10h00   Alessandro Torcini   (Istituto dei Sistemi Complessi - CNR, Firenze)
Synchronization of extended chaotic systems
11h20   Xavier Leoncini   (Centre de Physique Théorique, Marseille)
Chaos de lignes de champ et structures lagrangiennes
11h40   Jean-Marc Ginoux   (Laboratoire PROTEE, Université du Sud Toulon Var)
Slow Invariant Manifold of Chaotic Dynamical Systems

Résumé du séminaire de Alessandro Torcini

The main mechanisms leading to synchronization in chaotic extended systems with short-range interactions are revised, while novel results concerning systems with long-range coupling are presented. In particular, the chaotic synchronization of replica of coupled map lattices is considered. The spatial extension allows to interpret the synchronization transitions (STs) as nonequilibrium critical phenomena. Within this framework two different kind of continuous transitions have been identified for nearest-neighbour coupling : one ruled by linear mechanisms and one by nonlinear effects [1,2]. In the first case the ST belongs to the multiplicative noise universality class, while in the nonlinear dominated case the critical exponents coincide with those measured for directed percolation (DP) [1,3]. More recently spatially extended chaotic systems with power-law decaying interactions have been considered [4]. Also in this situation the ST appears to be continuous, while the critical indexes vary with continuity with the power law exponent characterizing the interaction. Moreover, strong numerical evidences indicate that the transition belongs to the "anomalous directed percolation" family of universality classes found for Levy-flight spreading of epidemic processes [5].

References:
[1] L. Baroni, R. Livi, and A. Torcini, Physical Review E 63 6226 (2001); V. Ahlers & A. Pikovsky, Phys. Rev. Lett. 88, 254101 (2002)
[2] M. Cencini & A. Torcini. Physical Review E 63 056201 (2001); Physica D 208 191 (2005)
[3] M.A. Munoz & R. Pastor-Satorrass, Phys. Rev. Lett. 90, 204101 (2003); F. Ginelli et al., Phys. Rev. E 68, 065102(R) (2003); F. Ginelli et. al., Phys. Rev. E 67, 046217 (2003)
[4] C. J. Tessone, M. Cencini, and A. Torcini, to appear in Phys. Rev. Lett. (2006) (eprint arXiv:nlin/0607068)
[5] H. Hinrichsen & M. Howard, The European Physical Journal B , 7, 635 (1999)

Résumé du séminaire de Xavier Leoncini

Nous nous intéressons à la structures tri-dimensionnelles des lignes de champs à divergence nulle. Dans un premier temps nous ferons un bref rappel sur la nature hamiltonienne du problème et le caractère chaotique de ces lignes. Puis nous illustrons comment utiliser ces propriétés d'une part pour construire un algorithme d'intégration de ces lignes et d'autre part pour caractériser et localiser des structures "cohérentes". Ceci sera illustré à l'aide d'un champ de type ABC et le champ magnétique issu d'une simulation MHD.

Résumé du séminaire de Jean-Marc Ginoux

The aim of this presentation is to show that Differential Geometry local metric properties of curvature and torsion of the trajectory curves of chaotic dynamical systems provides the slow invariant manifold associated with such systems. Then, using Darboux 's theory and the so-called tangent linear system approximation, it is established that such manifold is local first integral of this dynamical systems.