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June 2007

Volume 17, Issue 2,  Articles (02xxxx)


Cover image from Adilson E. Motter and Zoltan Toroczkai, Chaos 17, 026101 (2007).

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REGULAR ARTICLES

Quantitative and qualitative characterization of zigzag spatiotemporal chaos in a system of amplitude equations for nematic electroconvection

Iuliana Oprea, Ioana Triandaf, Gerhard Dangelmayr, and Ira B. Schwartz

Chaos 17, 023101 (2007) (12 pages)

Online Publication Date: 5 April 2007

Full Text: PDF (3724 kB)

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It has been suggested by experimentalists that a weakly nonlinear analysis of the recently introduced equations of motion for the nematic electroconvection by M. Treiber and L. Kramer [Phys. Rev. E 58, 1973 (1998)] has the potential to reproduce the dynamics of the zigzag-type extended spatiotemporal chaos and localized solutions observed near onset in experiments [M. Dennin, D. S. Cannell, and G. Ahlers, Phys. Rev. E 57, 638 (1998); J. T. Gleeson (private communication)]. In this paper, we study a complex spatiotemporal pattern, identified as spatiotemporal chaos, that bifurcates at the onset from a spatially uniform solution of a system of globally coupled complex Ginzburg-Landau equations governing the weakly nonlinear evolution of four traveling wave envelopes. The Ginzburg-Landau system can be derived directly from the weak electrolyte model for electroconvection in nematic liquid crystals when the primary instability is a Hopf bifurcation to oblique traveling rolls. The chaotic nature of the pattern and the resemblance to the observed experimental spatiotemporal chaos in the electroconvection of nematic liquid crystals are confirmed through a combination of techniques including the Karhunen-Loève decomposition, time-series analysis of the amplitudes of the dominant modes, statistical descriptions, and normal form theory, showing good agreement between theory and experiments.
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05.45.-a, 02.30.Uu, 61.30.-v

Tail resonances of Fermi-Pasta-Ulam q-breathers and their impact on the pathway to equipartition

Tiziano Penati and Sergej Flach

Chaos 17, 023102 (2007) (16 pages)

Online Publication Date: 5 April 2007

Full Text: PDF (254 kB)

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Upon initial excitation of a few normal modes the energy distribution among all modes of a nonlinear atomic chain (the Fermi-Pasta-Ulam model) exhibits exponential localization on large time scales. At the same time, resonant anomalies (peaks) are observed in its weakly excited tail for long times preceding equipartition. We observe a similar resonant tail structure also for exact time-periodic Lyapunov orbits, coined q-breathers due to their exponential localization in modal space. We give a simple explanation for this structure in terms of superharmonic resonances. The resonance analysis agrees very well with numerical results and has predictive power. We extend a previously developed perturbation method, based essentially on a Poincaré-Lindstedt scheme, in order to account for these resonances, and in order to treat more general model cases, including truncated Toda potentials. Our results give a qualitative and semiquantitative account for the superharmonic resonances of q-breathers and natural packets.
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05.45.-a, 05.50.+q

Chaotic spatial bifurcation by complex coupling

Vladimir D. Shalfeev, Mikhail V. Ivanchenko, and Gian L. Forti

Chaos 17, 023103 (2007) (4 pages)

Online Publication Date: 5 April 2007

Full Text: PDF (125 kB)

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A spatial bifurcation (a transition from stationary to oscillatory regime) in a chain of unidirectionally coupled phase systems is studied. It is shown that complication of coupling terms can make this bifurcation spatially chaotic in contrast to the previously observed “regular” and “predictable” type. It is demonstrated that the found type of spatial bifurcation corresponds to a smooth (predictable) manifold in the parameter space, while its spatial location gets actually unpredictable being governed by regularities of chaotic behavior. We infer that complex collective dynamics may arise in networks with plain architecture and simple dynamics of individual elements if nontrivial coupling is realized.
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05.45.-a

Insights into the algebraic structure of Lorenz-like systems using feedback circuit analysis and piecewise affine models

Christophe Letellier, Gleison F. V. Amaral, and Luis A. Aguirre

Chaos 17, 023104 (2007) (11 pages)

Online Publication Date: 18 April 2007

Full Text: PDF (827 kB)

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The characterization of chaotic attractors has been a widely addressed problem and there are now many different techniques to define their nature in a rather accurate way, at least in the case of a three-dimensional system. Nevertheless, the link between the structure of the ordinary differential equations and the topology of their solutions is still missing and only a few necessary conditions on the algebraic structure are known today. By using a feedback circuit analysis, it is shown that it is possible to identify the relevant terms of the equations, that is, the terms that really contribute to the structure of the phase portrait. Such analysis also provides some guidelines for constructing piecewise affine models. Moreover, equivalence classes can be defined on the basis of the active feedback circuits involved.
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05.45.-a, 02.10.-v, 02.30.Hq, 02.40.Pc

Instabilities in buoyant flows under localized heating

M. C. Navarro, A. M. Mancho, and H. Herrero

Chaos 17, 023105 (2007) (10 pages)

Online Publication Date: 18 April 2007

Full Text: PDF (672 kB)

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We study, from the numerical point of view, instabilities developed in a fluid layer with a free surface in a cylindrical container which is nonhomogeneously heated from below. In particular, we consider the case in which the applied heat is localized around the origin. An axisymmetric basic state appears as soon as a nonzero horizontal temperature gradient is imposed. The basic state may bifurcate to different solutions depending on vertical and lateral temperature gradients and on the shape of the heating function. We find different kinds of instabilities: extended patterns growing on the whole domain, which include those known as targets, and spiral waves. Spirals are present even for infinite Prandtl number. Localized structures both at the origin and at the outer part of the cylinder may appear either as Hopf or stationary bifurcations. An overview of the developed instabilities as functions of the dimensionless parameters is presented in this article.
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47.20.Bp, 47.20.Ky, 47.10.-g, 47.35.-i

Complete and generalized synchronization in a class of noise perturbed chaotic systems

Zhang Chen, Wei Lin, and Jie Zhou

Chaos 17, 023106 (2007) (8 pages)

Online Publication Date: 9 May 2007

Full Text: PDF (593 kB)

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In the paper, in light of the LaSalle-type invariance principle for stochastic differential equations, chaos synchronization is investigated for a class of chaotic systems dissatisfying a globally Lipschitz condition with noise perturbation. Sufficient criteria for both complete synchronization and generalized synchronization are rigorously established and thus successfully applied to realize chaos synchronization in the coupled unified chaotic systems. Furthermore, concrete examples as well as their numerical simulations are provided to illustrate the possible application of the established criteria.
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05.45.Xt, 05.45.Pq, 05.40.Ca, 02.50.Ey, 02.60.-x

Gauge theory for finite-dimensional dynamical systems

Pini Gurfil

Chaos 17, 023107 (2007) (13 pages)

Online Publication Date: 9 May 2007

Full Text: PDF (874 kB)

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Gauge theory is a well-established concept in quantum physics, electrodynamics, and cosmology. This concept has recently proliferated into new areas, such as mechanics and astrodynamics. In this paper, we discuss a few applications of gauge theory in finite-dimensional dynamical systems. We focus on the concept of rescriptive gauge symmetry, which is, in essence, rescaling of an independent variable. We show that a simple gauge transformation of multiple harmonic oscillators driven by chaotic processes can render an apparently “disordered” flow into a regular dynamical process, and that there exists a strong connection between gauge transformations and reduction theory of ordinary differential equations. Throughout the discussion, we demonstrate the main ideas by considering examples from diverse fields, including quantum mechanics, chemistry, rigid-body dynamics, and information theory.
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11.15.-q, 05.45.-a, 03.65.Ge, 02.30.Hq, 02.30.Uu

Energy-based analysis of frequency entrainment described by van der Pol and phase-locked loop equations

Yoshihiko Susuki, Yuuichi Yokoi, and Takashi Hikihara

Chaos 17, 023108 (2007) (8 pages)

Online Publication Date: 9 May 2007

Full Text: PDF (708 kB)

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This paper analyzes frequency entrainment described by van der Pol and phase-locked loop (PLL) equations. The PLL equation represents the dynamics of a PLL circuit that appear in typical phase-locking phenomena. These two equations describe frequency entrainment by a periodic force. The entrainment originates from two different types of limit cycles: libration for the van der Pol equation and rotation for the PLL one. To explore the relationship between the geometry of limit cycles and the mechanism of entrainment, we investigate the entrainment using an energy balance relation. This relation is equivalent to the energy conservation law of dynamical systems with dissipation and input terms. We show response curves for the dc component, harmonic amplitude, phase difference, and energy supplied by a periodic force. The obtained curves indicate that the entrainments for the two equations have different features of supplied energy, and that the entrainment for the PLL equation possibly has the same mechanism as does the regulation of the phase difference for the van der Pol equation.
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05.45.Xt, 02.40.-k

A nonlinear dynamics method for signal identification

T. L. Carroll

Chaos 17, 023109 (2007) (7 pages)

Online Publication Date: 9 May 2007

Full Text: PDF (138 kB)

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When a radio frequency signal is radiated by a transmitter, the properties of the transmitter itself affect the properties of the signal. These transmitter-induced changes are known as unintentional modulation, to differentiate them from intentional modulation used to add information to the signal. The unintentional modulation can be used to identify which transmitter produced a signal. This paper shows how phase space analysis based on nonlinear dynamics ideas can be used to determine which of several amplifiers produced a signal.
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84.40.Xb, 84.40.Ua

Hyperlabyrinth chaos: From chaotic walks to spatiotemporal chaos

Konstantinos E. Chlouverakis and J. C. Sprott

Chaos 17, 023110 (2007) (8 pages)

Online Publication Date: 21 May 2007

Full Text: PDF (363 kB)

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In this paper we examine a very simple and elegant example of high-dimensional chaos in a coupled array of flows in ring architecture that is cyclically symmetric and can also be viewed as an N-dimensional spatially infinite labyrinth (a “hyperlabyrinth”). The scaling laws of the largest Lyapunov exponent, the Kaplan–Yorke dimension, and the metric entropy are investigated in the high-dimensional limit (3<N<=101) together with its routes to chaos. It is shown that by tuning the single bifurcation parameter b that governs the dissipation and the number of coupled systems N, the attractor dimension can span the entire range of 0 to N including Hamiltonian (conservative) hyperchaos in the limit of b=0 and, furthermore, spatiotemporal chaotic behavior. Finally, stability analysis reveals interesting and important changes in the dynamics, whether N is even or odd.
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05.45.Jn, 05.40.Fb, 05.70.Ce, 02.30.Hq

Complex network study of Asian Go players

Xinping Xu, Junhui Hu, and Feng Liu

Chaos 17, 023111 (2007) (9 pages)

Online Publication Date: 21 May 2007

Full Text: PDF (141 kB)

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Tournaments of the game of Go can be represented as a directed network in which the vertices are players and a directed link is pointing from the winner to the loser for each game. In this article, we present some interesting results for the network of Asian Go players, which is composed of 756 Go players and 9473 tournaments. It is found that the topological structure of this network displays the small-world property and a significant rich-club phenomenon where high-degree nodes are tightly interconnected. In addition, we consider the weighted version of the network, and find the weights obey power-law distributions, while the strengths follow stretched exponential distributions. The time evolution of the network structure is studied and the corresponding results discussed. The analysis of this work provides a deeper understanding for the competition network structure of Asian Go players.
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02.50.Le, 89.75.Hc, 87.23.Ge

A simple Lorenz circuit and its radio frequency implementation

Jonathan N. Blakely, Michael B. Eskridge, and Ned J. Corron

Chaos 17, 023112 (2007) (5 pages)

Online Publication Date: 21 May 2007

Full Text: PDF (132 kB)

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A remarkably simple electronic circuit design based on the chaotic Lorenz system is described. The circuit consists of just two active nonlinear elements (high-speed analog multipliers) and a few passive linear elements. Experimental implementations of the circuit exhibit the classic butterfly attractor and the hysteretic transition from steady state to chaos observed in the Lorenz equations. The simplicity of the circuit makes it suitable for radio frequency applications. The power spectrum of the observed oscillations displays a peak frequency as high as 930  kHz and significant power beyond 1  MHz.
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84.40.Ua, 05.45.Vx, 84.30.-r

Robust anti-synchronization of a class of delayed chaotic neural networks

Juan Meng and Xing-yuan Wang

Chaos 17, 023113 (2007) (6 pages)

Online Publication Date: 21 May 2007

Full Text: PDF (283 kB)

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This paper deals with the anti-synchronization problem of a class of delayed neural networks. Based on the Lyapunov stability theory and the Halanay inequality lemma, a kind of controller is designed. It is proved that this kind of controller can achieve anti-synchronization of neural networks with delays. Numerical simulations demonstrate the effectiveness and robustness of the proposed anti-synchronization scheme.
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05.45.Gg, 05.45.Pq, 05.45.Xt

Identification of network modules by optimization of ratio association

L. Angelini, S. Boccaletti, D. Marinazzo, M. Pellicoro, and S. Stramaglia

Chaos 17, 023114 (2007) (6 pages)

Online Publication Date: 21 May 2007

Full Text: PDF (191 kB)

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We introduce a novel method for identifying the modular structures of a network based on the maximization of an objective function: the ratio association. This cost function arises when the communities detection problem is described in the probabilistic autoencoder frame. An analogy with kernel k-means methods allows us to develop an efficient optimization algorithm, based on the deterministic annealing scheme. The performance of the proposed method is shown on real data sets and on simulated networks.
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89.75.-k, 05.45.Xt, 87.18.Sn

Selective image encryption using a spatiotemporal chaotic system

Tao Xiang, Kwok-wo Wong, and Xiaofeng Liao

Chaos 17, 023115 (2007) (12 pages)

Online Publication Date: 21 May 2007

Full Text: PDF (4425 kB)

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A universal selective image encryption algorithm, in which the spatiotemporal chaotic system is utilized, is proposed to encrypt gray-level images. In order to resolve the tradeoff between security and performance, the effectiveness of selective encryption is discussed based on simulation results. The scheme is then extended to encrypt RGB color images. Security analyses for both scenarios show that the proposed schemes achieve high security and efficiency.
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05.45.Vx, 84.40.Ua

Numerical experiments on quantum chaotic billiards

D. D. de Menezes, M. Jar e Silva, and F. M. de Aguiar

Chaos 17, 023116 (2007) (10 pages)

Online Publication Date: 21 May 2007

Full Text: PDF (2579 kB)

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A recently proposed numerical technique for generation of high-quality unstructured meshes is combined with a finite-element method to solve the Helmholtz equation that describes the quantum mechanics of a particle confined in two-dimensional cavities. Different shapes are treated on equal footing, including Sinai, stadium, annular, threefold symmetric, mushroom, cardioid, triangle, and coupled billiards. The results are shown to be in excellent agreement with available measurements in flat microwave resonator counterparts with nonintegrable geometries.
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05.45.Mt, 05.45.Gg, 03.65.Ta, 02.70.Dh, 02.30.Jr

Nonlinear dynamics in combinatorial games: Renormalizing Chomp

Eric J. Friedman and Adam Scott Landsberg

Chaos 17, 023117 (2007) (14 pages)

Online Publication Date: 11 June 2007

Full Text: PDF (441 kB)

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We develop a new approach to combinatorial games that reveals connections between such games and some of the central ideas of nonlinear dynamics: scaling behaviors, complex dynamics and chaos, universality, and aggregation processes. We take as our model system the combinatorial game Chomp, which is one of the simplest in a class of “unsolved” combinatorial games that includes Chess, Checkers, and Go. We discover that the game possesses an underlying geometric structure that “grows” (reminiscent of crystal growth), and show how this growth can be analyzed using a renormalization procedure adapted from physics. In effect, this methodology allows one to transform a combinatorial game like Chomp into a type of dynamical system. Not only does this provide powerful insights into the game of Chomp (yielding a complete probabilistic description of optimal play in Chomp and an answer to a longstanding question about the nature of the winning opening move), but more generally, it offers a mathematical framework for exploring this unexpected relationship between combinatorial games and modern dynamical systems theory.
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05.45.-a, 02.50.Le, 05.10.Cc

Multiparameter estimation using only a chaotic time series and its applications

Debin Huang, Guojing Xing, and Diek W. Wheeler

Chaos 17, 023118 (2007) (9 pages)

Online Publication Date: 11 June 2007

Full Text: PDF (180 kB)

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An important extension to the techniques of synchronization-based parameter estimation is presented. Based on adaptive chaos synchronization, several methods are proposed to dynamically estimate multiple parameters using only a scalar chaotic time series. In comparison with previous schemes, the presented methods decrease the cost of parameter estimation and are more applicable in practice. Numerical examples are used to demonstrate the effectiveness and robustness of the presented methods. As an example application, an implementation of multichannel digital communication is proposed, where multiparameter modulation is used to simultaneously transmit more than one digital message. From a theoretical perspective, such an encoding increases the difficulty to directly read out the message from the transmitted signal and decreases the implementation cost.
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84.40.Ua, 05.45.Xt, 05.45.Vx, 05.45.Tp

Hash function based on chaotic map lattices

Shihong Wang and Gang Hu

Chaos 17, 023119 (2007) (8 pages)

Online Publication Date: 11 June 2007

Full Text: PDF (590 kB)

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A new hash function system, based on coupled chaotic map dynamics, is suggested. By combining floating point computation of chaos and some simple algebraic operations, the system reaches very high bit confusion and diffusion rates, and this enables the system to have desired statistical properties and strong collision resistance. The chaos-based hash function has its advantages for high security and fast performance, and it serves as one of the most highly competitive candidates for practical applications of hash function for software realization and secure information communications in computer networks.
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05.45.Xt

On the existence of chaotic circumferential waves in spinning disks

Arzhang Angoshtari and Mir Abbas Jalali

Chaos 17, 023120 (2007) (8 pages)

Online Publication Date: 11 June 2007

Full Text: PDF (193 kB)

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We use a third-order perturbation theory and Melnikov's method to prove the existence of chaos in spinning circular disks subject to a lateral point load. We show that the emergence of transverse homoclinic and heteroclinic points lead, respectively, to a random reversal in the traveling direction of circumferential waves and a random phase shift of magnitude pi for both forward and backward wave components. These long-term phenomena occur in imperfect low-speed disks sufficiently far from fundamental resonances.
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05.45.-a, 05.40.-a, 02.50.-r

Improvement of speech recognition by nonlinear noise reduction

Krzysztof Urbanowicz and Holger Kantz

Chaos 17, 023121 (2007) (6 pages)

Online Publication Date: 11 June 2007

Full Text: PDF (123 kB)

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The success of nonlinear noise reduction applied to a single channel recording of human voice is measured in terms of the recognition rate of a commercial speech recognition program in comparison to the optimal linear filter. The overall performance of the nonlinear method is shown to be superior. We hence demonstrate that an algorithm that has its roots in the theory of nonlinear deterministic dynamics possesses a large potential in a realistic application.
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05.45.-a, 43.72.-p

Adaptive dynamical networks via neighborhood information: Synchronization and pinning control

Wenlian Lu

Chaos 17, 023122 (2007) (18 pages)

Online Publication Date: 13 June 2007

Full Text: PDF (871 kB)

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In this paper, we introduce a model of an adaptive dynamical network by integrating the complex network model and adaptive technique. In this model, the adaptive updating laws for each vertex in the network depend only on the state information of its neighborhood, besides itself and external controllers. This suggests that an adaptive technique be added to a complex network without breaking its intrinsic existing network topology. The core of adaptive dynamical networks is to design suitable adaptive updating laws to attain certain aims. Here, we propose two series of adaptive laws to synchronize and pin a complex network, respectively. Based on the Lyapunov function method, we can prove that under several mild conditions, with the adaptive technique, a connected network topology is sufficient to synchronize or stabilize any chaotic dynamics of the uncoupled system. This implies that these adaptive updating laws actually enhance synchronizability and stabilizability, respectively. We find out that even though these adaptive methods can succeed for all networks with connectivity, the underlying network topology can affect the convergent rate and the terminal average coupling and pinning strength. In addition, this influence can be measured by the smallest nonzero eigenvalue of the corresponding Laplacian. Moreover, we provide a detailed study of the influence of the prior parameters in this adaptive laws and present several numerical examples to verify our theoretical results and further discussion.
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05.45.Xt, 05.45.Gg, 89.75.Hc, 02.10.Ud, 02.30.Hq

Stability of piecewise affine systems with application to chaos stabilization

Chuandong Li, Guanrong Chen, and Xiaofeng Liao

Chaos 17, 023123 (2007) (12 pages)

Online Publication Date: 14 June 2007

Full Text: PDF (727 kB)

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This paper addresses the stability issue for a class of piecewise affine (PWA) systems, where the state spaces are assumed to be dividable into a certain number of hypercuboid subspaces. By constructing appropriate piecewise continuous Lyapunov functions, several numerically tractable stability criteria are developed for four subclasses of such PWA systems, which allow to recast the switching control problem for the PWA systems as a convex optimization problem. Moreover, the proposed method is applied to switching controller design for (globally and locally) stabilizing the unstable equilibrium points of PWA chaotic systems. Numerical simulations on the chaotic Chua's circuit are presented to verify the theoretical results.
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05.45.Gg, 05.45.Pq, 02.30.Yy

Robust output synchronization of phase planar systems

David I. Rosas Almeida and Joaquin Alvarez

Chaos 17, 023124 (2007) (12 pages)

Online Publication Date: 14 June 2007

Full Text: PDF (761 kB)

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We propose a technique to synchronize, under the master/slave synchronization scheme, two planar systems represented by phase state variables; we name them phase planar systems. The coupling signal has a discontinuous term that produces a closed-loop system having good characteristics of robustness with respect to bounded disturbances and parametric variations, and guarantees exponential convergence to the synchronization state. In general, the coupling signal needs the full state vector of both systems, but because we assume that only the system outputs are available, we include a robust observer. This observer also guarantees exponential convergence to the state of the plant in spite of the existence of bounded disturbances and parametric variations; this characteristic facilitates the stability analysis of the closed-loop system. The performance of the synchronization technique is illustrated with experimental results.
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05.45.Xt, 84.30.Ng, 45.80.+r

Alternans amplification following a two-stimulus protocol in a one-dimensional cardiac ionic model of reentry: From annihilation to double-wave quasiperiodic reentry

P. Comtois and A. Vinet

Chaos 17, 023125 (2007) (13 pages)

Online Publication Date: 19 June 2007

Full Text: PDF (1614 kB)

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Electrical pacing is a common procedure in both experimental and clinical settings to study and/or annihilate anatomical reentry. A previous study [Comtois and Vinet, Chaos 12, 903 (2002)] has described new ways to terminate reentry in a one-dimensional loop model by a protocol consisting of only two stimulations. Annihilation of the reentrant activity was much more likely with these new scenarios than through a unidirectional block. This paper investigates the sensitivity of these scenarios of annihilation to the length of the pathway. It shows that double-pulse stimulation can stop the reentry if the circuit is shorter than a limiting length. Beyond this upper limit, stimulation rather yields sustained double-wave reentry. The same dynamical mechanism, labeled alternans amplification, is found to be responsible for these two types of post-stimulus dynamics.
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87.19.Hh, 87.19.Nn

The infinitesimal operator for the semigroup of the Frobenius-Perron operator from image sequence data: Vector fields and transport barriers from movies

N. Santitissadeekorn and E. M. Bollt

Chaos 17, 023126 (2007) (13 pages)

Online Publication Date: 19 June 2007

Full Text: PDF (2741 kB)

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In this paper, we present an approach to approximate the Frobenius-Perron transfer operator from a sequence of time-ordered images, that is, a movie dataset. Unlike time-series data, successive images do not provide a direct access to a trajectory of a point in a phase space; more precisely, a pixel in an image plane. Therefore, we reconstruct the velocity field from image sequences based on the infinitesimal generator of the Frobenius-Perron operator. Moreover, we relate this problem to the well-known optical flow problem from the computer vision community and we validate the continuity equation derived from the infinitesimal operator as a constraint equation for the optical flow problem. Once the vector field and then a discrete transfer operator are found, then, in addition, we present a graph modularity method as a tool to discover basin structure in the phase space. Together with a tool to reconstruct a velocity field, this graph-based partition method provides us with a way to study transport behavior and other ergodic properties of measurable dynamical systems captured only through image sequences.
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42.30.Va, 42.30.Wb, 42.30.Tz

Stochastic web as a generator of three-dimensional quasicrystal symmetry

G. M. Zaslavsky and M. Edelman

Chaos 17, 023127 (2007) (7 pages)

Online Publication Date: 19 June 2007

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It is shown that two coupled oscillators perturbed by periodic kicks generate a thin stochastic web in the four-dimensional phase space, which differs from the Arnold web. Under some resonance-type condition the web possesses a quasicrystal-type symmetry. In three-dimensional coordinate space, the web's symmetry corresponds to the icosahedral one and, due to that, the original four-dimensional map can be considered as a dynamical generator of the quasicrystal-type tiling of three-dimensional space.
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05.45.Xt, 05.45.Ra, 02.50.Ey, 05.40.-a

Isochronous synchronization in mutually coupled chaotic circuits

Alexandre Wagemakers, Javier M. Buldú, and Miguel A. F. Sanjuán

Chaos 17, 023128 (2007) (7 pages)

Online Publication Date: 26 June 2007

Full Text: PDF (1095 kB)

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This paper examines the robustness of isochronous synchronization in simple arrays of bidirectionally coupled systems. First, the achronal synchronization of two mutually chaotic circuits, which are coupled with delay, is analyzed. Next, a third chaotic circuit acting as a relay between the previous two circuits is introduced. We observe that, despite the delay in the coupling path, the outer dynamical systems show isochronous synchronization of their outputs, i.e., display the same dynamics at exactly the same moment. Finally, we give here the first experimental evidence that the central relaying system is not required to be of the same kind of its outer counterparts.
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05.45.Xt

Empirical analysis of the ship-transport network of China

Xinping Xu, Junhui Hu, and Feng Liu

Chaos 17, 023129 (2007) (9 pages)

Online Publication Date: 26 June 2007

Full Text: PDF (164 kB)

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Structural properties of the ship-transport network of China (STNC) are studied in the light of recent investigations of complex networks. STNC is composed of a set of routes and ports located along the sea or river. Network properties including the degree distribution, degree correlations, clustering, shortest path length, centrality, and betweenness are studied in different definitions of network topology. It is found that geographical constraint plays an important role in the network topology of STNC. We also study the traffic flow of STNC based on the weighted network representation, and demonstrate the weight distribution can be described by power-law or exponential function depending on the assumed definition of network topology. Other features related to STNC are also investigated.
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89.40.Cc, 89.75.Hc, 89.65.-s, 89.75.Fb

Digital transmission for improved synchronization of analog chaos generators in communications systems

C. Robilliard, E. H. Huntington, and M. R. Frater

Chaos 17, 023130 (2007) (7 pages)

Online Publication Date: 26 June 2007

Full Text: PDF (535 kB)

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We present a method for synchronization of chaos generators based on transmission of an analog chaotic waveform in a digital form. Experimental comparisons between digital and analog transmission of chaos from a delayed differential feedback system are performed. Synchronization is demonstrated to be between 18 and 39  dB (or equivalently 63 to 7943 times) better for digital transmission than analog. Coherent chaotic modulation and demodulation is demonstrated in a situation where there is no effective synchronization using analog transmission.
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84.30.Ng, 84.40.Ua

Periodic orbit analysis at the onset of the unstable dimension variability and at the blowout bifurcation

R. F. Pereira, S. E. de S. Pinto, R. L. Viana, S. R. Lopes, and C. Grebogi

Chaos 17, 023131 (2007) (13 pages)

Online Publication Date: 27 June 2007

Full Text: PDF (221 kB)

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Many chaotic dynamical systems of physical interest present a strong form of nonhyperbolicity called unstable dimension variability (UDV), for which the chaotic invariant set contains periodic orbits possessing different numbers of unstable eigendirections. The onset of UDV is usually related to the loss of transversal stability of an unstable fixed point embedded in the chaotic set. In this paper, we present a new mechanism for the onset of UDV, whereby the period of the unstable orbits losing transversal stability tends to infinity as we approach the onset of UDV. This mechanism is unveiled by means of a periodic orbit analysis of the invariant chaotic attractor for two model dynamical systems with phase spaces of low dimensionality, and seems to depend heavily on the chaotic dynamics in the invariant set. We also described, for these systems, the blowout bifurcation (for which the chaotic set as a whole loses transversal stability) and its relation with the situation where the effects of UDV are the most intense. For the latter point, we found that chaotic trajectories off, but very close to, the invariant set exhibit the same scaling characteristic of the so-called on-off intermittency.
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05.45.-a

Critical behavior of blind spots in sensor networks

Liang Huang, Ying-Cheng Lai, Kwangho Park, Junshan Zhang, and Zhifeng Hu

Chaos 17, 023132 (2007) (8 pages)

Online Publication Date: 29 June 2007

Full Text: PDF (503 kB)

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Blind spots in sensor networks, i.e., individual nodes or small groups of nodes isolated from the rest of the network, are of great concern as they may significantly degrade the network's ability to collect and process information. As the operations of many types of sensors in realistic applications rely on short-lifetime power supplies (e.g., batteries), once they are used up (“off”), replacements become necessary (“on”). This off-and-on process can lead to blind spots. An issue of both theoretical and practical interest concerns the dynamical process and the critical behavior associated with the occurrence of blind spots. In particular, there can be various network topologies, and the off-and-on process can be characterized by the probability that a node functions normally, or the occupying probability of a node in the network. The question to be addressed in this paper concerns how the dynamics of blind spots depend on the network topology and on the occupying probability. For regular, random, and mixed networks, we provide theoretical formulas relating the probability of blind spots to the occupying probability, from which the critical point for the occurrence of blind spots can be determined. For scale-free networks, we present a procedure to estimate the critical point. While our theoretical and numerical analyses are presented in the framework of sensor networks, we expect our results to be generally applicable to network partitioning issues in other networks, such as the wireless cellular network, the Internet, or transportation networks, where the issue of blind spots may be of concern.
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84.40.Ua

Cutting process dynamics by nonlinear time series and wavelet analysis

Asok K. Sen, Grzegorz Litak, and Arkadiusz Syta

Chaos 17, 023133 (2007) (8 pages)

Online Publication Date: 29 June 2007

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We have modeled the dynamics of a cutting process by a two-degree-of-freedom mass-spring system with dry friction. Using nonlinear time series and wavelet analysis, we have investigated the vibrational instabilities of the system for different values of the cutting force. By constructing the phase portraits and calculating the Lyapunov exponents we have delineated the conditions for which a periodic or chaotic motion can occur. The results are verified by means of a time-scale representation of the wavelet power spectrum.
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05.45.Tp, 02.50.-r
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FOCUS ISSUE: OPTIMIZATION IN NETWORKS

Introduction: Optimization in networks

Adilson E. Motter and Zoltan Toroczkai

Chaos 17, 026101 (2007) (3 pages)

Online Publication Date: 28 June 2007

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The recent surge in the network modeling of complex systems has set the stage for a new era in the study of fundamental and applied aspects of optimization in collective behavior. This Focus Issue presents an extended view of the state of the art in this field and includes articles from a large variety of domains in which optimization manifests itself, including physical, biological, social, and technological networked systems.
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05.45.-a, 87.23.Ge, 89.75.-k

Transport optimization on complex networks

Bogdan Danila, Yong Yu, John A. Marsh, and Kevin E. Bassler

Chaos 17, 026102 (2007) (9 pages)

Online Publication Date: 28 June 2007

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We present a comparative study of the application of a recently introduced heuristic algorithm to the optimization of transport on three major types of complex networks. The algorithm balances network traffic iteratively by minimizing the maximum node betweenness with as little path lengthening as possible. We show that by using this optimal routing, a network can sustain significantly higher traffic without jamming than in the case of shortest path routing. A formula is proved and tested with numerical simulation that allows quick computation of the average number of hops along the path and of the average travel times once the betweennesses of the nodes are computed. Using this formula, we show that routing optimization preserves the small-world character exhibited by networks under shortest path routing, and that it significantly reduces the average travel time on congested networks with only a negligible increase in the average travel time at low loads. Finally, we study the correlation between the weights of the links in the case of optimal routing and the betweennesses of the nodes connected by them.
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84.40.Ua

Complex systems analysis of series of blackouts: Cascading failure, critical points, and self-organization

Ian Dobson, Benjamin A. Carreras, Vickie E. Lynch, and David E. Newman

Chaos 17, 026103 (2007) (13 pages)

Online Publication Date: 28 June 2007

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We give an overview of a complex systems approach to large blackouts of electric power transmission systems caused by cascading failure. Instead of looking at the details of particular blackouts, we study the statistics and dynamics of series of blackouts with approximate global models. Blackout data from several countries suggest that the frequency of large blackouts is governed by a power law. The power law makes the risk of large blackouts consequential and is consistent with the power system being a complex system designed and operated near a critical point. Power system overall loading or stress relative to operating limits is a key factor affecting the risk of cascading failure. Power system blackout models and abstract models of cascading failure show critical points with power law behavior as load is increased. To explain why the power system is operated near these critical points and inspired by concepts from self-organized criticality, we suggest that power system operating margins evolve slowly to near a critical point and confirm this idea using a power system model. The slow evolution of the power system is driven by a steady increase in electric loading, economic pressures to maximize the use of the grid, and the engineering responses to blackouts that upgrade the system. Mitigation of blackout risk should account for dynamical effects in complex self-organized critical systems. For example, some methods of suppressing small blackouts could ultimately increase the risk of large blackouts.
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84.70.+p, 02.50.-r

Extreme fluctuations in noisy task-completion landscapes on scale-free networks

H. Guclu, G. Korniss, and Z. Toroczkai

Chaos 17, 026104 (2007) (13 pages)

Online Publication Date: 28 June 2007

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We study the statistics and scaling of extreme fluctuations in noisy task-completion landscapes, such as those emerging in synchronized distributed-computing networks, or generic causally constrained queuing networks, with scale-free topology. In these networks the average size of the fluctuations becomes finite (synchronized state) and the extreme fluctuations typically diverge only logarithmically in the large system-size limit ensuring synchronization in a practical sense. Provided that local fluctuations in the network are short tailed, the statistics of the extremes are governed by the Gumbel distribution. We present large-scale simulation results using the exact algorithmic rules, supported by mean-field arguments based on a coarse-grained description.
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89.75.Hc, 05.40.Ca, 05.45.Xt

Optimization in gradient networks

Natali Gulbahce

Chaos 17, 026105 (2007) (4 pages)

Online Publication Date: 28 June 2007

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Gradient networks can be used to model the dominant structure of complex networks. Previous work has focused on random gradient networks. Here we study gradient networks that minimize jamming on substrate networks with scale-free and Erdo-double_acute s-Rényi structure. We introduce structural correlations and strongly reduce congestion occurring on the network by using a Monte Carlo optimization scheme. This optimization alters the degree distribution and other structural properties of the resulting gradient networks. These results are expected to be relevant for transport and other dynamical processes in real network systems.
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89.75.Hc, 05.40.-a, 02.50.Ng, 05.60.-k

Nature-inspired interconnects for self-assembled large-scale network-on-chip designs

Christof Teuscher

Chaos 17, 026106 (2007) (12 pages)

Online Publication Date: 28 June 2007

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Future nanoscale electronics built up from an Avogadro number of components need efficient, highly scalable, and robust means of communication in order to be competitive with traditional silicon approaches. In recent years, the networks-on-chip (NoC) paradigm emerged as a promising solution to interconnect challenges in silicon-based electronics. Current NoC architectures are either highly regular or fully customized, both of which represent implausible assumptions for emerging bottom-up self-assembled molecular electronics that are generally assumed to have a high degree of irregularity and imperfection. Here, we pragmatically and experimentally investigate important design tradeoffs and properties of an irregular, abstract, yet physically plausible three–dimensional (3D) small-world interconnect fabric that is inspired by modern network-on-chip paradigms. We vary the framework's key parameters, such as the connectivity, number of switch nodes, and distribution of long- versus short-range connections, and measure the network's relevant communication characteristics. We further explore the robustness against link failures and the ability and efficiency to solve a simple toy problem, the synchronization task. The results confirm that (1) computation in irregular assemblies is a promising and disruptive computing paradigm for self-assembled nanoscale electronics and (2) that 3D small-world interconnect fabrics with a power-law decaying distribution of shortcut lengths are physically plausible and have major advantages over local two–dimensional and 3D regular topologies.
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85.40.Ls, 85.35.-p, 85.40.Bh

Optimal flux patterns in cellular metabolic networks

Eivind Almaas

Chaos 17, 026107 (2007) (7 pages)

Online Publication Date: 28 June 2007

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The availability of whole-cell-level metabolic networks of high quality has made it possible to develop a predictive understanding of bacterial metabolism. Using the optimization framework of flux balance analysis, I investigate the metabolic response and activity patterns to variations in the availability of nutrient and chemical factors such as oxygen and ammonia by simulating 30 000 random cellular environments. The distribution of reaction fluxes is heavy tailed for the bacteria H. pylori and E. coli, and the eukaryote S. cerevisiae. While the majority of flux balance investigations has relied on implementations of the simplex method, it is necessary to use interior-point optimization algorithms to adequately characterize the full range of activity patterns on metabolic networks. The interior-point activity pattern is bimodal for E. coli and S. cerevisiae, suggesting that most metabolic reactions are either in frequent use or are rarely active. The trimodal activity pattern of H. pylori indicates that a group of its metabolic reactions (20%) are active in approximately half of the simulated environments. Constructing the high-flux backbone of the network for every environment, there is a clear trend that the more frequently a reaction is active, the more likely it is a part of the backbone. Finally, I briefly discuss the predicted activity patterns of the central carbon metabolic pathways for the sample of random environments.
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87.16.Yc, 87.18.-h, 87.15.Rn, 87.14.Ee

Content-based networks: A pedagogical overview

Duygu Balcan and Ayse Erzan

Chaos 17, 026108 (2007) (14 pages)

Online Publication Date: 28 June 2007

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Complex interactions call for the sharing of information between different entities. In a recent paper, we introduced a combinatoric model which concretizes this idea via a string-matching rule. The model was shown to lend itself to analysis regarding certain topological features of the network. In this paper, we will introduce a statistical physics description of this network in terms of a Potts model. We will give an explicit mean-field treatment of a special case that has been proposed as a model for gene regulatory networks, and derive closed-form expressions for the topological coefficients. Simulations of the hidden variable network are then compared with numerically integrated results.
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87.10.+e, 02.10.Ox, 89.75.-k

Propagation of external regulation and asynchronous dynamics in random Boolean networks

H. Mahmoudi, A. Pagnani, M. Weigt, and R. Zecchina

Chaos 17, 026109 (2007) (12 pages)

Online Publication Date: 28 June 2007

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Boolean networks and their dynamics are of great interest as abstract modeling schemes in various disciplines, ranging from biology to computer science. Whereas parallel update schemes have been studied extensively in past years, the level of understanding of asynchronous updates schemes is still very poor. In this paper we study the propagation of external information given by regulatory input variables into a random Boolean network. We compute both analytically and numerically the time evolution and the asymptotic behavior of this propagation of external regulation (PER). In particular, this allows us to identify variables that are completely determined by this external information. All those variables in the network that are not directly fixed by PER form a core which contains, in particular, all nontrivial feedback loops. We design a message-passing approach allowing to characterize the statistical properties of these cores in dependence of the Boolean network and the external condition. At the end we establish a link between PER dynamics and the full random asynchronous dynamics of a Boolean network.
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05.40.-a, 02.50.-r, 02.10.Ox, 02.30.Sa

Multiple attractors, long chaotic transients, and failure in small-world networks of excitable neurons

Hermann Riecke, Alex Roxin, Santiago Madruga, and Sara A. Solla

Chaos 17, 026110 (2007) (15 pages)

Online Publication Date: 28 June 2007

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We study the dynamical states of a small-world network of recurrently coupled excitable neurons, through both numerical and analytical methods. The dynamics of this system depend mostly on both the number of long-range connections or “shortcuts”, and the delay associated with neuronal interactions. We find that persistent activity emerges at low density of shortcuts, and that the system undergoes a transition to failure as their density reaches a critical value. The state of persistent activity below this transition consists of multiple stable periodic attractors, whose number increases at least as fast as the number of neurons in the network. At large shortcut density and for long enough delays the network dynamics exhibit exceedingly long chaotic transients, whose failure times follow a stretched exponential distribution. We show that this functional form arises for the ensemble-averaged activity if the failure time for each individual network realization is exponentially distributed.
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05.45.-a, 02.50.Ng

Agreement dynamics on interaction networks with diverse topologies

Alain Barrat, Andrea Baronchelli, Luca Dall'Asta, and Vittorio Loreto

Chaos 17, 026111 (2007) (10 pages)

Online Publication Date: 28 June 2007

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We review the behavior of a recently introduced model of agreement dynamics, called the “Naming Game.” This model describes the self-organized emergence of linguistic conventions and the establishment of simple communication systems in a population of agents with pairwise local interactions. The mechanisms of convergence towards agreement strongly depend on the network of possible interactions between the agents. In particular, the mean-field case in which all agents communicate with all the others is not efficient, since a large temporary memory is requested for the agents. On the other hand, regular lattice topologies lead to a fast local convergence but to a slow global dynamics similar to coarsening phenomena. The embedding of the agents in a small-world network represents an interesting tradeoff: a local consensus is easily reached, while the long-range links allow to bypass coarsening-like convergence. We also consider alternative adaptive strategies which can lead to faster global convergence.
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02.50.Le, 89.75.Hc

Balls-in-boxes condensation on networks

L. Bogacz, Z. Burda, W. Janke, and B. Waclaw

Chaos 17, 026112 (2007) (6 pages)

Online Publication Date: 28 June 2007

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We discuss two different regimes of condensate formation in zero-range processes on networks: on a q-regular network, where the condensate is formed as a result of a spontaneous symmetry breaking, and on an irregular network, where the symmetry of the partition function is explicitly broken. In the latter case we consider a minimal irregularity of the q-regular network introduced by a single Q node with degree Q>q. The statics and dynamics of the condensation depend on the parameter alpha=ln  Q/q, which controls the exponential falloff of the distribution of particles on regular nodes and the typical time scale for melting of the condensate on the Q node, which increases exponentially with the system size N. This behavior is different than that on a q-regular network, where alpha=0 and where the condensation results from the spontaneous symmetry breaking of the partition function, which is invariant under a permutation of particle occupation numbers on the q nodes of the network. In this case the typical time scale for condensate melting is known to increase typically as a power of the system size.
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64.70.Fx, 64.70.Dv, 64.60.Cn

Spatial updating, spatial transients, and regularities of a complex automaton with nonperiodic architecture

Joana G. Freire, Owen J. Brison, and Jason A. C. Gallas

Chaos 17, 026113 (2007) (9 pages)

Online Publication Date: 28 June 2007

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We study the dynamics of patterns exhibited by rule 52, a totalistic cellular automaton displaying intricate behaviors and wide regions of active/inactive synchronization patches. Systematic computer simulations involving 230 initial configurations reveal that all complexity in this automaton originates from random juxtaposition of a very small number of interfaces delimiting active/inactive patches. Such interfaces are studied with a sidewise spatial updating algorithm. This novel tool allows us to prove that the interfaces found empirically are the only interfaces possible for these periods, independently of the size of the automata. The spatial updating algorithm provides an alternative way to determine the dynamics of automata of arbitrary size, a way of taking into account the complexity of the connections in the lattice.
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05.45.Xt, 05.45.Gg

A statistical mechanics approach for scale-free networks and finite-scale networks

Ginestra Bianconi

Chaos 17, 026114 (2007) (6 pages)

Online Publication Date: 28 June 2007

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We present a statistical mechanics approach for the description of complex networks. We first define an energy and an entropy, associated with a degree distribution, which have a geometrical interpretation. Next we evaluate the distribution that extremizes the free energy of the network. We find two important limiting cases: a scale-free degree distribution and a finite-scale degree distribution. The size of the space of allowed simple networks given these distributions is evaluated in the large network limit. Results are compared with simulations of algorithms generating these networks.
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89.75.Hc, 89.75.Fb, 89.75.Da, 05.40.-a, 05.20.-y, 05.70.Ce

Spectral densities of scale-free networks

D. Kim and B. Kahng

Chaos 17, 026115 (2007) (6 pages)

Online Publication Date: 28 June 2007

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The spectral densities of the weighted Laplacian, random walk, and weighted adjacency matrices associated with a random complex network are studied using the replica method. The link weights are parametrized by a weight exponent beta. Explicit results are obtained for scale-free networks in the limit of large mean degree after the thermodynamic limit, for arbitrary degree exponent and beta.
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05.40.Fb, 05.70.Ce, 89.75.Hc, 02.10.Yn, 02.10.Ox

A box-covering algorithm for fractal scaling in scale-free networks

J. S. Kim, K.-I. Goh, B. Kahng, and D. Kim

Chaos 17, 026116 (2007) (6 pages)

Online Publication Date: 28 June 2007

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A random sequential box-covering algorithm recently introduced to measure the fractal dimension in scale-free (SF) networks is investigated. The algorithm contains Monte Carlo sequential steps of choosing the position of the center of each box; thereby, vertices in preassigned boxes can divide subsequent boxes into more than one piece, but divided boxes are counted once. We find that such box-split allowance in the algorithm is a crucial ingredient necessary to obtain the fractal scaling for fractal networks; however, it is inessential for regular lattice and conventional fractal objects embedded in the Euclidean space. Next, the algorithm is viewed from the cluster-growing perspective that boxes are allowed to overlap; thereby, vertices can belong to more than one box. The number of distinct boxes a vertex belongs to is, then, distributed in a heterogeneous manner for SF fractal networks, while it is of Poisson-type for the conventional fractal objects.
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05.45.Df, 05.40.-a, 05.50.+q, 89.75.Hc, 02.50.Ng

Optimization and scale-freeness for complex networks

Petter Minnhagen and Sebastian Bernhardsson

Chaos 17, 026117 (2007) (7 pages)

Online Publication Date: 28 June 2007

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Complex networks are mapped to a model of boxes and balls where the balls are distinguishable. It is shown that the scale-free size distribution of boxes maximizes the information associated with the boxes provided configurations including boxes containing a finite fraction of the total amount of balls are excluded. It is conjectured that for a connected network with only links between different nodes, the nodes with a finite fraction of links are effectively suppressed. It is hence suggested that for such networks the scale-free node-size distribution maximizes the information encoded on the nodes. The noise associated with the size distributions is also obtained from a maximum entropy principle. Finally, explicit predictions from our least bias approach are found to be borne out by metabolic networks.
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05.40.Ca, 05.70.Ce, 02.50.Ng

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