIBRILLATION AS AN INTERMITTENCY

Atrial fibrillation is a chronic disease that affects the electrical activity of the heart and, consequently, its beating. Once this pathology is introduced, it will be explained how it can be viewed as an intermittency. Indeed, the route to chaos (or turbulence) is an interesting conceptual framework for understanding the slow evolution of the chronic disease from compensation (stable) to decompensation (acute). The heart variability will thus be investigated from tachograms, that is, from RR-intervals (as data from a Poincar´e section). A first-return map on ∆RR will be then used and some guidelines to read them will be introduced. A session on computers will be devoted to the treatment of some tachograms.

BIBLIOGRAPHY

[1] R. Kalman. On the general theory of control systems. IFAC Proceedings Volumes, 1(1):491–502, 1960. 1st International IFAC Congress on Automatic and Remote Control, Moscow, USSR, 1960.

[2] C.-T. Lin. Structural controllability. IEEE Transactions on Automatic Control, 19(3):201–208, 1974.

[3] F. Takens. Detecting strange attractors in turbulence. Lectures Notes in Mathematics, 898:366–381, 1981.