Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ahlers G. and R.P. Behringer: Prog. Theor. Phys. Suppl. 64, 186 (1979).
Auerbach D., P. Cvitanović, J.-P. Eckmann, G. Gunaratne, and I. Procaccia: Exploring chaotic motions through periodic orbits. Phys. Rev. Lett 58, 2387–2389 (1987).
Aronson, D. and H. Weinberger: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978).
Bodenschatz E., W. Pesch and L. Kramer: Physica 32D, 135 (1988).
Collet P. and J.-P. Eckmann: Instabilities and Fronts in Extended Systems, Princeton University Press (1990).
Coullet P., L. Gil and F. Lega: Phys. Rev. Lett. 62, 1619 (1989).
Eckmann J.-P., G. Goren and I. Procaccia: Nonequilibrium nucleation of topological defects as a deterministic phenomenon. Preprint. (Weizmann Institute 1991).
Eckmann J.-P. and I. Procaccia: The onset of defect mediated turbulence. Preprint. (Weizmann Institute 1990).
Eckmann J.-P. and I. Procaccia: The generation of spatio-temporal chaos in large aspect ratio hydrodynamics. Preprint. (Weizmann Institute 1990).
Eckmann J.-P. and C.E. Wayne: Propagating fronts and the center manifold theorem. Preprint. (University of Geneva 1990).
Eckmann J.-P. and M. Zamora: Stationary solutions for the Swift-Hohenberg equation in nonuniform backgrounds. Preprint. (University of Geneva 1990).
Goren G., I. Procaccia, S. Rasenat, and V. Steinberg: Interactions and dynamics of topological defects: Theory and experiments near the onset of weak turbulence. Phys. Rev. Lett. 63, 1237–1240 (1989).
Greenside H.S. and M.C. Cross: Stability analysis of two-dimensional models of three-dimensional convection. Phys. Rev. A 31, 2492–2501 (1985).
E. Hairer, S.P. Nørsett, and G. Wanner: Solving Ordinary Differential Equations I, Berlin, Heidelberg, New York, Springer (1987).
M.W. Hirsch, C.C. Pugh, and M. Shub: Invariant Manifolds, Lecture Notes in Mathematics Vol. 583, Berlin, Heidelberg, New York, Springer (1977).
Kirchgässner K.: Nonlinearly Resonant Surface Waves and Homoclinic Bifurcation. Adv. Appl. Mech. 26, 135–181 (1988).
Kramer L., H.R. Schober, and W. Zimmermann: Pattern competition and the decay of unstable patterns in quasi-one-dimensional systems. Physica D31, 212–226 (1988).
Langer J.S.: Theory of the condensation point. Ann. Physics 41, 108–157 (1967).
Langer J.S. and M.E. Fisher: Intrinsic critical velocity of a superfluid. Phys. Rev. Lett. 19, 560–563 (1967).
Llave R. and C.E. Wayne: Whiskered and low dimensional tori for nearly integrable hamiltonian systems. (To appear). Nonlinearity.
Lowe M. and J.P. Gollub: Pattern selection near the onset of convection: The Eckhaus instability. Phys. Rev. Lett. 55, 2575–2578 (1985).
Mielke A.: Reduction of Quasilinear Elliptic Equations in Cylindrical Domains with Applications. Math. Meth. Appl. Sci. 10, 51–66 (1988).
Newell A.: (To appear). Phys. Rev. Lett.
Moser J.: Stabie and random motions in dynamical systems: with special emphasis on mechanics, Princeton University Press (1973).
Pocheau A., V. Croquette, and O. LeGal: Phys. Rev. Lett. 55, 1099 (1985).
Rehberg I., S. Rasenat, and V. Steinberg: Phys. Rev. Lett. 62, 756 (1989).
Ribotta R., A. Joets: In Cellular Structures and Instabilities, (Wesfreid J.E. and S. Zaleski eds.). Springer (1984).
Sevryuk M.B.: Reversible Systems, Lecture Notes in Mathematics, Vol. 1211, Springer (1986).
Swift J. and P.C. Hohenberg: Phys. Rev. A15, 319 (1977).
Zippelius A. and E.D.Siggia: Stability of finite-amplitude convection. Phys. Fluids 26, 2905–2915 (1983).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Plenum Press, New York
About this chapter
Cite this chapter
Eckmann, JP., Procaccia, I. (1991). Spatio-Temporal Chaos. In: Artuso, R., Cvitanović, P., Casati, G. (eds) Chaos, Order, and Patterns. NATO ASI Series, vol 280. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-0172-2_6
Download citation
DOI: https://doi.org/10.1007/978-1-4757-0172-2_6
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-0174-6
Online ISBN: 978-1-4757-0172-2
eBook Packages: Springer Book Archive