51
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Garstecki P, Whitesides GM. Flowing crystals: nonequilibrium structure of foam. PHYSICAL REVIEW LETTERS 2006; 97:024503. [PMID: 16907453 DOI: 10.1103/physrevlett.97.024503] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/01/2005] [Indexed: 05/11/2023]
Abstract
Bubbles pushed through a quasi-two-dimensional channel self-organize into a variety of periodic lattices. The structures of these lattices correspond to local minima of the interfacial energy. The "flowing crystals" are long-lived metastable states, a small subset of possible local minima of confined quasi-two-dimensional foams [P. Garstecki and G. M. Whitesides, Phys. Rev. E 73, 031603 (2006)10.1103/PhysRevE.73.031603]. Experimental results suggest that the choice of the structures that we observe is dictated by the dynamic stability of the cyclic processes of their formation. Thus, the dynamic system that we report provides a unique example of nonequilibrium self-organization that results in structures that correspond to local minima of the relevant energy functional.
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Affiliation(s)
- Piotr Garstecki
- Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Cambridge, Massachusetts 02138, USA.
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52
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Zhang X, Fu M, Xiao J, Hu G. Self-organization of chaos synchronization and pattern formation in coupled chaotic oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:015202. [PMID: 16907144 DOI: 10.1103/physreve.74.015202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2006] [Indexed: 05/11/2023]
Abstract
Pattern formation in spatiotemporal chaotic systems is investigated. Temporally chaotic and spatially ordered patterns are observed by varying the coupling strength. Spatial orderings emerge spontaneously due to self-organization of partial and nonlocal chaos synchronization, governed by various types of spatial symmetries. The first and secondary bifurcations from spatially disordered chaos to chaos with different levels of spatial orderings are observed and the scaling behaviors associated with these bifurcations are statistically analyzed.
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Affiliation(s)
- Xiaoming Zhang
- Department of Physics, Beijing Normal University, Beijing 100875, China
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53
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Jayaraman A, Scheel JD, Greenside HS, Fischer PF. Characterization of the domain chaos convection state by the largest Lyapunov exponent. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:016209. [PMID: 16907179 DOI: 10.1103/physreve.74.016209] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2005] [Indexed: 05/11/2023]
Abstract
Using numerical integrations of the Boussinesq equations in rotating cylindrical domains with realistic boundary conditions, we have computed the value of the largest Lyapunov exponent lambda1 for a variety of aspect ratios and driving strengths. We study in particular the domain chaos state, which bifurcates supercritically from the conducting fluid state and involves extended propagating fronts as well as point defects. We compare our results with those from Egolf, [Nature 404, 733 (2000)], who suggested that the value of lambda1 for the spiral defect chaos state of a convecting fluid was determined primarily by bursts of instability arising from short-lived, spatially localized dislocation nucleation events. We also show that the quantity lambda1 is not intensive for aspect ratios Gamma over the range 20<Gamma<40 and that the scaling exponent of lambda1 near onset is consistent with the value predicted by the amplitude equation formalism.
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Affiliation(s)
- A Jayaraman
- Department of Physics, Duke University, Durham, NC 27708, USA
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54
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Fishman MP, Egolf DA. Revealing the building blocks of spatiotemporal chaos: deviations from extensivity. PHYSICAL REVIEW LETTERS 2006; 96:054103. [PMID: 16486935 DOI: 10.1103/physrevlett.96.054103] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2005] [Indexed: 05/06/2023]
Abstract
We have performed high-precision computational studies of the fractal dimension as a function of system length for spatiotemporal chaotic states of the one-dimensional complex Ginzburg-Landau equation. Our data show deviations from extensivity on a length scale consistent with the chaotic length scale, indicating that this spatiotemporal chaotic system is composed of weakly interacting building blocks, each containing about 2 degrees of freedom. Our results also suggest an explanation of some of the "windows of periodicity" found in spatiotemporal systems of moderate size.
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Affiliation(s)
- Matthew P Fishman
- Department of Physics, Georgetown University, Washington, DC 20057, USA
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55
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Yang HL, Radons G. Lyapunov instabilities of Lennard-Jones fluids. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:036211. [PMID: 15903550 DOI: 10.1103/physreve.71.036211] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/19/2004] [Revised: 11/16/2004] [Indexed: 05/02/2023]
Abstract
Recent work on many-particle systems reveals the existence of regular collective perturbations corresponding to the smallest positive Lyapunov exponents (LEs), called hydrodynamic Lyapunov modes. Until now, however, these modes have been found only for hard-core systems. Here we report results on Lyapunov spectra and Lyapunov vectors (LVs) for Lennard-Jones fluids. By considering the Fourier transform of the coordinate fluctuation density u((alpha)) (x,t) , it is found that the LVs with lambda approximately equal to 0 are highly dominated by a few components with low wave numbers. These numerical results provide strong evidence that hydrodynamic Lyapunov modes do exist in soft-potential systems, although the collective Lyapunov modes are more vague than in hard-core systems. In studying the density and temperature dependence of these modes, it is found that, when the value of the Lyapunov exponent lambda((alpha)) is plotted as function of the dominant wave number k(max) of the corresponding LV, all data from simulations with different densities and temperatures collapse onto a single curve. This shows that the dispersion relation lambda((alpha)) vs k(max) for hydrodynamical Lyapunov modes appears to be universal for the low-density cases studied here. Despite the wavelike character of the LVs, no steplike structure exists in the Lyapunov spectrum of the systems studied here, in contrast to the hard-core case. Further numerical simulations show that the finite-time LEs fluctuate strongly. We have also investigated localization features of LVs and propose a length scale to characterize the Hamiltonian spatiotemporal chaotic states.
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Affiliation(s)
- Hong-liu Yang
- Institute of Physics, Chemnitz University of Technology, D-09107 Chemnitz, Germany.
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56
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Brunet P, Limat L. Defects and spatiotemporal disorder in a pattern of falling liquid columns. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:046207. [PMID: 15600495 DOI: 10.1103/physreve.70.046207] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2004] [Revised: 06/02/2004] [Indexed: 05/24/2023]
Abstract
Disordered regimes of a one-dimensional pattern of liquid columns hanging below an overflowing circular dish are investigated experimentally. The interaction of two basic dynamical modes (oscillations and drift) combined with the occurrence of defects (birth of new columns, disappearances by coalescences of two columns) leads to spatiotemporal chaos. When the flow rate is progressively increased, a continuous transition between transient and permanent chaos is pointed into evidence. We introduce the rate of defects as the sole relevant quantity to quantify this "turbulence" without ambiguity. Statistics on both transient and endlessly chaotic regimes enable to define a critical flow rate around which exponents are extracted. Comparisons are drawn with other interfacial pattern-forming systems, where transition towards chaos follows similar steps. Qualitatively, careful examinations of the global dynamics show that the contamination processes are nonlocal and involve the propagation of blocks of elementary laminar states (such as propagative domains or local oscillations), emitted near the defects, which turn out to be essential ingredients of this self-sustained disorder.
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Affiliation(s)
- Philippe Brunet
- Laboratoire de Physique et Mécanique des Milieux Hétérogènes UMR 7636 CNRS - Ecole Supérieure de Physique et Chimie Industrielles 10, rue Vauquelin, 75231 Paris Cedex 05, France.
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57
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Huepe C, Riecke H, Daniels KE, Bodenschatz E. Statistics of defect trajectories in spatio-temporal chaos in inclined layer convection and the complex Ginzburg-Landau equation. CHAOS (WOODBURY, N.Y.) 2004; 14:864-874. [PMID: 15446997 DOI: 10.1063/1.1778495] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
For spatio-temporal chaos observed in numerical simulations of the complex Ginzburg-Landau equation (CGL) and in experiments on inclined-layer convection (ILC) we report numerical and experimental data on the statistics of defects and of defect loops. These loops consist of defect trajectories in space-time that are connected to each other through the pairwise annihilation or creation of the associated defects. While most such loops are small and contain only a few defects, the loop distribution functions decay only slowly with the quantities associated with the loop size, consistent with power-law behavior. For the CGL, two of the three power-law exponents are found to agree, within our computational precision, with those from previous investigations of a simple lattice model. In certain parameter regimes of the CGL and ILC, our results for the single-defect statistics show significant deviations from the previously reported findings that the defect dynamics are consistent with those of random walkers that are created with fixed probability and annihilated through random collisions.
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Affiliation(s)
- Cristián Huepe
- Engineering Science and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA
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58
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Chiam KH, Lai MC, Greenside HS. Efficient algorithm on a nonstaggered mesh for simulating Rayleigh-Bénard convection in a box. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:026705. [PMID: 14525146 DOI: 10.1103/physreve.68.026705] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/26/2003] [Indexed: 11/07/2022]
Abstract
An efficient semi-implicit second-order-accurate finite-difference method is described for studying incompressible Rayleigh-Bénard convection in a box, with sidewalls that are periodic, thermally insulated, or thermally conducting. Operator-splitting and a projection method reduce the algorithm at each time step to the solution of four Helmholtz equations and one Poisson equation, and these are solved by fast direct methods. The method is numerically stable even though all field values are placed on a single nonstaggered mesh commensurate with the boundaries. The efficiency and accuracy of the method are characterized for several representative convection problems.
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Affiliation(s)
- K-H Chiam
- Nonlinear and Statistical Physics, California Institute of Technology, Mail Code 114-36, Pasadena, California 91125-3600, USA
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59
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Zhang C, Zhang H, Ouyang Q, Hu B, Gunaratne GH. Transition from spiral waves to defect-mediated turbulence induced by gradient effects in a reaction-diffusion system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:036202. [PMID: 14524863 DOI: 10.1103/physreve.68.036202] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2003] [Indexed: 05/24/2023]
Abstract
The transition from spiral waves to defect-mediated turbulence was studied in a spatial open reactor using Belousov-Zhabotinsky reaction. The experimental results show a new mechanism of the transition from spirals to spatiotemporal chaos, in which the gradient effects in the three-dimensional system are essential. The transition scenario consists of two stages: first, the effects of gradients in the third dimension cause a splitting of the spiral tip and a deletion of certain wave segments, generating new wave sources; second, the waves sent by the new wave sources undergo a backfire instability, and the back waves are laterally unstable. As a result, defects are automatically generated and fill all over the system. The result of numerical simulation using the FitzHugh-Nagumo model essentially agrees with the experimental observation.
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Affiliation(s)
- Chunxia Zhang
- Department of Physics, Peking University, Beijing 100871, China
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60
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Chiam KH, Paul MR, Cross MC, Greenside HS. Mean flow and spiral defect chaos in Rayleigh-Bénard convection. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:056206. [PMID: 12786249 DOI: 10.1103/physreve.67.056206] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/05/2002] [Indexed: 05/24/2023]
Abstract
We describe a numerical procedure to construct a modified velocity field that does not have any mean flow. Using this procedure, we present two results. First, we show that, in the absence of the mean flow, spiral defect chaos collapses to a stationary pattern comprising textures of stripes with angular bends. The quenched patterns are characterized by mean wave numbers that approach those uniquely selected by focus-type singularities, which, in the absence of the mean flow, lie at the zigzag instability boundary. The quenched patterns also have larger correlation lengths and are comprised of rolls with less curvature. Secondly, we describe how the mean flow can contribute to the commonly observed phenomenon of rolls terminating perpendicularly into lateral walls. We show that, in the absence of the mean flow, rolls begin to terminate into lateral walls at an oblique angle. This obliqueness increases with the Rayleigh number.
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Affiliation(s)
- K-H Chiam
- Nonlinear and Statistical Physics, Mail Code 114-36, California Institute of Technology, Pasadena, CA 91125-3600, USA.
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61
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Young YN, Riecke H. Penta-hepta defect chaos in a model for rotating hexagonal convection. PHYSICAL REVIEW LETTERS 2003; 90:134502. [PMID: 12689294 DOI: 10.1103/physrevlett.90.134502] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2002] [Indexed: 05/24/2023]
Abstract
In a model for rotating non-Boussinesq convection with mean flow, we identify a regime of spatiotemporal chaos that is based on a hexagonal planform and is sustained by the induced nucleation of dislocations by penta-hepta defects. The probability distribution function for the number of defects deviates substantially from the usually observed Poisson-type distribution. It implies strong correlations between the defects in the form of density-dependent creation and annihilation rates of defects. We extract these rates from the distribution function and also directly from the defect dynamics.
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Affiliation(s)
- Yuan-Nan Young
- Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA
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62
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Tajima S, Greenside HS. Microextensive chaos of a spatially extended system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:017205. [PMID: 12241518 DOI: 10.1103/physreve.66.017205] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/03/2001] [Revised: 04/26/2002] [Indexed: 05/23/2023]
Abstract
By analyzing chaotic states of the one-dimensional Kuramoto-Sivashinsky equation for system sizes L in the range 79 < or = L < or = 93, we show that the Lyapunov fractal dimension D scales microextensively, increasing linearly with L even for increments Delta L that are small compared to the average cell size of 9 and to various correlation lengths. This suggests that a spatially homogeneous chaotic system does not have to increase its size by some characteristic amount to increase its dynamical complexity.
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Affiliation(s)
- Shigeyuki Tajima
- Department of Physics, Duke University, Durham, North Carolina 27708-0305, USA
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63
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Schmitz R, Pesch W, Zimmermann W. Spiral-defect chaos: Swift-Hohenberg model versus Boussinesq equations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:037302. [PMID: 11909322 DOI: 10.1103/physreve.65.037302] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/19/2001] [Indexed: 05/23/2023]
Abstract
Spiral-defect chaos (SDC) in Rayleigh-Bénard convection is a well-established spatio-temporal complex pattern, which competes with stationary rolls near the onset of convection. The characteristic properties of SDC are accurately described on the basis of the standard three-dimensional Boussinesq equations. As a much simpler and attractive two-dimensional model for SDC generalized Swift-Hohenberg (SH) equations have been extensively used in the literature from the early beginning. Here, we show that the description of SDC by SH models has to be considered with care, especially regarding its long-time dynamics. For parameters used in previous SH simulations, SDC occurs only as a transient in contrast to the experiments and the rigorous solutions of the Boussinesq equations. The small-scale structure of the vorticity field at the spiral cores, which might be crucial for persistent SDC, is presumably not perfectly captured in the SH model.
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Affiliation(s)
- Rainer Schmitz
- Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany
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64
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Eguíluz VM, Hernández-García E, Piro O. Complex Ginzburg-Landau equation in the presence of walls and corners. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:036205. [PMID: 11580420 DOI: 10.1103/physreve.64.036205] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/08/2000] [Revised: 05/10/2001] [Indexed: 05/23/2023]
Abstract
We investigate the influence of walls and corners (with Dirichlet and Neumann boundary conditions) in the evolution of two-dimensional autooscillating fields described by the complex Ginzburg-Landau equation. Analytical solutions are found, and arguments provided, to show that Dirichlet walls introduce strong selection mechanisms for the wave pattern. Corners between walls provide additional synchronization mechanisms and associated selection criteria. The numerical results fit well with the theoretical predictions in the parameter range studied.
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Affiliation(s)
- V M Eguíluz
- Center for Chaos and Turbulence Studies, The Niels Bohr Institute, Blegdamsvej 17, DK2100 Copenhagen Ø, Denmark.
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65
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de Bruyn JR, Lewis BC, Shattuck MD, Swinney HL. Spiral patterns in oscillated granular layers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:041305. [PMID: 11308836 DOI: 10.1103/physreve.63.041305] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2000] [Indexed: 05/23/2023]
Abstract
Cell-filling spiral patterns are observed in a vertically oscillated layer of granular material when the oscillation amplitude is suddenly increased from below the onset of pattern formation into the region where stripe patterns appear for quasistatic increases in amplitude. These spirals are transients and decay to stripe patterns with defects. A transient spiral defect chaos state is also observed. We describe the behavior of the spirals, and the way in which they form and decay. Our results are compared with those for similar spiral patterns in Rayleigh-Bénard convection in fluids.
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Affiliation(s)
- J R de Bruyn
- Department of Physics and Physical Oceanography, Memorial University of Newfoundland, St. John's, Newfoundland, Canada A1B 3X7.
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66
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Matley RG, Wong WY, Thurlow MS, Lucas PG, Lees MJ, Griffiths OJ, Woodcraft AL. Flow pattern dynamics in convecting liquid helium. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:045301. [PMID: 11308899 DOI: 10.1103/physreve.63.045301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/20/2000] [Indexed: 05/23/2023]
Abstract
We present experimental data which correlate thermal measurements and flow visualization in convecting liquid 4He. For a small range R(C)<R<R1 of the Rayleigh number R above the convection threshold value R(C) a robust stationary pattern is observed. This pattern exhibits periodic pulsation when R>R1, generating thermal oscillations as in earlier reports. At higher R values the time dependence becomes aperiodic with the surprising appearance of spiral-defect chaos at an aspect ratio smaller than has previously been reported.
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Affiliation(s)
- R G Matley
- Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
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67
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