Spatio-temporal chaos

Spatially extended systems serve as a basis for the study and modeling of spatio-temporal behaviour in a large class of physical, chemical and biological systems. Their behaviour includes periodic patterns, traveling interfaces, spirals, synchronization, etc. Often, the complex interaction between time and space gives rise to spatio-temporal chaos (e.g. turbulent behaviour).

I am interested in the characterization and study of spatio-temporal chaos. I deal both with a) dynamical systems whose equations of motion are given or known and with b) systems whose dynamics is not known and one has to resort to time series observations. One of the most common and useful tools to characterize chaotic behaviour is given by the Lyapunov exponents. In particular, from the Lyapunov spectrum (LS) (collection of all Lyapunov exponents) it is possible to estimate bounds for the effective number of degrees of freedom of the system [11] (i.e. the dimension of the attractor) and the mean growth rate of infinitesimal perturbations [12].

Lyapunov Spectrum from subsystem information [a.5]
The computation of the LS involves matrix manipulation techniques that soon become prohibitive (computing time and memory storage) as the number of system variables increases [13]. For typical extended dynamical systems, the number of variables is too large to apply conventional LS computation techniques. To avoid this problem, it is often possible to estimate the whole LS (of the original system) from a subsystem LS based upon a relatively small subset of the system variables. Thus, it is possible to obtain a LS density (LS per unit volume). I developed new methods, both to extract the subsystem LS and to rescale this subsystem LS to obtain a good approximation of the original LS.

Lyapunov Spectrum from time series [t.3,a.9,a.10]
As an important extension of the above, I am interested in developing spatio-temporal embedding techniques, in order to reconstruct the LS using time series. The aim is to reconstruct the whole LS when the sole information about the system's dynamics is a collection of measurements at discrete spatial locations. In practice it is extremely difficult to reconstruct spatial features from one-dimensional data. However, by considering multi-dimensional data acquired at different spatial locations, it is possible to resolve some of the complex spatio-temporal structure.

Stable core for extended dynamical systems [t.2]
I am interested in the transition to chaos as the size of the considered subsystem is increased. This transition from stable to chaotic behaviour suggests that some spatially extended systems possess an inherent stable core. This stable core is responsible for a shift on intensive quantities related to the Lyapunov spectra (dimension density, entropy density, etc...).

Thermodynamic limit and boundary effects [a.7]
I am involved in addressing some aspects of the thermodynamic limit (infinite number of degrees of freedom, i.e. infinite domain size) for spatio-temporal systems. In particular the possibility of reproducing some thermodynamic invariants (probability densities, power spectra, two-point correlations) with the use of a truncated (finite) version of the original system with proper inputs at the boundaries.

Future directions: reconstruction of continuous extended dynamical systems
Most of the above results where obtained and tested for discrete spatio-temporal systems. It is important to extend these ideas to continuous space-time models. Continuous spatio-temporal systems (models or real physical systems) typically produce an enormous amount of data. An important practical question arises: what is the most efficient way to probe (space and time) in order to achieve the best reconstructions? On a deeper quest, one should address the validity and limitations of spatio-temporal embeddings. Takens' theorem [14] provides the formal basis when reconstructing generic dynamical systems and observations. Most spatio-temporal systems lack the genericity conditions (cf. translational invariance). Surprisingly however, it is often possible to achieve a reasonably good reconstruction.