Abstract
We investigate the construction of low-dimensional spatiallylocalized models of extended systems. Specifically, theKuramoto–Sivashinsky (KS) equation on large one-dimensional domainsdisplays spatiotemporally complex dynamics that are remarkablywell-localized in both real and Fourier space, as demonstrated by a(spline) wavelet representation. We show how wavelet projectionsmay be used to construct various localized, relativelylow-dimensional models of KS spatiotemporal chaos. There ispersuasive evidence that short, periodized systems, internally forcedat their largest scales, form minimal models for chaotic dynamics inarbitrarily large domains. Such models assist in the understandingof extended systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aubry, N., Holmes, P., Lumley, J. L., and Stone, E., 'The dynamics of coherent structures in the wall region of a turbulent boundary layer', Journal of Fluid Mechanics 192, 1988, 115–173.
Barabási, A.-L. and Stanley, H. E., Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, 1995.
Berkooz, G., Elezgaray, J., and Holmes, P., 'Coherent structures in random media and wavelets', Physica D 61, 1992, 47–58.
Browning, G. L., Henshaw, W. D., and Kreiss, H.-O., 'A numerical investigation of the interaction between the large and small scales of the two-dimensional incompressible Navier-Stokes equations', preprint, April 1998; UCLA CAM Report 98-23, 1998.
Dankowicz, H., Holmes, P., Berkooz, G., and Elezgaray, J., 'Local models of spatio-temporally complex fields', Physica D 90, 1996, 387–407.
Elezgaray, J., Berkooz, G., and Holmes, P., 'Wavelet analysis of the motion of coherent structures', in Progress in Wavelet Analysis and Applications, Y. Meyer and S. Roques (eds.), Editions Frontières, Gif-sur-Yvette, 1993, pp. 471–476.
Elezgaray, J., Berkooz, G., and Holmes, P., 'Large-scale statistics of the Kuramoto-Sivashinsky equation: A wavelet-based approach', Physical Review E 54, 1996, 224–230.
Halpin-Healy, T. and Zhang, Y.-C., 'Kinetic roughening phenomena, stochastic growth, directed polymers and all that', Physics Reports 254, 1995, 215–414.
Hayot, F., Jayaprakash, C., and Josserand, C., 'Long-wavelength properties of the Kuramoto-Sivashinsky equation', Physical Review E 47, 1993, 911–915.
Holmes, P., Lumley, J. L., and Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, 1996.
Hyman, J. M., Nicolaenko, B., and Zaleski, S., 'Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interfaces', Physica D 23, 1986, 265–292.
Kardar, M., Parisi, G., and Zhang, Y.-C., 'Dynamic scaling of growing interfaces', Physical Review Letters 56, 1986, 889–892.
L'vov, V. S., Lebedev, V. V., Paton, M., and Procaccia, I., 'Proof of scale invariant solutions in the Kardar-Parisi-Zhang and Kuramoto-Sivashinsky equations in 1+1 dimensions: Analytical and numerical results', Nonlinearity 6, 1993, 25–47.
Myers, M., Holmes, P., Elezgaray, J., and Berkooz, G., 'Wavelet projections of the Kuramoto-Sivashinsky equation I. Heteroclinic cycles and modulated traveling waves for short systems', Physica D 86, 1995, 396–427.
Perrier, V. and Basdevant, C., 'Periodical wavelet analysis, a tool for inhomogeneous field investigation. Theory and algorithms', La Recherche Aérospatiale 1989(3), 1989, 53–67.
Pomeau, Y., Pumir, A., and Pelce, P., 'Intrinsic stochasticity with many degrees of freedom', Journal of Statistical Physics 37, 1984, 39–49.
Pumir, A., 'Statistical properties of an equation describing fluid interfaces', Journal de Physique 46, 1985, 511–522.
Sneppen, K., Krug, J., Jensen, M., Jayaprakash, C., and Bohr, T., 'Dynamic scaling and crossover analysis for the Kuramoto-Sivashinsky equation', Physical Review A 46, 1992, R7351–R7354.
Toh, S., 'Statistical model with localized structures describing the spatio-temporal chaos of Kuramoto-Sivashinsky equation', Journal of the Physical Society of Japan 56, 1987, 949–962.
Wittenberg, R.W., 'Local dynamics and spatiotemporal chaos. The Kuramoto-Sivashinsky equation: A case study', Ph.D. Thesis, Princeton University, 1998.
Wittenberg, R. W., 'Dissipativity, analyticity and shocks in the (de)stabilized Kuramoto-Sivashinsky equation', 2000, in preparation.
Wittenberg, R. W. and Holmes, P., 'Scale and space localization in the Kuramoto-Sivashinsky equation', Chaos 9, 1999, 452–465.
Yakhot, V., 'Large-scale properties of unstable systems governed by the Kuramoto-Sivashinski equation', Physical Review A 24, 1981, 642–644.
Zaleski, S., 'A stochastic model for the large scale dynamics of some fluctuating interfaces', Physica D 34, 1989, 427–438.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wittenberg, R.W., Holmes, P. Spatially Localized Models of Extended Systems. Nonlinear Dynamics 25, 111–132 (2001). https://doi.org/10.1023/A:1012902732610
Issue Date:
DOI: https://doi.org/10.1023/A:1012902732610