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Chaotic mode competition in parametrically forced surface waves

Published online by Cambridge University Press:  20 April 2006

S. Ciliberto  and
J. P. Gollub
S. Ciliberto
Affiliation:
Department of Physics, Haverford College, Haverford, PA 19041 U.S.A., and Department of Physics, University of Pennsylvania, Philadelphia, PA 19104 U.S.A. Permanent address: Istituto Nazionale di Ottica, 50125 Arcetri-Firenze, Largo Enrico Fermi 6, Italy.
J. P. Gollub
Affiliation:
Department of Physics, Haverford College, Haverford, PA 19041 U.S.A., and Department of Physics, University of Pennsylvania, Philadelphia, PA 19104 U.S.A.
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Abstract

Vertical forcing of a fluid layer leads to standing waves by means of a subharmonic instability. When the driving amplitude and frequency are chosen to be near the intersection of the stability boundaries of two nearly degenerate modes, we find that they can compete with each other to produce either periodic or chaotic motion on a slow timescale. We utilize digital image-processing methods to determine the time-dependent amplitudes of the competing modes, and local-sampling techniques to study the onset of chaos in some detail. Reconstruction of the attractors in phase space shows that in the chaotic regime the dimension of the attractor is fractional and at least one Lyapunov exponent is positive. The evidence suggests that a theory incorporating four coupled slow variables will be sufficient to account for the mode competition.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 158 , September 1985 , pp. 381 - 398
Copyright
© 1985 Cambridge University Press

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References

Brandstater A., Swift J., Swinney H. L., Wolf A., Farmer J. D., Jen, E. & Crutchfield, J. P. 1983 Low dimensional chaos in a hydrodynamic system. Phys. Rev. Lett. 51, 1442–1445.Google Scholar
Brandstater, A. & Swinney H. L. 1984 Distinguishing low-dimensional chaos from random noise in a hydrodynamic experiment. In Fluctuations and Sensitivity in Nonequilibrium Systems (ed. W. Horsthemke & D. Kondepudi). Springer.
Ben-Mizrachi A., Procaccia, I. & Grassberger P. 1984 The characterization of experimental (noisy) strange attractors Phys. Rev. A 29, 975–977.Google Scholar
Benjamin, T. B. & Ursell F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion Proc. R. Soc. Lond. A 225, 505–515.Google Scholar
Ciliberto, S. & Gollub J. P. 1984 Pattern competition leads to chaos. Phys. Rev. Lett. 52, 922–925.Google Scholar
Ciliberto, S. & Gollub J. P. 1985 - Phenomenological model of chaotic mode competition in surface waves. Nuovo Cim. B, in press.Google Scholar
Curry J. H. 1978 A generalized Lorenz system Commun. Math. Phys. 60, 193–204.Google Scholar
Faraday M. 1831 On the forms and states assumed by fluids in contact with vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 319–340.Google Scholar
Farmer J. D., Ott, E. & Yorke J. A. 1983 The dimension of chaotic attractors Physica D 7, 153–180.Google Scholar
Gollub, J. P. & Meyer C. W. 1983 Symmetry-breaking instabilities on a fluid surface Physica D 6, 337–346.Google Scholar
Grassberger, P. & Procaccia I. 1983a Measuring the strangeness of strange attractors Physica D 9, 189–208.Google Scholar
Grassberger, P. & Procaccia I. 1983b Estimation of the Kolmogorov entropy for a chaotic signal Phys. Rev. A 28, 2591–2593.Google Scholar
Guckenheimer, J. & Buzyna G. 1983 Dimension measurements for geostrophic turbulence. Phys. Rev. Lett. 51, 1438–1441.Google Scholar
Lanford O. E. 1981 Strange attractors and turbulence. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub), pp. 7–26. Springer.
Malraison B., Atten P., Bergé, P. & Dubois M. 1983 Dimension of strange attractors: an experimental determination for the chaotic regime of two chaotic systems. J. Phys. Lett. (Paris) 44, L897–L902.Google Scholar
Miles J. 1967 Surface-wave damping in closed basins Proc. R. Soc. Lond. A 297, 459–475.Google Scholar
Miles J. 1984a Strange attractors in fluid dynamics. Adv. Appl. Mech. 24, 189–214.Google Scholar
Miles J. 1984b Nonlinear Faraday resonance. J. Fluid Mech. 146, 285–302.Google Scholar
Packard N. H., Crutchfield J. P., Farmer, J. D. & Shaw, R. S. 1980 Geometry from a time series Phys. Rev. Lett. 45, 712–715.Google Scholar
Ruelle D. 1980 Strange attractors. Math. Intelligencer 2, 126–137.Google Scholar
Roux J.-C., Simoyi, R. H. & Swinney H. L. 1983 Observation of a strange attractor Physica D 8, 257–266.Google Scholar
Takens F. 1981 Detecting strange attractors in turbulence (ed. D. A. Rand & L. S. Young). Lecture Notes in Mathematics vol. 898, Springer.