Journée de Dynamique Non Linéaire

10h30   Mark Freidlin (University of Maryland)
Stochastic resonance and related problems

11h40   Alessandro Torcini (Istituto Nazionale di Ottica Applicata, Florence)
A mechanism for superspiral formation and breakup

Résumé du séminaire de M. Freidlin

If a dynamical system has more than one asymptotically stable equilibrium or other attractors, random perturbations of such a system may lead to non-random, in a sense, oscillations or stable equilibriums, which are not availible in the non-perturbed system. These effects are a manifestation of the large deviation laws. The general framework for such results is consideration of stochastic perturbations of dynamical systems with fast and slow components. In particular, in the non-autonomous systems, the time can play the part of slow component. Many popular in applications models can be incorporated in these scheme and considered from a common point of view. Some results related to this talk can be found in the following references.
[1] L.Gammaitoni at al, Stochastic resonance, Rev. of Modern Physics, 70(1), 223-288 (1998).
[2] M.Freidlin, Quasi-deterministic approximation, metastability, and stochastic resonance, Physica D, 137, 333-352 (2000).
[3] M.Freidlin, On stable oscillations and equilibriums induces by small noise, Journal of Statistical Physics, 103,1/2,283-300 (2001).
[4] M.Freidlin, On stochastic perturbations of dynamical systems with fast and slow components, Stochastics and Dynamics, 1(2), 261-281 (2001).

Résumé du séminaire de A. Torcini

The Eckhaus instability of travelling waves in the one dimensional complex Ginzburg-Landau equation (CGLE) produces modulated amplitude waves (MAWs) that are parametrized by the spatial period of the modulation P and the average local wave number (winding number). First, we relate the transition from phase to defect turbulence in the CGLE to a saddle-node bifurcation of these coherent structures (MAWs). In particular, the period P of MAWs is shown to be limited by a maximum value P_{SN}, which depends on the CGLE coefficients; MAW-like initial conditions with P > P_{SN} evolve to defects. Moreover in the phase chaotic regime, slowly evolving ,,near MAWs'' of various periods p occur naturally. Anytime that p becomes larger than P_{SN} defects are generated in the system. Second, the analysis of the existence and stability domains of the MAWs suggests a new interpretation of recent experiments, exhibiting a transition from rotating spirals to modulated ,,superspirals'' and finally to spiral breakup in a pattern forming chemical reaction (a Belousov-Zhabotinsky reaction).
[1] L. Brusch et al., Phys. Rev. Lett. 85, 86 (2000); Physica D 160 (2001) 127-148.
[2] A. Torcini, Phys. Rev. Lett., 77, 1047 (1996).
[3] L. Q. Zhou and Q. Ouyang, Phys. Rev. Lett. 85, 1600 (2000).