Dynamics of nearly inviscid Faraday waves in almost circular containers

Parametrically driven surface gravity-capillary waves in an elliptically distorted circular cylinder are studied. In the nearly inviscid regime, the waves couple to a streaming flow driven in oscillatory viscous boundary layers. In a cylindrical container, the streaming flow couples to the spatial phase of the waves, but in a distorted cylinder, it couples to their amplitudes as well. This coupling may destabilize pure standing oscillations, and lead to complex time-dependent dynamics at onset. Among the new dynamical behavior that results are relaxation oscillations involving abrupt transitions between standing and quasiperiodic oscillations, and exhibiting ‘canards’. © 2004 Elsevier B.V. All rights reserved.

[1]  Edgar Knobloch,et al.  Coupled Amplitude-Streaming Flow Equations for Nearly Inviscid Faraday Waves in Small Aspect Ratio Containers , 2002, J. Nonlinear Sci..

[2]  Jeff Moehlis,et al.  Canards in a Surface Oxidation Reaction , 2002, J. Nonlinear Sci..

[3]  S. Baer,et al.  Sungular hopf bifurcation to relaxation oscillations , 1986 .

[4]  J. Moehlis,et al.  Forced Symmetry Breaking as a Mechanism for Bursting , 1998 .

[5]  J. Rinzel,et al.  Oscillatory and bursting properties of neurons , 1998 .

[6]  David Terman,et al.  Chaotic spikes arising from a model of bursting in excitable membranes , 1991 .

[7]  Arshad Kudrolli,et al.  Superlattice patterns in surface waves , 1998, chao-dyn/9803016.

[8]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[9]  Michael Selwyn Longuet-Higgins,et al.  Mass transport in water waves , 1953, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[10]  Jerry P. Gollub,et al.  Patterns and spatiotemporal chaos in parametrically forced surface waves: A systematic survey at lar , 1996 .

[11]  Edgar Knobloch,et al.  Direction-reversing traveling waves , 1991 .

[12]  Mitsuaki Funakoshi,et al.  Surface waves due to resonant horizontal oscillation , 1988, Journal of Fluid Mechanics.

[13]  M. Silber,et al.  Broken symmetries and pattern formation in two-frequency forced faraday waves. , 2002, Physical review letters.

[14]  Philip Holmes,et al.  The limited effectiveness of normal forms: a critical review and extension of local bifurcation studies of the Brusselator PDE , 1997 .

[15]  P. Glendinning,et al.  Imperfect homoclinic bifurcations. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  E. Knobloch,et al.  Simulations of oscillatory binary fluid convection in large aspect ratio containers. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Coupling of identical chemical oscillators , 1990 .

[18]  Edgar Knobloch,et al.  Hopf bifurcation with broken circular symmetry , 1991 .

[19]  S. Fauve,et al.  Oscillatory Phase Modulation of Parametrically Forced Surface Waves , 1989 .

[20]  J. Fineberg,et al.  Pattern formation in two-frequency forced parametric waves. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  W. S. Edwards,et al.  Patterns and quasi-patterns in the Faraday experiment , 1994, Journal of Fluid Mechanics.

[22]  M. Silber,et al.  Spatial period-multiplying instabilities of hexagonal Faraday waves , 2000, nlin/0005066.

[23]  Carlos Martel,et al.  Surface wave damping in a brimful circular cylinder , 1998, Journal of Fluid Mechanics.

[24]  Vivien Kirk,et al.  Breaking of symmetry in the saddle-node Hopf bifurcation , 1991 .

[25]  C. Martel,et al.  Weakly dissipative Faraday waves in 2D large aspect ratio annuli , 2002 .

[26]  John Guckenheimer,et al.  Asymptotic analysis of subcritical Hopf-homoclinic bifurcation , 2000 .

[27]  Stephen Wiggins,et al.  Global Bifurcations and Chaos , 1988 .

[28]  Steven H. Strogatz,et al.  Nonlinear dynamics of a solid-state laser with injection , 1998 .

[29]  John Guckenheimer,et al.  Numerical Computation of Canards , 2000, Int. J. Bifurc. Chaos.

[30]  Sebastian Wieczorek,et al.  Accumulating regions of winding periodic orbits in optically driven lasers , 2002 .

[31]  E. Knobloch,et al.  Heteroclinic dynamics in the nonlocal parametrically driven nonlinear Schrödinger equation , 2002 .

[32]  Analysis of the shearing instability in nonlinear convection and magnetoconvection , 1996 .

[33]  E. M. Ortega,et al.  Drift instability of standing Faraday waves , 2002, Journal of Fluid Mechanics.

[34]  P. Hirschberg,et al.  Sbil'nikov-Hopf bifurcation , 1993 .

[35]  E. Knobloch,et al.  Natural doubly diffusive convection in three-dimensional enclosures , 2002 .

[36]  Vivien Kirk,et al.  Merging of resonance tongues , 1993 .

[37]  John Guckenheimer,et al.  On a codimension two bifurcation , 1981 .

[38]  Global structure of homoclinic bifurcations: a combinatorial approach , 1989 .

[39]  Wiktor Eckhaus,et al.  Relaxation oscillations including a standard chase on French ducks , 1983 .

[40]  K. Bar-Eli,et al.  Canard explosion and excitation in a model of the Belousov-Zhabotinskii reaction , 1991 .

[41]  E. Knobloch,et al.  Nearly inviscid Faraday waves in annular containers of moderately large aspect ratio , 2001 .

[42]  John W. Miles,et al.  Internally resonant surface waves in a circular cylinder , 1984, Journal of Fluid Mechanics.