Double Phase Slips and Spatiotemporal Chaos in a Model for Parametrically Excited Standing Waves

We present results of numerical simulations of coupled one-dimensional Ginzburg--Landau equations that describe parametrically excited waves. We focus on a new regime in which the Eckhaus sideband instability does not lead to an overall change in the wavelength via the occurrence of a single phase slip but instead leads to "double phase slips." They are characterized by the phase slips occurring in sequential pairs, with the second phase slip quickly following and negating the first. The resulting dynamics range from transient excursions from a fixed point resembling those seen in excitable media, to periodic solutions of varying complexity and chaotic solutions. In larger systems we find in addition localized spatiotemporal chaos, where the solution consists of a chaotic region with quiescent regions on each side. We explain the localization using an effective phase diffusion equation which can be viewed as arising from a homogenization of the chaotic state.

[1]  D. Bailin.  Field theory , 1979, Nature.

[2]  Thomas P. Witelski,et al.  The structure of internal layers for unstable nonlinear diffusion equations , 1996 .

[3]  John Whitehead,et al.  Finite bandwidth, finite amplitude convection , 1969, Journal of Fluid Mechanics.

[4]  Stéphane Zaleski,et al.  A stochastic model for the large scale dynamics of some fluctuating interfaces , 1989 .

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

[6]  T. Brooke Benjamin,et al.  The disintegration of wave trains on deep water Part 1. Theory , 1967, Journal of Fluid Mechanics.

[7]  Hegseth,et al.  Temporal modulation of traveling waves in the flow between rotating cylinders with broken azimuthal symmetry. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  Weber,et al.  Core instability and spatiotemporal intermittency of spiral waves in oscillatory media. , 1994, Physical review letters.

[9]  L. Tuckerman,et al.  Bifurcation analysis of the Eckhaus instability , 1990 .

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

[11]  Riecke,et al.  Phase Diffusion in Localized Spatio-Temporal Amplitude Chaos. , 1996, Physical review letters.

[12]  D. Thouless,et al.  Ordering, metastability and phase transitions in two-dimensional systems , 1973 .

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

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

[15]  Ordered and disordered defect chaos , 1997, chao-dyn/9704020.

[16]  Hermann Riecke,et al.  Stable wave-number kinks in parametrically excited standing waves , 1990 .

[17]  Paul Manneville,et al.  Dissipative Structures and Weak Turbulence , 1995 .

[18]  Paul Manneville,et al.  Phase turbulence in the two-dimensional complex Ginzburg-Landau equation , 1996 .

[19]  Victor Martin-Mayor,et al.  Field Theory, the Renormalization Group and Critical Phenomena , 1984 .

[20]  D. Walgraef,et al.  Melting of 2D nonequilibrium structures at low and high constraints , 1982 .

[21]  Hermann Riecke,et al.  PARAMETRIC FORCING OF WAVES WITH A NONMONOTONIC DISPERSION RELATION : DOMAIN STRUCTURES IN FERROFLUIDS , 1996, patt-sol/9601002.

[22]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[23]  Formation of dynamical domains in a circular Couette system. , 1986 .

[24]  David R. Nelson,et al.  Dislocation-mediated melting in two dimensions , 1979 .

[25]  Yoshiki Kuramoto,et al.  Diffusion-Induced Chaos in Reaction Systems , 1978 .

[26]  P. Bergé,et al.  Pattern Domains in Rayleigh-Bénard Slot Convection , 1992 .

[27]  Tsimring,et al.  Rotating spirals in a Faraday experiment. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[28]  Large-scale properties of unstable systems governed by the Kuramoto-Sivashinksi equation , 1981 .

[29]  J. Gollub,et al.  Order-disorder transition in capillary ripples. , 1989, Physical review letters.

[30]  Edgar Knobloch,et al.  Amplitude equations for travelling wave convection , 1990 .

[31]  Domain structures in fourth-order phase and Ginzburg-Landau equations , 1994, patt-sol/9402003.

[32]  R. Occelli,et al.  Order in convective structures , 1983 .

[33]  Hermann Riecke,et al.  The Stability of Standing Waves , 1998 .

[34]  On the Stability of Parametrically Excited Standing Waves , 1990 .

[35]  W. Eckhaus.  Studies in Non-Linear Stability Theory , 1965 .

[36]  Zhang,et al.  Dynamic scaling of growing interfaces. , 1986, Physical review letters.

[37]  D. Walgraef.  External forcing of spatio-temporal patterns , 1988 .

[38]  L. Kramer,et al.  On the Eckhaus instability for spatially periodic patterns , 1985 .

[39]  Hermann Riecke,et al.  Symmetry-breaking Hopf bifurcation in anisotropic systems , 1992 .

[40]  Steinberg,et al.  Temporal modulation of traveling waves. , 1988, Physical review letters.

[41]  Local oscillations, traveling waves, and chaos in Rayleigh-Bénard convection. , 1987, Physical review letters.

[42]  A.,et al.  Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation , 1992 .

[43]  B. Matkowsky,et al.  Coupled nonlocal complex Ginzburg-Landau equations in gasless combustion , 1992 .

[44]  Brand,et al.  Confined states in phase dynamics. , 1989, Physical review letters.

[45]  J. Vega.  On the amplitude equations arising at the onset of the oscillatory instability in pattern formation , 1993 .

[46]  R. D. Pierce,et al.  On the modulational stability of traveling and standing water waves , 1994 .

[47]  P. C. Hohenberg,et al.  Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation , 1982 .

[48]  Boccaletti,et al.  Domain coexistence in two-dimensional optical patterns. , 1996, Physical review letters.

[49]  Gollub,et al.  Localized spatiotemporal chaos in surface waves. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[50]  Hermann Riecke,et al.  Double Phase Slips and Bound Defect Pairs in Parametrically Driven Waves , 1997, chao-dyn/9707013.

[51]  Jayaprakash,et al.  Universal properties of the two-dimensional Kuramoto-Sivashinsky equation. , 1993, Physical review letters.

[52]  Roland Faller,et al.  Phase chaos in the anisotropic complex Ginzburg-Landau equation , 1998 .

[53]  Lee A. Segel,et al.  Distant side-walls cause slow amplitude modulation of cellular convection , 1969, Journal of Fluid Mechanics.

[54]  Knobloch,et al.  Time-modulated oscillatory convection. , 1988, Physical review letters.

[55]  Egolf,et al.  Characterization of the transition from defect to phase turbulence. , 1994, Physical review letters.

[56]  H. Suhl.  The theory of ferromagnetic resonance at high signal powers , 1957 .