Design of Parametrically Forced Patterns and Quasipatterns

The Faraday wave experiment is a classic example of a system driven by parametric forcing, and it produces a wide range of complex patterns, including superlattice patterns and quasipatterns. Nonlinear three-wave interactions between driven and weakly damped modes play a key role in determining which patterns are favored. We use this idea to design single and multifrequency forcing functions that produce examples of superlattice patterns and quasipatterns in a new model PDE with parametric forcing. We make quantitative comparisons between the predicted patterns and the solutions of the PDE. Unexpectedly, the agreement is good only for parameter values very close to onset. The reason that the range of validity is limited is that the theory requires strong damping of all modes apart from the driven pattern-forming modes. This is in conflict with the requirement for weak damping if three-wave coupling is to influence pattern selection effectively. We distinguish the two different ways that three-wave interac...

[1]  W. Water,et al.  Patterns of Faraday waves , 2003, Journal of Fluid Mechanics.

[2]  Thomas Brooke Benjamin,et al.  The stability of the plane free surface of a liquid in vertical periodic motion , 1954, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[3]  Wenbin Zhang,et al.  Numerical study of pattern formation in weakly damped parametric surface waves , 1998 .

[4]  M. Silber,et al.  Parametrically excited surface waves: two-frequency forcing, normal form symmetries, and pattern selection. , 1998, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  Enhanced Faraday pattern stability with three-frequency driving. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[7]  Ron Lifshitz,et al.  THEORETICAL MODEL FOR FARADAY WAVES WITH MULTIPLE-FREQUENCY FORCING , 1997 .

[8]  Kenneth R. Meyer,et al.  Singularities and Groups in Bifurcation Theory. Volume II (Martin Golubitsky, Ian Stewart, and David G. Schaeffer) , 1989, SIAM Rev..

[9]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

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

[11]  Hermann Riecke,et al.  Superlattice Patterns in the Complex Ginzburg-Landau Equation with Multiresonant Forcing , 2008, SIAM Journal on Applied Dynamical Systems.

[12]  V. D. Water,et al.  Effect of Depth on the Pattern Formation of Faraday Waves , 1997 .

[13]  Fineberg,et al.  Two-mode rhomboidal states in driven surface waves , 2000, Physical review letters.

[14]  Anne C. Skeldon,et al.  Stability results for steady, spatially periodic planforms , 1995, patt-sol/9509004.

[15]  Lev Tsimring,et al.  Localized and Cellular Patterns in a Vibrated Granular Layer , 1997 .

[16]  P. Matthews.  Hexagonal patterns in finite domains , 1997, patt-sol/9703002.

[17]  Swinney,et al.  Pattern formation in the presence of symmetries. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[18]  M. Levinsen,et al.  Ordered capillary-wave states: Quasicrystals, hexagons, and radial waves , 1992 .

[19]  H. Müller,et al.  Quasiperiodic patterns in Rayleigh-Benard convection under gravity modulation , 1997 .

[20]  J. Swift,et al.  Hydrodynamic fluctuations at the convective instability , 1977 .

[21]  Giovanna Guidoboni,et al.  Pattern Selection for Faraday Waves in an Incompressible Viscous Fluid , 2007, SIAM J. Appl. Math..

[22]  Complex-ordered patterns in shaken convection. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  S. Cox,et al.  Exponential Time Differencing for Stiff Systems , 2002 .

[24]  M. Silber,et al.  Two-frequency forced Faraday waves: weakly damped modes and pattern selection , 2000, nlin/0002041.

[25]  J. Cahn,et al.  Metallic Phase with Long-Range Orientational Order and No Translational Symmetry , 1984 .

[26]  Stephen M. Cox,et al.  Instability and localisation of patterns due to a conserved quantity , 2003 .

[27]  Super-réseaux et quasi-cristaux dans les systèmes , 2001 .

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

[29]  M. Silber,et al.  Resonances and superlattice pattern stabilization in two-frequency forced Faraday waves , 2001, nlin/0111039.

[30]  W. Lange,et al.  Twelvefold Quasiperiodic Patterns in a Nonlinear Optical System with Continuous Rotational Symmetry , 1999 .

[31]  Jeff Porter,et al.  Multifrequency control of Faraday wave patterns. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  H. Diamant,et al.  Soft quasicrystals–Why are they stable? , 2006, cond-mat/0611115.

[33]  M. Silber,et al.  Resonant triad dynamics in weakly damped Faraday waves with two-frequency forcing , 2003, nlin/0301015.

[34]  W. Rucklidge,et al.  Convergence properties of the 8, 10 and 12 mode representations of quasipatterns , 2002, nlin/0209034.

[35]  Stefan C. Müller,et al.  Evolution of spontaneous structures in dissipative continuous systems , 1998 .

[36]  M. Silber,et al.  Pattern control via multifrequency parametric forcing. , 2004, Physical review letters.

[37]  K. Knorr,et al.  Pattern formation at the bicritical point of the Faraday instability. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Müller.  Model equations for two-dimensional quasipatterns. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[39]  Kjartan Pierre Emilsson,et al.  Strong resonances of spatially distributed oscillators: a laboratory to study patterns and defects , 1992 .

[40]  W. Water,et al.  Nonlinear pattern formation of Faraday waves , 1997 .

[41]  Peter V. E. McClintock,et al.  Pattern formation : an introduction to methods. , 2006 .

[42]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[43]  Tuckerman,et al.  Two-frequency parametric excitation of surface waves. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[44]  Wenbin Zhang,et al.  Pattern formation in weakly damped parametric surface waves , 1996, Journal of Fluid Mechanics.

[45]  Zhang,et al.  Square patterns and quasipatterns in weakly damped Faraday waves. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[46]  Cross,et al.  Stability analysis of two-dimensional models of three-dimensional convection. , 1985, Physical review. A, General physics.

[47]  P. Coullet,et al.  Pattern formation in forced reaction diffusion systems with nearly degenerate bifurcations. , 2007, Chaos.

[48]  David Mark Winterbottom,et al.  Pattern formation with a conservation law , 2006 .

[49]  M. Silber,et al.  Quasipatterns in parametrically forced systems. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[50]  Y. Pomeau,et al.  Turbulent crystals in macroscopic systems , 1993 .

[51]  I. Melbourne.  Derivation of the Time-Dependent Ginzburg-Landau Equation on the Line , 1998 .

[52]  P. Santini,et al.  Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve μ 2 = ν n − 1 , n ∈ Z : ergodicity , isochrony , periodicity and fractals , 2006 .

[53]  M. Silber,et al.  Nonlinear Competition between Small and Large Hexagonal Patterns , 1997, patt-sol/9710004.

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

[55]  H Germany,et al.  Analytic Stability Theory for Faraday Waves and the Observation of the Harmonic Surface Response , 1996, patt-sol/9610006.

[56]  Jonathan H.P. Dawes,et al.  Reducible actions of D4⋉T2: superlattice patterns and hidden symmetries , 2003 .

[57]  L. Tuckerman,et al.  Parametric instability of the interface between two fluids , 1994, Journal of Fluid Mechanics.

[58]  K. Kumar.  Linear theory of Faraday instability in viscous liquids , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

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

[60]  Mermin,et al.  Mean-field theory of quasicrystalline order. , 1985, Physical review letters.

[61]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

[62]  Gunaratne.  Complex spatial patterns on planar continua. , 1993, Physical review letters.

[63]  Hermann Riecke,et al.  Quasipatterns in a model for chemical oscillations forced at multiple resonance frequencies. , 2007, Physical review letters.

[64]  P. J. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[65]  Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve μ2 = νn - 1, n ∈ Z: Ergodicity, isochrony and fractals , 2006, nlin/0607031.

[66]  Hermann Riecke,et al.  Sideband instabilities and defects of quasipatterns , 2000, nlin/0012031.

[67]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[68]  PlanformsBenoit,et al.  Stability Results for Steady , Spatially { Periodic , 1996 .

[69]  R. Friedrich,et al.  Theoretical and Experimental Investigations of the Faraday Instability , 1998 .

[70]  W. Steurer.  Boron-based quasicrystals with sevenfold symmetry , 2007 .

[71]  E. Knobloch,et al.  Coupled Mean Flow‐Amplitude Equations for Nearly Inviscid Parametrically Driven Surface Waves , 2002, Annals of the New York Academy of Sciences.