Research Groups

1  Control and Synchronization in Extended Systems

R. Lima (CNRS, Marseille), M. Pettini (Firenze)

Irregular and chaotic dynamics have been identified in the most varied fields of engineering, biology, social sciences, and economics. In some cases the suppression of irregular behavior is desirable. In others cases the perspective of using the increased dynamical richness and flexibility of nonlinear unstable systems opens up a whole new field of technological applications. The challenging problems lie mostly in the domain of extended systems where the large number of degrees of freedom makes the traditional control-of-chaos techniques to be unapplicable.

Spatially extended dynamical systems are typical of many experimental situations, examples being found in hydrodynamics, plasma physics, chemical reactions and biological systems. Simple models, that exhibit the essential features of space-time complexity, are the coupled map lattices (CML) [15] [10]. Problems that have been studied in this field are:

Previous studies on these subjects are to be pursued in this project, having in mind, in particular, the extension of the work done on intermittence in local turbulence [26].

Two related issues in CML, for which very little is known, are the problem of travelling solutions with a given shape and that of synchronized dynamics [1]. Several authors have studied the relationships between synchronization and control of chaos [24] [13]. Usually synchronization of coupled dynamical systems has been studied from two main points of view. In one case, the analysis concerns a small number of coupled systems [7] where analytic methods can be applied, whereas in the other, it involves large systems where numerical studies are often the only possible technique. Recently, coupled map lattice models have received considerable attention and offer a third option between these extremes. In this field it is worthwhile to mention the recent understanding of the mechanisms by which synchronization takes place in networks of identical maps which are globally coupled [12]. Control of this synchronization mechanism can be used in a wide range of situations, running from safety in the encoding and transmission of information to the control of physical real systems in engineering or fluid dynamics.

Turbulence modeling and turbulence control in fluid dynamics is a paradigmatic problem and successful techniques in this domain may serve as models to deal with other extended systems. Therefore, this will also be one of main themes of this research group. The problem of turbulence means different things for the physicist and for the engineering designer. The physicist, as usual, is mainly interested in general laws. Therefore he prefers to study fully developed turbulence away from any borders because only there will he expect to find universal features, independent of the nature of the fluid and boundary conditions. Small scale phenomena, presumably independent of the boundary conditions, are the comfortable arena chosen by the theoretical physicist. For the engineer, on the other hand, boundary conditions and large scale properties of the devices are everything. This situation is now bound to change if further progress is to be achieved and a real understanding of turbulence requires calculations and models that are globally sensitive to all length scales. This, in particular, brings back the question of the relation of the phenomenological models to actual solutions of the Navier-Stokes equation, and the even older question of whether this equation is able to capture all the features of the turbulence phenomenon. On the engineering side a good control of the interplay of the instabilities at several scales is even more important if active turbulence control methods are attempted. As a conclusion, phenomenological models cannot lead to much further progress and because a global understanding of turbulence at all scales seems to be required, the two fields, theoretical physics and fluid engineering, are condemned to merge if real progress is to be achieved.

It is clear that the development of this subproject needs a collaboration with a large number of other people as well as a permanent interdisciplinary effort, which is one of the main characteristics of this project. In particular we highlight two points: (i) experimentalists will be involved in the project (P. LeGal, Marseille and L. Fronzoni, Pisa) who are exploring the possibility to carry out a more refined laboratory experiment in Bielefeld on liquid crystals; a deeper understanding of both synchronization and control is expected to come out in this context from the close interaction among experimentalists and theoreticians; (ii) learning, either unsupervised or supervised, is a common feature of many systems which are present in other topics of this project.



  1. V.S. Afraimovich, S.N. Chow and J.K. Hale, Physica D 103 (1997) 442.
  2. I.S. Aronson, V.S. Afraimovich and M.I. Rabinovich, Nonlinearity 3 (1990) 639.
  3. N. Aubry, R. Guyonnet and R. Lima, J. Nonlinear Science 2 (1992).
  4. L. Battiston, L.A. Bunimovich and R. Lima, Complex Systems 5 (1991) 415.
  5. L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1 (1988) 491.
  6. L.A. Bunimovich, A. Lambert and R. Lima, J. Stat. Phys. 61 (1990) 253.
  7. T.L. Carrol and L.M. Pecora, Int. Journal of Bif. and Chaos, 2 (1992), 669.
  8. S. Ciliberto and P. Bigazzi, Phys.Rev. Lett. 60 (1988) 286.
  9. H. Chaté and P. Manneville, Physica D 32 (1988) 409; 37 (1989) 33.14.
  10. J. Crutchfield and K. Kaneko, in Directions in Chaos, World Scientific, Singapore, 1987.
  11. B. Fernandez, J. Stat. Phys. 72 (1996).
  12. A.L. Gelover-Santiago, R. Lima, G. Martinez-Mekler, preprint Marseille 1997.
  13. R.O. Grigoriev, M.C. Cross and H.G. Schuster, Phys. Rev. Letters. 79, n. 15, 2795.
  14. R. Guyonnet and R. Lima, proceedings of Conference on Nonlinear Dynamics in Plasma Physics, Algier 1990.
  15. K. Kaneko, Prog. Theor. Phys. 72 (1984) 480; 74 (1985) 1033.
  16. K. Kaneko, Collapse of Tori and Genesis of Chaos in Dissipative Systems, World Scientific, Singapore, 1986.
  17. K. Kaneko, Physica D 37 (1989) 60.
  18. J.D. Keller and J.D. Farmer, Physica D 23 (1986) 413.
  19. S.P. Kuznetsov and A.S. Pikovsky, Physica D 19 (1986) 384.
  20. A. Lambert and R. Lima, Physica D71 (1994) 390-411.
  21. R. Lima, Chaos 2 (1992) 315.
  22. P. Manneville, Dissipative Structures and Weak Turbulence (Academic Press, New York,1990).
  23. G.L. Oppo and R. Kapral, Phys Rev. A 33 (1986) 4219.
  24. J.W. Shuai, K.W. Wong and L.M. Cheng, Physical Review E 56, n. 2, 2272.25.
  25. K.R. Sreenivasan and P. Kailasnath, Phys. of Fluids A5, 512, (1993).
  26. E. Ugalde, R. Lima, Physica D (1996).
  27. I. Waller and R. Kapral; Phys. Rev. A30 (1984) 2074.
  28. T. Yamada and H. Fujisaka, Progr. Theoretical Phys. 72 (1984) 885.


With special support by Ecos-Nord


People of the Group at ZiF on 25-Feb-2001

People of the Group at ZiF on 25-Feb-2001
First row (left to right): Patrice Le Gal, Albert Cordonet, Ricardo Lima, Anne Cros, Marion Chazottes, Alix Chazottes;
Second row (left to right): Roberto Livi, Antonio Politi, Jesús Urías, Elena Floriani, Edgardo Ugalde, Jean-René Chazottes, Galina Erochenkova, Sébastien Jaeger, Valentin Afraimovich, Dima Volchenkov, Karl Kürten.