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Symmetry-breaking bifurcations in resonant surface waves

Published online by Cambridge University Press:  26 April 2006

Z. C. Feng  and
P. R. Sethna
Z. C. Feng
Affiliation:
Deparment of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
P. R. Sethna
Affiliation:
Deparment of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
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Abstract

Surface waves in a nearly square container subjected to vertical oscillations are studied. The theoretical results are based on the analysis of a derived set of normal form equations, which represent perturbations of systems with 1:1 internal resonance and with D4 symmetry. Bifurcation analysis of these equations shows that the system is capable of periodic and quasi-periodic standing as well as travelling waves. The analysis also identifies parameter values at which chaotic behaviour is to be expected. The theoretical results are verified with the aid of some experiments.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 199 , February 1989 , pp. 495 - 518
Copyright
© 1989 Cambridge University Press

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References

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 A255, 505–517.Google Scholar
Bridges, T. J.: 1987 Secondary bifurcation and change of type of three-dimensional standing waves in a finite depth. J. Fluid Mech. 179, 137–153.Google Scholar
Chow, S.-N. & Hale, J. K. 1982 Methods of Bifurcation Theory. Springer.
Ciliberto, S. & Gollub, J. P., 1984 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381–398.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, 319–346.Google Scholar
Golubitsky, M. & Stewart, I., 1985 Hopf bifurcation in the presence of symmetry. Arch. Rat. Mech. Anal. 87, 107–165.Google Scholar
Golubitsky, M. & Stewart, I., 1986 Hopf bifurcation with dihedral group symmetry: coupled nonlinear oscillators. In Multiparameter Bifurcation Theory. Contemporary Mathematics, vol. 56 (ed. M. Golubitsky & J. Guckenheimer), pp. 131–173. American Mathematical Society.
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 Bifurcations of Vector Fields. Springer.
Holmes, P.: 1986 Chaotic motions in a weakly nonlinear model for surface waves. J. Fluid Mech. 162, 365–388.Google Scholar
Kit, E., Shemer, L. & Miloh, T., 1987 Experimental and theoretical investigation of nonlinear sloshing waves in a rectangular channel. J. Fluid Mech. 181, 265–291.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
Miles, J. W.: 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297, 459–475.Google Scholar
Miles, J. W.: 1984a Internally resonant surface waves in a circular cylinder. J. Fluid Mech. 149, 1–14.Google Scholar
Miles, J. W.: 1984b Resonantly forced surface waves in a circular cylinder. J. Fluid Mech. 149, 15–31.Google Scholar
Miles, J. W.: 1984c Nonlinear Faraday resonance. J. Fluid Mech. 146, 285–302.Google Scholar
Sethna, P. R. & Bajaj, A. K., 1978 Bifurcations in dynamical systems with internal resonance. Trans. ASME E: J. Appl. Mech. 45, 895–902.Google Scholar
Simonelli, F. & Gollub, J. P., 1989 Surface wave mode interactions: effects of symmetry and degeneracy. J. Fluid Mech. 199, 471–494.Google Scholar
Swift, J. W.: 1988 Hopf bifurcation with the symmetry of the square. To appear.
Verhulst, F.: 1979 Discrete symmetric dynamic systems at the main resonances with applications to axi-symmetric galaxies. Phil. Trans. R. Soc. Lond. 290, 435–465.Google Scholar
Verma, G. R. & Keller, J. B., 1962 Three-dimensional surface waves of finite amplitude. Phys. Fluids 5, 52–56.Google Scholar
Virnig, J. C., Berman, A. S. & Sethna, P. R., 1987 On three-dimensional nonlinear subharmonic resonant surface waves in a fluid. Part II: Experiment. Trans. ASME E: J. Appl. Mech. 55, 220–224.Google Scholar