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Allahverdyan AE, Steeg GV, Galstyan A. Memory-induced mechanism for self-sustaining activity in networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062824. [PMID: 26764761 DOI: 10.1103/physreve.92.062824] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/20/2015] [Indexed: 06/05/2023]
Abstract
We study a mechanism of activity sustaining on networks inspired by a well-known model of neuronal dynamics. Our primary focus is the emergence of self-sustaining collective activity patterns, where no single node can stay active by itself, but the activity provided initially is sustained within the collective of interacting agents. In contrast to existing models of self-sustaining activity that are caused by (long) loops present in the network, here we focus on treelike structures and examine activation mechanisms that are due to temporal memory of the nodes. This approach is motivated by applications in social media, where long network loops are rare or absent. Our results suggest that under a weak behavioral noise, the nodes robustly split into several clusters, with partial synchronization of nodes within each cluster. We also study the randomly weighted version of the models where the nodes are allowed to change their connection strength (this can model attention redistribution) and show that it does facilitate the self-sustained activity.
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Affiliation(s)
- A E Allahverdyan
- Yerevan Physics Institute, Alikhanian Brothers Street 2, Yerevan 375036, Armenia
| | - G Ver Steeg
- USC Information Sciences Institute, 4676 Admiralty Way, Marina del Rey, California 90292, USA
| | - A Galstyan
- USC Information Sciences Institute, 4676 Admiralty Way, Marina del Rey, California 90292, USA
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2
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Pecora LM, Carroll TL. Synchronization of chaotic systems. CHAOS (WOODBURY, N.Y.) 2015; 25:097611. [PMID: 26428564 DOI: 10.1063/1.4917383] [Citation(s) in RCA: 65] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.
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Affiliation(s)
- Louis M Pecora
- U.S. Naval Research Laboratory, Washington, District of Columbia 20375, USA
| | - Thomas L Carroll
- U.S. Naval Research Laboratory, Washington, District of Columbia 20375, USA
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Qiao W, Sipahi R. Rules and limitations of building delay-tolerant topologies for coupled systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:016104. [PMID: 22400622 DOI: 10.1103/physreve.85.016104] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/27/2011] [Indexed: 05/31/2023]
Abstract
This paper investigates the equilibrium behavior of broadly studied synchronization dynamics of coupled systems, among which shared information is delayed. The underlying relationship is established between graph structures and the largest amount of delay the dynamics can withstand without losing stability. In particular, based on Cartesian product of graphs, we present the rules and limitations for synthesizing the graphs of large-scale systems that can remain stable for as large delays as possible. Examples are provided to demonstrate the results.
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Affiliation(s)
- Wei Qiao
- Department of Mechanical and Industrial Engineering, Northeastern University, Boston, Massachusetts 02115, USA
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Kagawa Y, Takamatsu A. Synchronization and spatiotemporal patterns in coupled phase oscillators on a weighted planar network. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:046216. [PMID: 19518321 DOI: 10.1103/physreve.79.046216] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2008] [Indexed: 05/27/2023]
Abstract
To reveal the relation between network structures found in two-dimensional biological systems, such as protoplasmic tube networks in the plasmodium of true slime mold, and spatiotemporal oscillation patterns emerged on the networks, we constructed coupled phase oscillators on weighted planar networks and investigated their dynamics. Results showed that the distribution of edge weights in the networks strongly affects (i) the propensity for global synchronization and (ii) emerging ratios of oscillation patterns, such as traveling and concentric waves, even if the total weight is fixed. In-phase locking, traveling wave, and concentric wave patterns were, respectively, observed most frequently in uniformly weighted, center weighted treelike, and periphery weighted ring-shaped networks. Controlling the global spatiotemporal patterns with the weight distribution given by the local weighting (coupling) rules might be useful in biological network systems including the plasmodial networks and neural networks in the brain.
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Affiliation(s)
- Yuki Kagawa
- Department of Electrical Engineering and Bioscience, Waseda University, Tokyo 169-8555, Japan.
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de Souza Pinto SE, Lunardi JT, Saleh AM, Batista AM. Some aspects of the synchronization in coupled maps. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:037206. [PMID: 16241623 DOI: 10.1103/physreve.72.037206] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2005] [Revised: 06/07/2005] [Indexed: 05/05/2023]
Abstract
We numerically study the synchronization behavior of a coupled map lattice consisting of a chain of chaotic logistic maps exhibiting power law interactions. We report two main results. First, we find a practical lower bound in the lattice size in order that this system could be considered in the thermodynamic limit in numerical simulations. Second, we observe the existence of a strong correlation between the Lyapunov dimension and the averaged synchronization time.
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Affiliation(s)
- Sandro E de Souza Pinto
- Grupo de Física Teórica, Departamento de Matemática e Estatística, Universidade Estadual de Ponta Grossa, Avenida Gal. Carlos Cavalcanti 4748. CEP 84032-900, Ponta Grossa, Paraná, Brazil.
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Jalan S, Amritkar RE, Hu CK. Synchronized clusters in coupled map networks. I. Numerical studies. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:016211. [PMID: 16090070 DOI: 10.1103/physreve.72.016211] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2004] [Indexed: 05/03/2023]
Abstract
We study the synchronization of coupled maps on a variety of networks including regular one- and two-dimensional networks, scale-free networks, small world networks, tree networks, and random networks. For small coupling strengths nodes show turbulent behavior but form phase synchronized clusters as coupling increases. When nodes show synchronized behavior, we observe two interesting phenomena. First, there are some nodes of the floating type that show intermittent behavior between getting attached to some clusters and evolving independently. Second, we identify two different ways of cluster formation, namely self-organized clusters which have mostly intracluster couplings and driven clusters which have mostly intercluster couplings. The synchronized clusters may be of dominant self-organized type, dominant driven type, or mixed type depending on the type of network and the parameters of the dynamics. We define different states of the coupled dynamics by considering the number and type of synchronized clusters. For the local dynamics governed by the logistic map we study the phase diagram in the plane of the coupling constant (epsilon) and the logistic map parameter (mu). For large coupling strengths and nonlinear coupling we find that the scale-free networks and the Caley tree networks lead to better cluster formation than the other types of networks with the same average connectivity. For most of our study we use the number of connections of the order of the number of nodes. As the number of connections increases the number of nodes forming clusters and the size of the clusters in general increase.
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Affiliation(s)
- Sarika Jalan
- Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India.
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Atay FM, Biyikoğlu T. Graph operations and synchronization of complex networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:016217. [PMID: 16090076 DOI: 10.1103/physreve.72.016217] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/23/2004] [Indexed: 05/03/2023]
Abstract
The effects of graph operations on the synchronization of coupled dynamical systems are studied. The operations range from addition or deletion of links to various ways of combining networks and generating larger networks from simpler ones. Methods from graph theory are used to calculate or estimate the eigenvalues of the Laplacian operator, which determine the synchronizability of continuous or discrete time dynamics evolving on the network. Results are applied to explain numerical observations on random, scale-free, and small-world networks. An interesting feature is that, when two networks are combined by adding links between them, the synchronizability of the resulting network may worsen as the synchronizability of the individual networks is improved. Similarly, adding links to a network may worsen its synchronizability, although it decreases the average distance in the graph.
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Affiliation(s)
- Fatihcan M Atay
- Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig, Germany.
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Stefański A, Wojewoda J, Kapitaniak T, Yanchuk S. Simple estimation of synchronization threshold in ensembles of diffusively coupled chaotic systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:026217. [PMID: 15447575 DOI: 10.1103/physreve.70.026217] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2003] [Accepted: 04/13/2004] [Indexed: 05/24/2023]
Abstract
In this paper, we define a simple criterion of the synchronization threshold in the set of coupled chaotic systems (flows or maps) with diagonal diffusive coupling. The condition of chaotic synchronization is determined only by two "parameters of order," i.e., the largest Lyapunov exponent and the coupling coefficient. Our approach can be applied for both regular chaotic networks and arrays or lattices of chaotic oscillators with irregular, arbitrarily assumed structure of coupling. The main idea of the synchronization stability criterion is based on linear analysis of the ensembles of simplest dynamical systems. Numerical simulations confirm that such a linear approach approximates the synchronization threshold with high precision.
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Affiliation(s)
- Andrzej Stefański
- Division of Dynamics, Technical University of Łódź, Stefanowskiego 1/15, 90-924 Łódź, Poland.
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Atay FM, Jost J, Wende A. Delays, connection topology, and synchronization of coupled chaotic maps. PHYSICAL REVIEW LETTERS 2004; 92:144101. [PMID: 15089542 DOI: 10.1103/physrevlett.92.144101] [Citation(s) in RCA: 91] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/20/2003] [Indexed: 05/06/2023]
Abstract
We consider networks of coupled maps where the connections between units involve time delays. We show that, similar to the undelayed case, the synchronization of the network depends on the connection topology, characterized by the spectrum of the graph Laplacian. Consequently, scale-free and random networks are capable of synchronizing despite the delayed flow of information, whereas regular networks with nearest-neighbor connections and their small-world variants generally exhibit poor synchronization. On the other hand, connection delays can actually be conducive to synchronization, so that it is possible for the delayed system to synchronize where the undelayed system does not. Furthermore, the delays determine the synchronized dynamics, leading to the emergence of a wide range of new collective behavior which the individual units are incapable of producing in isolation.
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Affiliation(s)
- Fatihcan M Atay
- Max Planck Institute for Mathematics in the Sciences, Leipzig 04103, Germany.
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Anteneodo C, Pinto SEDS, Batista AM, Viana RL. Analytical results for coupled-map lattices with long-range interactions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:045202. [PMID: 14682992 DOI: 10.1103/physreve.68.045202] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/18/2003] [Indexed: 05/24/2023]
Abstract
We obtain exact analytical results for lattices of maps with couplings that decay with distance as r(-alpha). We analyze the effect of the coupling range on the system dynamics through the Lyapunov spectrum. For lattices whose elements are piecewise linear maps, we get an algebraic expression for the Lyapunov spectrum. When the local dynamics is given by a nonlinear map, the Lyapunov spectrum for a completely synchronized state is analytically obtained. The critical line characterizing the synchronization transition is determined from the expression for the largest transversal Lyapunov exponent. In particular, it is shown that in the thermodynamical limit, such transition is only possible for sufficiently long-range interactions, namely, for alpha<alpha(c)=d, where d is the lattice dimension.
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Affiliation(s)
- Celia Anteneodo
- Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Rio de Janeiro, Brazil
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Tucci K, Cosenza MG, Alvarez-Llamoza O. Phase separation in coupled chaotic maps on fractal networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:027202. [PMID: 14525153 DOI: 10.1103/physreve.68.027202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/01/2003] [Indexed: 05/24/2023]
Abstract
The phase ordering dynamics of coupled chaotic maps on fractal networks is investigated. The statistical properties of the systems are characterized by means of the persistence probability of equivalent spin variables that define the phases. The persistence saturates and phase domains freeze for all values of the coupling parameter as a consequence of the fractal structure of the networks, in contrast to the phase transition behavior previously observed in regular Euclidean lattices. Several discontinuities and other features found in the saturation persistence curve as a function of the coupling are explained in terms of changes of stability of local phase configurations on the fractals.
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Affiliation(s)
- K Tucci
- SUMA-CESIMO, Universidad de Los Andes, Mérida, Venezuela
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Jalan S, Amritkar RE. Self-organized and driven phase synchronization in coupled maps. PHYSICAL REVIEW LETTERS 2003; 90:014101. [PMID: 12570614 DOI: 10.1103/physrevlett.90.014101] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/23/2002] [Indexed: 05/24/2023]
Abstract
We study the phase synchronization and cluster formation in coupled maps on different networks. We identify two different mechanisms of cluster formation: (a) self-organized phase synchronization which leads to clusters with dominant intracluster couplings and (b) driven phase synchronization which leads to clusters with dominant intercluster couplings. In the novel driven synchronization the nodes of one cluster are driven by those of the others. We also discuss the dynamical origin of these two mechanisms for small networks with two and three nodes.
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Affiliation(s)
- Sarika Jalan
- Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India.
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Cosenza MG, Tucci K. Pattern formation on trees. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:026208. [PMID: 11497678 DOI: 10.1103/physreve.64.026208] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/19/2001] [Indexed: 05/23/2023]
Abstract
Networks having the geometry and the connectivity of trees are considered as the spatial support of spatiotemporal dynamical processes. A tree is characterized by two parameters: its ramification and its depth. The local dynamics at the nodes of a tree is described by a nonlinear map, giving rise to a coupled map lattice system. The coupling is expressed by a matrix whose eigenvectors constitute a basis on which spatial patterns on trees can be expressed by linear combination. The spectrum of eigenvalues of the coupling matrix exhibit a nonuniform distribution that manifests itself in the bifurcation structure of the spatially synchronized modes. These models may describe reaction-diffusion processes and several other phenomena occurring on heterogeneous media with hierarchical structure.
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Affiliation(s)
- M G Cosenza
- Centro de Astrofísica Teórica, Universidad de Los Andes, Apartado Postal 26, La Hechicera, Mérida 5251, Venezuela
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Chai Wah Wu. Synchronization in arrays of coupled nonlinear systems: passivity, circle criterion, and observer design. ACTA ACUST UNITED AC 2001. [DOI: 10.1109/81.956024] [Citation(s) in RCA: 95] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/05/2022]
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Gade PM, Hu CK. Synchronous chaos in coupled map lattices with small-world interactions. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:6409-6413. [PMID: 11101976 DOI: 10.1103/physreve.62.6409] [Citation(s) in RCA: 36] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2000] [Revised: 07/27/2000] [Indexed: 05/23/2023]
Abstract
In certain physical situations, extensive interactions arise naturally in systems. We consider one such situation, namely, small-world couplings. We show that, for a fixed fraction of nonlocal couplings, synchronous chaos is always a stable attractor in the thermodynamic limit. We point out that randomness helps synchronization. We also show that there is a size dependent bifurcation in the collective behavior in such systems.
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Affiliation(s)
- PM Gade
- Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan
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Fink KS, Johnson G, Carroll T, Mar D, Pecora L. Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:5080-5090. [PMID: 11031550 DOI: 10.1103/physreve.61.5080] [Citation(s) in RCA: 52] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/1999] [Indexed: 05/23/2023]
Abstract
We show that the stability surface that governs the synchronization of a large class of arrays of identical oscillators can be probed with a simple array of just three identical oscillators. Experimentally this implies that it may be possible to probe the synchronization conditions of many arrays all at the same time. In the process of developing a theory of the three-oscillator probe, we also show that several regimes of asymptotic coupling can be derived for the array classes, including the case of large imaginary coupling, which apparently has not been explored.
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Affiliation(s)
- KS Fink
- Columbia University, New York, New York 10027, USA
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Gade PM, Hu CK. Synchronization and coherence in thermodynamic coupled map lattices with intermediate-range coupling. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:4966-9. [PMID: 11970366 DOI: 10.1103/physreve.60.4966] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/23/1999] [Indexed: 04/18/2023]
Abstract
In spatially extended systems, intermediate-range interactions arise naturally in some physical contexts. To study them, we investigate a model of coupled map lattices (CML's) with intermediate-range coupling, and derive analytic conditions for its synchronization. We find that in these CML's, if the range of coupling is fixed, the law of large numbers applies for the mean field. The total normalized power in nonzero components of the power spectrum of the mean field goes to zero in the thermodynamic limit. We also show that in the same limit the relevant parameter for synchronization and coherence is the fraction of sites coupled, and not their number.
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Affiliation(s)
- P M Gade
- Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan.
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Gade PM. Synchronization of oscillators with random nonlocal connectivity. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:64-70. [PMID: 9965047 DOI: 10.1103/physreve.54.64] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Chatterjee N, Gupte N. Synchronization in coupled sine circle maps. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:4457-4466. [PMID: 9964778 DOI: 10.1103/physreve.53.4457] [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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