Skip to main content Accessibility help

Chaotic mode competition in parametrically forced surface waves

  • S. Ciliberto (a1) (a2) and J. P. Gollub (a1)


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.



Hide All
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, 14421445.
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, 975977.
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, 505515.
Ciliberto, S. & Gollub J. P. 1984 Pattern competition leads to chaos. Phys. Rev. Lett. 52, 922925.
Ciliberto, S. & Gollub J. P. 1985 - Phenomenological model of chaotic mode competition in surface waves. Nuovo Cim. B, in press.
Curry J. H. 1978 A generalized Lorenz system Commun. Math. Phys. 60, 193204.
Faraday M. 1831 On the forms and states assumed by fluids in contact with vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 319340.
Farmer J. D., Ott, E. & Yorke J. A. 1983 The dimension of chaotic attractors Physica D 7, 153180.
Gollub, J. P. & Meyer C. W. 1983 Symmetry-breaking instabilities on a fluid surface Physica D 6, 337346.
Grassberger, P. & Procaccia I. 1983a Measuring the strangeness of strange attractors Physica D 9, 189208.
Grassberger, P. & Procaccia I. 1983b Estimation of the Kolmogorov entropy for a chaotic signal Phys. Rev. A 28, 25912593.
Guckenheimer, J. & Buzyna G. 1983 Dimension measurements for geostrophic turbulence. Phys. Rev. Lett. 51, 14381441.
Lanford O. E. 1981 Strange attractors and turbulence. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub), pp. 726. 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, L897L902.
Miles J. 1967 Surface-wave damping in closed basins Proc. R. Soc. Lond. A 297, 459475.
Miles J. 1984a Strange attractors in fluid dynamics. Adv. Appl. Mech. 24, 189214.
Miles J. 1984b Nonlinear Faraday resonance. J. Fluid Mech. 146, 285302.
Packard N. H., Crutchfield J. P., Farmer, J. D. & Shaw, R. S. 1980 Geometry from a time series Phys. Rev. Lett. 45, 712715.
Ruelle D. 1980 Strange attractors. Math. Intelligencer 2, 126137.
Roux J.-C., Simoyi, R. H. & Swinney H. L. 1983 Observation of a strange attractor Physica D 8, 257266.
Takens F. 1981 Detecting strange attractors in turbulence (ed. D. A. Rand & L. S. Young). Lecture Notes in Mathematics vol. 898, Springer.
MathJax is a JavaScript display engine for mathematics. For more information see

Chaotic mode competition in parametrically forced surface waves

  • S. Ciliberto (a1) (a2) and J. P. Gollub (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed