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Schmitzer B, Kinzel W, Kanter I. Pulses of chaos synchronization in coupled map chains with delayed transmission. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:047203. [PMID: 19905486 DOI: 10.1103/physreve.80.047203] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/17/2009] [Indexed: 05/28/2023]
Abstract
Pulses of synchronization in chaotic coupled map lattices are discussed in the context of transmission of information. Synchronization and desynchronization propagate along the chain with different velocities which are calculated analytically from the spectrum of convective Lyapunov exponents. Since the front of synchronization travels slower than the front of desynchronization, the maximal possible chain length for which information can be transmitted by modulating the first unit of the chain is bounded.
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Affiliation(s)
- Bernhard Schmitzer
- Institute for Theoretical Physics, University of Würzburg, Würzburg, Germany
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Qi GX, Huang HB, Shen CK, Wang HJ, Chen L. Predicting the synchronization time in coupled-map networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:056205. [PMID: 18643140 DOI: 10.1103/physreve.77.056205] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/03/2008] [Revised: 02/11/2008] [Indexed: 05/26/2023]
Abstract
An analytical expression for the synchronization time in coupled-map networks is given. By means of the expression, the synchronization time for any given network can be predicted accurately. Furthermore, for networks in which the distributions of nontrivial eigenvalues of coupling matrices have some unique characteristics, analytical results for the minimal synchronization time are given.
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Affiliation(s)
- G X Qi
- Department of Physics, Southeast University, Nanjing 210096, China.
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Zhan M, Gao J, Wu Y, Xiao J. Chaos synchronization in coupled systems by applying pinning control. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:036203. [PMID: 17930319 DOI: 10.1103/physreve.76.036203] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2006] [Revised: 07/01/2007] [Indexed: 05/25/2023]
Abstract
Chaos synchronization in coupled chaotic oscillator systems with diffusive and gradient couplings forced by only one local feedback injection signal (boundary pinning control) is studied. By using eigenvalue analysis, we obtain controllable regions directly in control parameter space for different types of coupling links (including diagonal coupling and nondiagonal couplings). The effects of both diffusive and gradient couplings on chaos synchronization become clear. Some relevant factors on control efficiency such as coupled system size, transient process, and feedback signal intensity are also studied.
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Affiliation(s)
- Meng Zhan
- Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China.
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He SH, Huang HB, Zhang X, Liu ZX, Xu DS, Shen CK. Chaotic synchronization in large map networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:057203. [PMID: 17280026 DOI: 10.1103/physreve.74.057203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/09/2006] [Indexed: 05/13/2023]
Abstract
The chaotic synchronization in n-dimensional large map networks with local coupling and their size stabilities in the node number N-->infinity are studied analytically and numerically. The analytical results show that the chaotic synchronization is stable for N-->infinity in the presence of the external driving or global coupling. The numerical calculations show that, as the driving or global interaction strength increases from zero, the network states have the whole route: spatiotemporal chaotic state --> cluster chaotic synchronous state --> complete chaotic synchronous state --> spatiotemporal pattern --> spatiotemporal chaotic state.
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Affiliation(s)
- S H He
- Department of Physics, Southeast University, Nanjing 210096, China
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Qi GX, Huang HB, Wang HJ, Xie X, Yang P, Zhang YJ. Layered synchronous propagation of noise-induced chaotic spikes in linear arrays. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:021916. [PMID: 16196613 DOI: 10.1103/physreve.72.021916] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/07/2005] [Revised: 05/16/2005] [Indexed: 05/04/2023]
Abstract
Stable propagation of noise-induced synchronous spiking in uncoupled linear neuron arrays is studied numerically. The chaotic neurons in the unidirectionally coupled linear array are modeled by Hindmarsh-Rose neurons. Stability analysis shows that the synchronous chaotic spiking can be successfully transmitted to cortical areas through layered synchronization in the neural network under certain conditions of the network structure.
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Affiliation(s)
- G X Qi
- Department of Physics, Southeast University, Nanjing 210096, China
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Kim SY, Lim W, Ott E, Hunt B. Dynamical origin for the occurrence of asynchronous hyperchaos and chaos via blowout bifurcations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:066203. [PMID: 14754293 DOI: 10.1103/physreve.68.066203] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/17/2003] [Revised: 07/29/2003] [Indexed: 05/24/2023]
Abstract
We investigate the dynamical origin for the occurrence of asynchronous hyperchaos and chaos via blowout bifurcations in coupled chaotic systems. An asynchronous hyperchaotic or chaotic attractor with a positive or negative second Lyapunov exponent appears through a blowout bifurcation. It is found that the sign of the second Lyapunov exponent of the newly born asynchronous attractor, exhibiting on-off intermittency, is determined through competition between its laminar and bursting components. When the "strength" (i.e., a weighted second Lyapunov exponent) of the bursting component is larger (smaller) than that of the laminar component, an asynchronous hyperchaotic (chaotic) attractor appears.
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Affiliation(s)
- Sang-Yoon Kim
- Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA.
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Lind PG, Corte-Real J, Gallas JAC. Modeling velocity in gradient flows with coupled-map lattices with advection. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:016219. [PMID: 12241473 DOI: 10.1103/physreve.66.016219] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/11/2002] [Indexed: 05/23/2023]
Abstract
We introduce a simple model to investigate large scale behavior of gradient flows based on a lattice of coupled maps which, in addition to the usual diffusive term, incorporates advection, as an asymmetry in the coupling between nearest neighbors. This diffusive-advective model predicts traveling patterns to have velocities obeying the same scaling as wind velocities in the atmosphere, regarding the advective parameter as a sort of geostrophic wind. In addition, the velocity and wavelength of traveling wave solutions are studied. In general, due to the presence of advection, two regimes are identified: for strong diffusion the velocity varies linearly with advection, while for weak diffusion a power law is found with a characteristic exponent proportional to the diffusion.
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Affiliation(s)
- Pedro G Lind
- Unidade de Meteorologia e Climatologia, Instituto de Ciência Aplicada e Tecnologia, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal
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Kim SY, Lim W. Effect of asymmetry on the loss of chaos synchronization. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:016211. [PMID: 11461371 DOI: 10.1103/physreve.64.016211] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/26/2000] [Indexed: 05/23/2023]
Abstract
We investigate the effect of asymmetry of coupling on the bifurcation mechanism for the loss of synchronous chaos in coupled systems. It is found that only when the symmetry-breaking pitchfork bifurcations take part in the process of the synchronization loss for the case of symmetric coupling, the asymmetry changes the bifurcation scenarios of the desynchronization. For the case of weak coupling, pitchfork bifurcations of asynchronous periodic saddles are replaced by saddle-node bifurcations, while for the case of strong coupling, pitchfork bifurcations of synchronous periodic saddles transform to transcritical bifurcations. The effects of the saddle-node and transcritical bifurcations for the weak asymmetry are similar to those of the pitchfork bifurcations for the symmetric-coupling case. However, with increasing the "degree" of the asymmetry, their effects change qualitatively, and eventually become similar to those for the extreme case of unidirectional asymmetric coupling.
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Affiliation(s)
- S Y Kim
- Department of Physics, Kangwon National University, Chunchon, Kangwon-Do 200-701, Korea
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Sanchez E, Matias M, Perez-Munuzuri V. Chaotic synchronization in small assemblies of driven Chua's circuits. ACTA ACUST UNITED AC 2000. [DOI: 10.1109/81.847871] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Lago-Fernández LF, Huerta R, Corbacho F, Sigüenza JA. Fast response and temporal coherent oscillations in small-world networks. PHYSICAL REVIEW LETTERS 2000; 84:2758-61. [PMID: 11017318 DOI: 10.1103/physrevlett.84.2758] [Citation(s) in RCA: 195] [Impact Index Per Article: 8.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/1999] [Indexed: 05/23/2023]
Abstract
We have investigated the role that different connectivity regimes play in the dynamics of a network of Hodgkin-Huxley neurons by computer simulations. The different connectivity topologies exhibit the following features: random topologies give rise to fast system response yet are unable to produce coherent oscillations in the average activity of the network; on the other hand, regular topologies give rise to coherent oscillations, but in a temporal scale that is not in accordance with fast signal processing. Finally, small-world topologies, which fall between random and regular ones, take advantage of the best features of both, giving rise to fast system response with coherent oscillations.
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Affiliation(s)
- L F Lago-Fernández
- Grupo de Neurocomputación Biológica, E.T.S. de Ingeniería Informática, Universidad Autónoma de Madrid, 28049 Madrid, Spain
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Rudzick O, Pikovsky A. Unidirectionally coupled map lattice as a model for open flow systems. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:5107-5115. [PMID: 9965690 DOI: 10.1103/physreve.54.5107] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Gade PM, Cerdeira HA, Ramaswamy R. Coupled maps on trees. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:2478-2485. [PMID: 9963691 DOI: 10.1103/physreve.52.2478] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Willeboordse FH, Kaneko K. Periodic lattices of chaotic defects. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:1516-1519. [PMID: 9963571 DOI: 10.1103/physreve.52.1516] [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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Raghavachari S, Glazier JA. Spatially coherent states in fractally coupled map lattices. PHYSICAL REVIEW LETTERS 1995; 74:3297-3300. [PMID: 10058161 DOI: 10.1103/physrevlett.74.3297] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Shinbrot T. Synchronization of coupled maps and stable windows. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:3230-3233. [PMID: 9962368 DOI: 10.1103/physreve.50.3230] [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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Ginzburg I, Sompolinsky H. Theory of correlations in stochastic neural networks. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:3171-3191. [PMID: 9962363 DOI: 10.1103/physreve.50.3171] [Citation(s) in RCA: 120] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Willeboordse FH, Kaneko K. Bifurcations and spatial chaos in an open flow model. PHYSICAL REVIEW LETTERS 1994; 73:533-536. [PMID: 10057471 DOI: 10.1103/physrevlett.73.533] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Flesselles JM, Croquette V, Jucquois S. Period doubling of a torus in a chain of oscillators. PHYSICAL REVIEW LETTERS 1994; 72:2871-2874. [PMID: 10056006 DOI: 10.1103/physrevlett.72.2871] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Auerbach D. Controlling extended systems of chaotic elements. PHYSICAL REVIEW LETTERS 1994; 72:1184-1187. [PMID: 10056644 DOI: 10.1103/physrevlett.72.1184] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Shinbrot T, Ottino JM. Geometric method to create coherent structures in chaotic flows. PHYSICAL REVIEW LETTERS 1993; 71:843-846. [PMID: 10055382 DOI: 10.1103/physrevlett.71.843] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Pikovsky AS. Local Lyapunov exponents for spatiotemporal chaos. CHAOS (WOODBURY, N.Y.) 1993; 3:225-232. [PMID: 12780031 DOI: 10.1063/1.165987] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Local Lyapunov exponents are proposed for characterization of perturbations in distributed dynamical systems with chaotic behavior. Their relation to usual and velocity-dependent exponents is discussed. Local Lyapunov exponents are analytically calculated for coupled map lattices using random field approximation. Boundary Lyapunov exponents describing reflection of perturbations at boundaries are also introduced and calculated.
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