Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-19T13:11:46.402Z Has data issue: false hasContentIssue false
  • English
  • Français

Surface wave mode interactions: effects of symmetry and degeneracy

Published online by Cambridge University Press:  26 April 2006

F. Simonelli  and
J. P. Gollub
F. Simonelli
Affiliation:
Physics Department, Haverford College, Haverford, PA 19041, USAand Physics Department, The University of Pennsylvania, Philadelphia, PA 19104, USA
J. P. Gollub
Affiliation:
Physics Department, Haverford College, Haverford, PA 19041, USAand Physics Department, The University of Pennsylvania, Philadelphia, PA 19104, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Parametrically excited surface wave modes on a fluid layer driven by vertical forcing can interact with each other when more than one spatial mode is excited. We have investigated the dynamics of the interaction of two modes that are degenerate in a square layer, but non-degenerate in a rectangular one. Novel experimental techniques were developed for this purpose, including the real-time measurement of all relevant slowly varying mode amplitudes, investigation of the phase-space structure by means of transient studies starting from a variety of initial conditions, and automated determination of stability boundaries as a function of driving amplitude and frequency. These methods allowed both stable and unstable fixed points (sinks, sources, and saddles) to be determined, and the nature of the bifurcation sequences to be clearly established. In most of the dynamical regimes, multiple attractors and repellers (up to 16) were found, including both pure and mixed modes. We found that the symmetry of the fluid cell has dramatic effects on the dynamics. The fully degenerate case (square cell) yields no time-dependent patterns, and is qualitatively understood in terms of third-order amplitude equations whose basic structure follows from symmetry arguments. In a slightly rectangular cell, where the two modes are separated in frequency by a small amount (about 1%), mode competition produces both periodic and chaotic states organized around unstable pure and mixed-state fixed points.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 199 , February 1989 , pp. 471 - 494
Copyright
© 1989 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ciliberto, S. & Gollub, J. P., 1984 Pattern competition leads to chaos. Phys. Rev. Lett. 52, 922–925.Google Scholar
Ciliberto, S. & Gollub, J. P., 1985a Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381–398.Google Scholar
Ciliberto, S. & Gollub, J. P., 1985b Phenomenological model of chaotic mode competition in surface waves. II Nuovo Cimento 6, 309–316.Google Scholar
Crawford, J. & Knobloch, E., 1988 Classification and unfolding of degenerate Hopf bifurcations with O(2) symmetry: no distinguished parameter. Physica D 31, 1–48.Google Scholar
Feng, Z. C. & Sethna, P. R., 1989 Symmetry-breaking bifurcations in resonant surface waves. J. Fluid Mech. 199, 495–518.Google Scholar
Funakoshi, M. & Inque, S., 1987 Chaotic behaviour of resonantly forced surface waves. Phys. Lett. A 121, 229–232.Google Scholar
Golubitsky, M. & Roberts, M., 1987 A classification of degenerate Hopf bifurcation with O(2) symmetry. J. Diffl. Equat. 69, 216–264.Google Scholar
Golubitsky, M. & Shaeffer, D. G., 1985 Singularities and Groups in Bifurcation Theory. Springer.
Gu, X. M., Sethna, P. R. & Narain, A., 1988 On three-dimensional non-linear subharmonic resonant surface waves in a fluid. Part I: Theory. Trans. ASME E: J. Appl. Mech. 55, 213–219.Google Scholar
Gu, X. M. & Sethna, P. R., 1987 Resonant surface waves and chaotic phenomena. J. Fluid Mech. 183, 543–565.Google Scholar
Guckenheimer, J. & Holmes, P., 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields. Springer.
Holmes, P. J.: 1986 Chaotic motion in a weakly nonlinear model for surface waves. J. Fluid Mech. 162, 365–388.Google Scholar
Meron, E.: 1987 Parametric excitation of multimode dissipative systems. Phys Rev. A 35, 4892–4895.Google Scholar
Meron, E. & Procaccia, I., 1986a Theory of chaos in surface waves: The reduction from hydrodynamics to few-dimensional dynamics. Phys Rev. Lett. 56, 1323–1326.Google Scholar
Meron, E. & Procaccia, I., 1986b Low dimensional chaos in surface waves: Theoretical analysis of an experiment. Phys. Rev. A 34, 3221–3237.Google Scholar
Meron, E. & Procaccia, I., 1987 Gluing bifurcations in critical flows: The route to chaos in parametrically excited surface waves. Phys. Rev. A 35, 4008–4011.Google Scholar
Miles, J. W.: 1967 Surface wave damping in closed basins. Proc. R. Soc. Lond. A 297, 459–475.Google Scholar
Miles, J. W.: 1976 Nonlinear surface waves in closed basins. J. Fluid Mech. 75, 419–448.Google Scholar
Miles, J. W.: 1984a Nonlinear Faraday resonance. J. Fluid Mech. 146, 285–302.Google Scholar
Miles, J. W.: 1984b Resonantly forced surface waves in a circular cylinder. J. Fluid Mech. 149, 15–31.Google Scholar
Silber, M. & Knobloch, E., 1988 Parametrically excited surface waves in square geometry. (submitted to Phys. Lett.).
Simonelli, F. & Gollub, J. P., 1987 The masking of symmetry by degeneracy in the dynamics of interacting modes. Nuc. Phys. B (Proc. Suppl.) 2, 87–96.Google Scholar
Simonelli, F. & Gollub, J. P., 1988 Stability boundaries and phase space measurements for spatially extended dynamical systems. Rev. Sci. Instrum. 59, 280–284.Google Scholar
Swinney, H. L. & Gollub, J. P., 1986 Characterization of hydrodynamic strange attractors. Physica 18D, 448–454.Google Scholar
Virnig, J. C., Berman, A. S. & Sethna, P. R., 1988 On three-dimensional non-linear subharmonic resonant surface waves in a fluid. Part II: Experiment. Trans. ASME E: J. Appl. Mech. 55, 220–224.Google Scholar