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Shlizerman E, Rom-Kedar V. Parabolic resonance: a route to hamiltonian spatiotemporal chaos. PHYSICAL REVIEW LETTERS 2009; 102:033901. [PMID: 19257354 DOI: 10.1103/physrevlett.102.033901] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/18/2008] [Indexed: 05/27/2023]
Abstract
We show that initial data near an unperturbed stable plane wave can evolve into a regime of spatiotemporal chaos in the slightly forced conservative periodic one-dimensional nonlinear Schrödinger equation. Statistical measures are employed to demonstrate that this spatiotemporal chaos is intermittent: there are windows in time for which the solution gains spatial coherence. The parameters and initial profiles that lead to such intermittency are predicted by utilizing a novel geometrical description of the integrable unforced equation.
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Affiliation(s)
- Eli Shlizerman
- Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Post Office Box 26, Rehovot 76100, Israel.
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Ramazza PL, Bortolozzo U, Boccaletti S. Experimental synchronization of spatiotemporal chaos in nonlinear optics. Phys Rev E 2006; 73:036213. [PMID: 16605637 DOI: 10.1103/physreve.73.036213] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/10/2005] [Revised: 09/14/2005] [Indexed: 11/07/2022]
Abstract
We demonstrate that a unidirectional coupling between a pattern forming system and its replića induces complete synchronization of the slave to the master system onto a spatiotemporal chaotic state.
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Affiliation(s)
- P L Ramazza
- Istituto Sistemi Complessi CNR, Largo E. Fermi 6, I 50125 Florence, Italy
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Pastur L, Gostiaux L, Bortolozzo U, Boccaletti S, Ramazza PL. Experimental targeting and control of spatiotemporal chaos in nonlinear optics. PHYSICAL REVIEW LETTERS 2004; 93:063902. [PMID: 15323632 DOI: 10.1103/physrevlett.93.063902] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/02/2004] [Indexed: 05/24/2023]
Abstract
We demonstrate targeting and control over spatiotemporal chaos in an optical feedback loop experiment. Different stationary target patterns are stabilized in real time by means of a two dimensional space extended perturbation field driven by an interfaced computer and applied in real space to a liquid crystal display device inserted within a control optical loop. The flexibility of the system in switching between different target patterns is also demonstrated.
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Affiliation(s)
- L Pastur
- Istituto Nazionale di Ottica Applicata, Largo E Fermi 6, 50125 Firenze, Italy
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D'Alessandro G, Papoff F, Louvergneaux E, Glorieux P. Average patterns and coherent phenomena in wide aperture lasers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:066212. [PMID: 15244713 DOI: 10.1103/physreve.69.066212] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2003] [Revised: 03/04/2004] [Indexed: 05/24/2023]
Abstract
Using a realistic model of wide aperture, weakly astigmatic lasers we develop a framework to analyze experimental average intensity patterns. We use the model to explain the appearance of patterns in terms of the modes of the cavity and to show that the breaking of the symmetry of the average intensity patterns is caused by overlaps in the frequency spectra of nonvanishing of modes with different parity. This result can be used even in systems with very fast dynamics to detect experimentally overlaps of frequency spectra of modes.
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Affiliation(s)
- G D'Alessandro
- Department of Mathematics, University of Southampton, Southampton, SO17 1BJ, England, United Kingdom
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Pastur L, Bortolozzo U, Ramazza PL. Transition to space-time chaos in an optical loop with translational transport. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:016210. [PMID: 14995696 DOI: 10.1103/physreve.69.016210] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2003] [Indexed: 05/24/2023]
Abstract
We describe the transition from regular patterns to space-time chaos in an optical system with nonlocal feedback. The nonlocality introduced breaks the rotational symmetry of the system, resulting in a scenario for the transition from regular patterns to a disordered dynamics. The different regimes are characterized both in terms of spatial and temporal correlation functions, and by means of a Kahrunen-Loeve decomposition. This allows the determination of the structures participating in the dynamics, and an estimation of the active degrees of freedom versus the control parameter.
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Affiliation(s)
- L Pastur
- Istituto Nazionale di Ottica Applicata, Largo Enrico Fermi 6, 50125 Firenze, Italy
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Sakaguchi H. Shock structures in time-averaged patterns for the kuramoto-sivashinsky equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:8817-9. [PMID: 11138191 DOI: 10.1103/physreve.62.8817] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/23/2000] [Indexed: 11/07/2022]
Abstract
The Kuramoto-Sivashinsky equation with fixed boundary conditions is numerically studied. Shocklike structures appear in the time-averaged patterns for some parameter range of the boundary values. Effective diffusion constant is estimated from the relation of the width and the height of the shock structures.
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Affiliation(s)
- H Sakaguchi
- Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan
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López C, Alvarez A, Hernández-García E. Forecasting confined spatiotemporal chaos with genetic algorithms. PHYSICAL REVIEW LETTERS 2000; 85:2300-2303. [PMID: 10977996 DOI: 10.1103/physrevlett.85.2300] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/29/2000] [Indexed: 05/23/2023]
Abstract
A technique to forecast spatiotemporal time series is presented. It uses a proper orthogonal or Karhunen-Loève decomposition to encode large spatiotemporal data sets in a few time series, and genetic algorithms to efficiently extract dynamical rules from the data. The method works very well for confined systems displaying spatiotemporal chaos, as exemplified here by forecasting the evolution of the one-dimensional complex Ginzburg-Landau equation in a finite domain.
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Affiliation(s)
- C López
- Instituto Mediterráneo de Estudios Avanzados, IMEDEA (CSIC-Universitat de les Illes Balears), 07071 Palma de Mallorca, Spain
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Meixner M, Zoldi SM, Bose S, Scholl E. Karhunen-Loeve local characterization of spatiotemporal chaos in a reaction-diffusion system. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:1382-1385. [PMID: 11046416 DOI: 10.1103/physreve.61.1382] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/10/1999] [Indexed: 05/23/2023]
Abstract
By computing the Karhunen-Loeve decomposition (KLD) correlation length xi(KLD) of a reaction-diffusion system in the extensive chaos regime, we show that it is a sensitive measure of spatial dynamical inhomogeneities. It reveals substantial spatial nonuniformity of the dynamics at the boundaries and can also detect slow spatial variations in system parameters. The intensive length xi(KLD) can be easily computed from small local subsystems and is found to have a similar parametric dependence as the two-point correlation length computed over the full system size.
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Affiliation(s)
- M Meixner
- Institut fur Theoretische Physik, Technische Universitat Berlin, Hardenbergstrasse 36, D-10623 Berlin, Germany
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Encinas-Sanz F, Leyva I, Guerra JM. Time resolved pattern evolution in a large aperture laser. PHYSICAL REVIEW LETTERS 2000; 84:883-886. [PMID: 11017396 DOI: 10.1103/physrevlett.84.883] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/1999] [Revised: 09/28/1999] [Indexed: 05/23/2023]
Abstract
We have measured quasiinstantaneous transverse patterns in a broad aperture laser. Nonordered patterns yielding to boundary determined regular structures in progressive time-integrated recording are observed. The linear analysis and numerical integration of the full Maxwell-Bloch equations allow us to interpret the features of the experiment. We show that this system being far from threshold cannot be fully understood with a perturbative model.
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Affiliation(s)
- F Encinas-Sanz
- Departamento de Optica, Facultad de CC. Fisicas, Universidad Complutense de Madrid, 28040 Madrid, Spain
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Rabinovich MI, Torres JJ, Varona P, Huerta R, Weidman P. Origin of coherent structures in a discrete chaotic medium. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:R1130-3. [PMID: 11969935 DOI: 10.1103/physreve.60.r1130] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/26/1998] [Revised: 02/11/1999] [Indexed: 04/18/2023]
Abstract
Using as an example a large lattice of locally interacting Hindmarsh-Rose chaotic neurons, we disclose the origin of ordered structures in a discrete nonequilibrium medium with fast and slow chaotic oscillations. The origin of the ordering mechanism is related to the appearance of a periodic average dynamics in the group of chaotic neurons whose individual slow activity is significantly synchronized by the group mean field. Introducing the concept of a "coarse grain" as a cluster of neuron elements with periodic averaged behavior allows consideration of the dynamics of a medium composed of these clusters. A study of this medium reveals spatially ordered patterns in the periodic and slow dynamics of the coarse grains that are controlled by the average intensity of the fast chaotic pulsation.
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Affiliation(s)
- M I Rabinovich
- Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0402, USA
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Cerisier P, Rahal S, Billia B. Extrinsic effects on the disorder dynamics of Bénard-Marangoni patterns. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:3508-3517. [PMID: 9965496 DOI: 10.1103/physreve.54.3508] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Abstract
This paper reviews recent efforts to describe complex patterns in isotropic fluids (Rayleigh-Benard convection) as well as in anisotropic liquid crystals (electro-hydrodynamic convection) when driven away from equilibrium. A numerical scheme for solving the full hydrodynamic equations is presented that allows surprisingly well for a detailed comparison with experiments. The approach can also be useful for a systematic construction of models (order parameter equations). (c) 1996 American Institute of Physics.
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Affiliation(s)
- W. Pesch
- Physikalisches Institut der Universitat Bayreuth, 95440 Bayreuth, Germany
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Surko CM. Dynamics of two-dimensional traveling-wave convection patterns. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:5916-5934. [PMID: 9964951 DOI: 10.1103/physreve.53.5916] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Sasa SI. Collective patterns arising out of spatio-temporal chaos. CHAOS (WOODBURY, N.Y.) 1996; 6:238-242. [PMID: 12780252 DOI: 10.1063/1.166169] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We present a simple mathematical model in which a time averaged pattern emerges out of spatio-temporal chaos as a result of the collective action of chaotic fluctuations. Our evolution equation possesses spatial translational symmetry under periodic boundary conditions. Thus the spatial inhomogeneity of the statistical state arises through spontaneous symmetry breaking. The transition from a state of homogeneous spatio-temporal chaos to one exhibiting spatial order is explained by introducing a collective viscosity which relates the averaged pattern with a correlation of the fluctuations. (c) 1996 American Institute of Physics.
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Affiliation(s)
- Shin-Ichi Sasa
- Department of Pure and Applied Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan
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Huyet G, Martinoni MC, Tredicce JR, Rica S. Spatiotemporal dynamics of lasers with a large fresnel number. PHYSICAL REVIEW LETTERS 1995; 75:4027-4030. [PMID: 10059796 DOI: 10.1103/physrevlett.75.4027] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Abstract
The development of complex states of fluid motion is illustrated by reviewing a series of experiments, emphasizing film flows, surface waves, and thermal convection. In one dimension, cellular patterns bifurcate to states of spatiotemporal chaos. In two dimensions, even ordered patterns can be surprisingly intricate when quasiperiodic patterns are included. Spatiotemporal chaos is best characterized statistically, and methods for doing so are evolving. Transport and mixing phenomena can also lead to spatial complexity, but the degree depends on the significance of molecular or thermal diffusion.
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Affiliation(s)
- J P Gollub
- Physics Department, Haverford College, Haverford, PA 19041, USA
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Gluckman BJ, Arnold CB, Gollub JP. Statistical studies of chaotic wave patterns. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:1128-1147. [PMID: 9962755 DOI: 10.1103/physreve.51.1128] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Quantitative investigation of spatiotemporal chaos in the ferroin-catalyzed Belousov-Zhabotinskii reaction in a batch reactor. THEOR EXP CHEM+ 1995. [DOI: 10.1007/bf00536135] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Feng Q, Moloney JV, Newell AC. Transverse patterns in lasers. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1994; 50:R3601-R3604. [PMID: 9911443 DOI: 10.1103/physreva.50.r3601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Dougherty A, Gunawardana A. Mean shape of three-dimensional dendrites: A comparison of pivalic acid and ammonium chloride. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:1349-1352. [PMID: 9962100 DOI: 10.1103/physreve.50.1349] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Olsen OH. Chaotic patterns in a Josephson junction model. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:182-187. [PMID: 9961955 DOI: 10.1103/physreve.50.182] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Bosch E, Lambermont H. Average patterns in Faraday waves. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:R3580-R3583. [PMID: 9961789 DOI: 10.1103/physreve.49.r3580] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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