Continuous physical systems, such as electromagnetic fields or fluids, are described dynamically by partial differential equations, or field theories. Thus they have infinitely many degrees of freedom; accordingly, it came as a surprise at the end of the seventies that such systems can exhibit low dimensional chaos which is characteristic of systems with few degrees of freedom. In the meanwhile this miracle has been fully understood. By keeping a fluid in a box which is not too large compared to some typical macroscopic scale (like the size of a convection roll) one can maintain spatial coherence but the system will become temporally chaotic. Only a few spatial modes get appreciably excited, and their amplitudes define a low-dimensional “phase-space” in which the chaotic dynamics takes place.
KeywordsStationary Solution Invariant Manifold Unstable Manifold Center Manifold Topological Defect
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- [CE]Collet P. and J.-P. Eckmann: Instabilities and Fronts in Extended Systems, Princeton University Press (1990).Google Scholar
- [EGP]Eckmann J.-P., G. Goren and I. Procaccia: Nonequilibrium nucleation of topological defects as a deterministic phenomenon. Preprint. (Weizmann Institute 1991).Google Scholar
- [EP1]Eckmann J.-P. and I. Procaccia: The onset of defect mediated turbulence. Preprint. (Weizmann Institute 1990).Google Scholar
- [EP2]Eckmann J.-P. and I. Procaccia: The generation of spatio-temporal chaos in large aspect ratio hydrodynamics. Preprint. (Weizmann Institute 1990).Google Scholar
- [EW]Eckmann J.-P. and C.E. Wayne: Propagating fronts and the center manifold theorem. Preprint. (University of Geneva 1990).Google Scholar
- [EZ]Eckmann J.-P. and M. Zamora: Stationary solutions for the Swift-Hohenberg equation in nonuniform backgrounds. Preprint. (University of Geneva 1990).Google Scholar
- [HPS]M.W. Hirsch, C.C. Pugh, and M. Shub: Invariant Manifolds, Lecture Notes in Mathematics Vol. 583, Berlin, Heidelberg, New York, Springer (1977).Google Scholar
- [LW]Llave R. and C.E. Wayne: Whiskered and low dimensional tori for nearly integrable hamiltonian systems. (To appear). Nonlinearity.Google Scholar
- [N]Newell A.: (To appear). Phys. Rev. Lett.Google Scholar
- [Mo]Moser J.: Stabie and random motions in dynamical systems: with special emphasis on mechanics, Princeton University Press (1973).Google Scholar
- [RJ]Ribotta R., A. Joets: In Cellular Structures and Instabilities, (Wesfreid J.E. and S. Zaleski eds.). Springer (1984).Google Scholar
- [Se]Sevryuk M.B.: Reversible Systems, Lecture Notes in Mathematics, Vol. 1211, Springer (1986).Google Scholar