Near-Resonant Steady Mode Interaction: Periodic, Quasi-periodic, and Localized Patterns

Motivated by the rich variety of complex periodic and quasi-periodic patterns found in systems such as two-frequency forced Faraday waves, we study the interaction of two spatially periodic modes that are nearly resonant. Within the framework of two coupled one-dimensional Ginzburg--Landau equations we investigate analytically the stability of the periodic solutions to general perturbations, including perturbations that do not respect the periodicity of the pattern, and which may lead to quasi-periodic solutions. We study the impact of the deviation from exact resonance on the destabilizing modes and on the final states. In regimes in which the mode interaction leads to the existence of traveling waves our numerical simulations reveal localized waves in which the wavenumbers are resonant and which drift through a steady background pattern that has an off-resonant wavenumber ratio.

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

[2]  Hermann Riecke,et al.  Structure and dynamics of modulated traveling waves in cellular flames , 1994, patt-sol/9402001.

[3]  Solitary modes and the Eckhaus instability in directional solidification. , 1988, Physical review letters.

[4]  Riecke,et al.  Stability and wave-vector restriction of axisymmetric Taylor vortex flow. , 1986, Physical review. A, General physics.

[5]  Comment on "Parity-breaking transitions of modulated patterns in hydrodynamic systems" , 1990, Physical review letters.

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

[7]  M. Umeki.  Faraday resonance in rectangular geometry , 1991, Journal of Fluid Mechanics.

[8]  Z. C. Feng,et al.  Symmetry-breaking bifurcations in resonant surface waves , 1989, Journal of Fluid Mechanics.

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

[10]  Wiener,et al.  Parity breaking and solitary waves in axisymmetric Taylor vortex flow. , 1992, Physical review letters.

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

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

[13]  Instabilities induced by a weak breaking of a strong spatial resonance , 2003, nlin/0311045.

[14]  F. Arecchi,et al.  Two-dimensional crystals and quasicrystals in nonlinear optics. , 1995, Physical review letters.

[15]  E. Curado,et al.  Effects of an almost resonant spatial thermal modulation in the Rayleigh-Bénard problem: quasiperiodic behaviour , 1987 .

[16]  J. A. Kuznecov.  Elements of applied bifurcation theory , 1998 .

[18]  E. Knobloch,et al.  Complex dynamics in the 1:3 spatial resonance , 2000 .

[19]  Gerhard Dangelmayr,et al.  Steady-state mode interactions in the presence of 0(2)-symmetry , 1986 .

[20]  Resonant interactions and traveling-solidification cells. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[21]  Goldstein,et al.  Parity-breaking transitions of modulated patterns in hydrodynamic systems. , 1989, Physical review letters.

[22]  Philip Holmes,et al.  Heteroclinic cycles and modulated travelling waves in a system with D 4 symmetry , 1992 .

[23]  Procaccia,et al.  Low-dimensional chaos in surface waves: Theoretical analysis of an experiment. , 1986, Physical review. A, General physics.

[24]  Convergence properties of the 8, 10 and 12 mode representations of quasipatterns , 2002, nlin/0209034.

[25]  Tom C. Lubensky,et al.  Commensurate and Incommensurate Structures in a Nonequilibrium System , 1983 .

[26]  Jay Fineberg,et al.  SPATIAL AND TEMPORAL DYNAMICS OF TWO INTERACTING MODES IN PARAMETRICALLY DRIVEN SURFACE WAVES , 1998 .

[27]  Gil,et al.  Hydrodynamic and interfacial patterns with broken space-time symmetry. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[28]  B. Malomed,et al.  Higher-order bifurcations in distributed kinetic systems with aperiodic instability , 1984 .

[29]  Broken symmetries and pattern formation in two-frequency forced faraday waves. , 2002, Physical review letters.

[30]  Coullet.  Commensurate-incommensurate transition in nonequilibrium systems. , 1986, Physical review letters.

[31]  Mark R. Proctor,et al.  The interaction of two spatially resonant patterns in thermal convection. Part 1. Exact 1:2 resonance , 1988, Journal of Fluid Mechanics.

[32]  M. Proctor,et al.  Strong spatial resonance and travelling waves in benard convection , 1987 .

[33]  John Guckenheimer,et al.  Heteroclinic cycles and modulated traveling waves in systems with O(2) symmetry , 1988 .

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

[35]  Riecke,et al.  Parity-breaking and Hopf bifurcations in axisymmetric Taylor vortex flow. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[36]  Arjen Doelman,et al.  Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms , 1997 .

[37]  E. Knobloch,et al.  Period-doubling mode interactions with circular symmetry , 1990 .

[38]  Edgar Knobloch,et al.  New type of complex dynamics in the 1:2 spatial resonance , 2001 .

[39]  Jerry P. Gollub,et al.  Surface wave mode interactions: effects of symmetry and degeneracy , 1989, Journal of Fluid Mechanics.

[40]  Couder,et al.  Dynamical regimes of directional viscous fingering: Spatiotemporal chaos and wave propagation. , 1990, Physical review letters.

[41]  K. Kawasaki,et al.  Forced Periodic and Quasi-Periodic Patterns in Anisotropic Systems , 1996 .

[42]  Eckhaus boundary and wave-number selection in rotating Couette-Taylor flow. , 1986, Physical review. A, General physics.

[43]  P. Coullet,et al.  Nature of spatial chaos. , 1987, Physical review letters.

[44]  Rappel,et al.  Parity breaking in directional solidification: Numerics versus amplitude equations. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[45]  P. Coullet,et al.  Resonance and phase solitons in spatially-forced thermal convection , 1986 .