Extended Systems Close to Threshold

 Members of the group (from left to right): Angelica Gelover Santiago, Dima Volchenkov, Ouerdia Ourrad, Jean Leray, Tounsia Benzekri, Ricardo Lima, Françoise Briolle; not on this picture: Galina Erochenkova and Elena Floriani; click here for another picture.

Extended dynamical systems may run close to a stability threshold in different ways:

1. either in time or space, or in both, exploring regions of the phase space with different behaviors,
2. or, due to intrinsic or external reasons, the parameters of the system may be changing and, as a consequence, mixing different regimes.

This group studies different physical situations where the above scenarios may lead to increasing complexity (click here for an idea of mathematical complexity) or even to the need of searching for different measures of complexity.

The simplest case is the behavior of a (simple!) system at the threshold of stability. A simple "toy" model for a system at a threshold of stability is proposed. We assumed that the system looses stability when the parameter characterizing its state exceeds a threshold value. We study the distribution of sojourn times below the threshold that characterizes the ergodic properties of a system and allows to measure its complexity. This distribution can be found analytically for the most physically meaningful cases in our model. We have treated both the state parameter and the threshold as random independent variables distributed on the unit interval with respect to different distribution laws. The natural control parameter in our toy model is a relative frequency of threshold changes. Varying this frequency, one can control the statistics of switching events (i.e. when the state parameter passes the stability threshold) tuning it from the truly exponential decay distribution of laminar phases to a power-law with some exponent. The crucial advantage of the proposed model is that, for most physically meaningful distributions, it can be solved analytically. In other cases, a numerical solution can readily be found. We have justified the result that the well-known -3/2-power law statistics observed in many real physical systems (on-off intermittency) corresponds the case of a Gaussian noise drive. We have shown, by the direct computation, that the statistics of quiescent length fits a -3/2-power law for intermediate times and departs from the power law to faster decay rates for long times. We also proved that, for different distributions, different exponents may be obtained or even, no scaling at all.

A nice deterministic counterpart to this model is given by a skew product dynamical system, a case for which Jean Leray, a student visiting our group, is preparing his mind with a work in progress.

The diffusion of a marker through irregular packed beds combines deterministic instability in extended systems with stochastic fluctuations. After a first mathematical rigorous study (ZiF publications 2001/070, 2000/003) and a companion more phenomenological approach (ZiF publication 2000/018), we are now numerically exploring the situation not covered in the previous theoretical framework. Ouerdia Ourrad is working strongly in that direction, and we may benefit from future help of other groups in the project.

Surface waves in presence of resonances is an other situation in fluid dynamics where extended dynamics is the central issue. In the framework of the Zakharov's formalism, this is an Hamiltonian system. In this context, solutions which are not traveling waves share a higher level of complexity since in this case it is no longer sufficient to describe the form and celerity to characterize them. Their existence, as well as a series expansion allowing for its practical computation is obtained for the first time (ZiF publication 2000/086) and Tounsia Benzekri is now working on the extension of this result to a model of 2 interacting resonances. Next, by adding a stochastic perturbation, which is physically reasonable to model a water-wind interaction, she may give a realistic insight to previous work done by our group (ZiF publication 2000/011).

Partial synchronization in extended systems has been studied by us for models of Globally Coupled Maps (ZiF publication 2001/085). By acting on a specific spatial position, the system is maintained close to the threshold of different cluster synchronizations and shows then the same complex behavior of the previous cases. By assigning to different synchronized regimes "normal" or "abnormal" qualities, Angelica Gelover-Santiago is working on more complex methodological tests for the work in progress with the Mechatronics Group on Immunity based techniques for anomaly detection in industrial systems.

How to actually measure the complexity in such different situations, being then able to compare them with the real word studied by the experimentalists ("The Experiment") is an open question that deserves the active attention of Françoise Briolle and Ricardo Lima.