Showing papers in "Chaos in 2006"
TL;DR: It is shown that two coupled identical neural networks with or without time-varying delay can achieve synchronization by enhancing the coupling strength dynamically.
Abstract: In this paper, based on the invariant principle of functional differential equations, a simple, analytical, and rigorous adaptive feedback scheme is proposed for the synchronization of almost all kinds of coupled identical neural networks with time-varying delay, which can be chaotic, periodic, etc. We do not assume that the concrete values of the connection weight matrix and the delayed connection weight matrix are known. We show that two coupled identical neural networks with or without time-varying delay can achieve synchronization by enhancing the coupling strength dynamically. The update gain of coupling strength can be properly chosen to adjust the speed of achieving synchronization. Also, it is quite robust against the effect of noise and simple to implement in practice. In addition, numerical simulations are given to show the effectiveness of the proposed synchronization method.
368 citations
TL;DR: A comprehensive study of centrality distributions over geographic networks of urban streets indicates that a spatial analysis, that is grounded not on a single centrality assessment but on a set of different centrality indices, allows an extended comprehension of the city structure.
Abstract: Centrality has revealed crucial for understanding the structural properties of complex relational networks. Centrality is also relevant for various spatial factors affecting human life and behaviors in cities. Here, we present a comprehensive study of centrality distributions over geographic networks of urban streets. Five different measures of centrality, namely degree, closeness, betweenness, straightness and information, are compared over 18 1-square-mile samples of different world cities. Samples are represented by primal geographic graphs, i.e., valued graphs defined by metric rather than topologic distance where intersections are turned into nodes and streets into edges. The spatial behavior of centrality indices over the networks is investigated graphically by means of color-coded maps. The results indicate that a spatial analysis, that we term multiple centrality assessment, grounded not on a single but on a set of different centrality indices, allows an extended comprehension of the city structure, nicely capturing the skeleton of most central routes and subareas that so much impacts on spatial cognition and on collective dynamical behaviors. Statistically, closeness, straightness and betweenness turn out to follow similar functional distribution in all cases, despite the extreme diversity of the considered cities. Conversely, information is found to be exponential in planned cities and to follow a power-law scaling in self-organized cities. Hierarchical clustering analysis, based either on the Gini coefficients of the centrality distributions, or on the correlation between different centrality measures, is able to characterize classes of cities.
304 citations
TL;DR: Results on the sync basin for a ring of n >> 1 identical phase oscillators with sinusoidal coupling are reported, revealing that their basin sizes obey a tantalizingly simple statistical law: the probability that the final state has q twists follows a Gaussian distribution with respect to q.
Abstract: We suggest a new line of research that we hope will appeal to the nonlinear dynamics community, especially the readers of this Focus Issue. Consider a network of identical oscillators. Suppose the synchronous state is locally stable but not globally stable; it competes with other attractors for the available phase space. How likely is the system to synchronize, starting from a random initial condition? And how does the probability of synchronization depend on the way the network is connected? On the one hand, such questions are inherently difficult because they require calculation of a global geometric quantity, the size of the “sync basin” (or, more formally, the measure of the basin of attraction for the synchronous state). On the other hand, these questions are wide open, important in many real-world settings, and approachable by numerical experiments on various combinations of dynamical systems and network topologies. To give a case study in this direction, we report results on the sync basin for a ri...
274 citations
TL;DR: An analytic approach is proposed to judge the prediction performance of multivariate seizure prediction methods, taking into account that prediction methods are applied to multiple time series and several seizures.
Abstract: Nonlinear time series analysis techniques have been proposed to detect changes in the electroencephalography dynamics prior to epileptic seizures. Their applicability in practice to predict seizure onsets is hampered by the present lack of generally accepted standards to assess their performance. We propose an analytic approach to judge the prediction performance of multivariate seizure prediction methods. Statistical tests are introduced to assess patient individual results, taking into account that prediction methods are applied to multiple time series and several seizures. Their performance is illustrated utilizing a bivariate seizure prediction method based on synchronization theory.
191 citations
TL;DR: In this paper, the authors considered a one-dimensional chain of coupled linear and nonlinear oscillators with long-range power-wise interaction and showed how their synchronization can appear as a result of bifurcation, and how the corresponding solutions depend on α.
Abstract: We consider a one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction. The corresponding term in dynamical equations is proportional to 1∕∣n−m∣α+1. It is shown that the equation of motion in the infrared limit can be transformed into the medium equation with the Riesz fractional derivative of order α, when 0<α<2. We consider a few models of coupled oscillators and show how their synchronization can appear as a result of bifurcation, and how the corresponding solutions depend on α. The presence of a fractional derivative also leads to the occurrence of localized structures. Particular solutions for fractional time-dependent complex Ginzburg-Landau (or nonlinear Schrodinger) equation are derived. These solutions are interpreted as synchronized states and localized structures of the oscillatory medium.
166 citations
TL;DR: A secure algorithm for direct encryption and decryption of digital images with chaotic map lattices that converts, pixel by pixel, the image color to chaotic logistic maps one-way coupled by initial conditions.
Abstract: We propose a secure algorithm for direct encryption and decryption of digital images with chaotic map lattices. The basic idea is to convert, pixel by pixel, the image color to chaotic logistic maps one-way coupled by initial conditions. After small numbers of iterations and cycles, the image becomes indistinguishable due to inherent properties of chaotic systems. Since the maps are coupled, the image can be completely recovered by the decryption algorithm if map parameters, number of iterations, number of cycles, and the image size are exactly known.
162 citations
TL;DR: A new method is used, which constructs a coupling scheme with cooperative and competitive weight-couplings, which is used to stabilize arbitrarily selected cluster synchronization patterns with several clusters for connected chaotic networks.
Abstract: In this article, a new method, which constructs a coupling scheme with cooperative and competitive weight-couplings, is used to stabilize arbitrarily selected cluster synchronization patterns with several clusters for connected chaotic networks. By the coupling scheme, a sufficient condition about the global stability of the selected cluster synchronization patterns is derived. That is to say, when the sufficient condition is satisfied, arbitrarily selected cluster synchronization patterns in connected chaotic networks can be achieved via an appropriate coupled scheme. The effectiveness of the method is illustrated by an example.
143 citations
TL;DR: This study makes the first attempt to use the 23-order fractional Laplacian modeling of Kolmogorov -53 scaling of fully developed turbulence and enhanced diffusing movements of random turbulent particles to suggest that the fractional calculus is an effective approach to modeling the chaotic fractal phenomena induced by nonlinear interactions.
Abstract: This study makes the first attempt to use the 2∕3-order fractional Laplacian modeling of Kolmogorov −5∕3 scaling of fully developed turbulence and enhanced diffusing movements of random turbulent particles. Nonlinear inertial interactions and molecular Brownian diffusivity are considered to be the bifractal mechanism behind multifractal scaling of moderate Reynolds number turbulence. Accordingly, a stochastic equation is proposed to describe turbulence intermittency. The 2∕3-order fractional Laplacian representation is also used to model nonlinear interactions of fluctuating velocity components, and then we conjecture a fractional Reynolds equation, underlying fractal spacetime structures of Levy 2∕3 stable distribution and the Kolmogorov scaling at inertial scales. The new perspective of this study is that the fractional calculus is an effective approach to modeling the chaotic fractal phenomena induced by nonlinear interactions.
141 citations
TL;DR: A hierarchical organization of the synchronization behavior with respect to the collective dynamics of the network is found, where oscillators with more connections are synchronized more closely by the collective Dynamics and constitute the dynamical core of thenetwork.
Abstract: We study synchronization behavior in networks of coupled chaotic oscillators with heterogeneous connection degrees. Our focus is on regimes away from the complete synchronization state, when the coupling is not strong enough, when the oscillators are under the influence of noise or when the oscillators are nonidentical. We have found a hierarchical organization of the synchronization behavior with respect to the collective dynamics of the network. Oscillators with more connections (hubs) are synchronized more closely by the collective dynamics and constitute the dynamical core of the network. The numerical observation of this hierarchical synchronization is supported with an analysis based on a mean field approximation and the master stability function.
127 citations
TL;DR: Numerical simulations of the stochastic models show that robust circadian oscillations can already occur with a limited number of mRNA and protein molecules, in the range of a few tens and hundreds, respectively, whereas the proximity of a bifurcation point leads to less robust oscillations.
Abstract: Circadian rhythms, characterized by a period of about 24 h, are the most widespread biological rhythms generated autonomously at the molecular level. The core molecular mechanism responsible for circadian oscillations relies on the negative regulation exerted by a protein on the expression of its own gene. Deterministic models account for the occurrence of autonomous circadian oscillations, for their entrainment by light-dark cycles, and for their phase shifting by light pulses. Stochastic versions of these models take into consideration the molecular fluctuations that arise when the number of molecules involved in the regulatory mechanism is low. Numerical simulations of the stochastic models show that robust circadian oscillations can already occur with a limited number of mRNA and protein molecules, in the range of a few tens and hundreds, respectively. Various factors affect the robustness of circadian oscillations with respect to molecular noise. Besides an increase in the number of molecules, entrainment by light-dark cycles, and cooperativity in repression enhance robustness, whereas the proximity of a bifurcation point leads to less robust oscillations. Another parameter that appears to be crucial for the coherence of circadian rhythms is the binding/unbinding rate of the inhibitory protein to the promoter of the clock gene. Intercellular coupling further increases the robustness of circadian oscillations.
112 citations
TL;DR: Two methods that can be used to estimate the impact of internal and external variability on nonlinear systems are discussed and their utility is demonstrated by comparing two experimentally implemented oscillatory genetic networks with different designs.
Abstract: We discuss two methods that can be used to estimate the impact of internal and external variability on nonlinear systems, and demonstrate their utility by comparing two experimentally implemented oscillatory genetic networks with different designs. The methods allow for rapid estimations of intrinsic and extrinsic noise and should prove useful in the analysis of natural genetic networks and when constructing synthetic gene regulatory systems.
TL;DR: The present article reports the first experimental verification of the direct transition to quasiperiodicity through a border-collision bifurcation in the two-dimensional piecewise-linear normal form map.
Abstract: Numerical studies of higher-dimensional piecewise-smooth systems have recently shown how a torus can arise from a periodic cycle through a special type of border-collision bifurcation. The present article investigates this new route to quasiperiodicity in the two-dimensional piecewise-linear normal form map. We have obtained the chart of the dynamical modes for this map and showed that border-collision bifurcations can lead to the birth of a stable closed invariant curve associated with quasiperiodic or periodic dynamics. In the parameter regions leading to the existence of an invariant closed curve, there may be transitions between an ergodic torus and a resonance torus, but the mechanism of creation for the resonance tongues is distinctly different from that observed in smooth maps. The transition from a stable focus point to a resonance torus may lead directly to a new focus of higher periodicity, e.g., a period-5 focus. This article also contains a discussion of torus destruction via a homoclinic bifurcation in the piecewise-linear normal map. Using a dc–dc converter with two-level control as an example, we report the first experimental verification of the direct transition to quasiperiodicity through a border-collision bifurcation.
TL;DR: The emergence of collective synchronization in large directed networks of heterogeneous oscillators by generalizing the classical Kuramoto model of globally coupled phase oscillators to more realistic networks was studied in this paper.
Abstract: We study the emergence of collective synchronization in large directed networks of heterogeneous oscillators by generalizing the classical Kuramoto model of globally coupled phase oscillators to more realistic networks We extend recent theoretical approximations describing the transition to synchronization in large undirected networks of coupled phase oscillators to the case of directed networks We also consider the case of networks with mixed positive-negative coupling strengths We compare our theory with numerical simulations and find good agreement
TL;DR: The connection graph stability method is extended to directed graphs with node balance, the property that all nodes in the network have equal input and output weight sums, and the same upper bound for synchronization in asymmetrically connected networks is obtained, provided that the condition of node balance is satisfied.
Abstract: We study global stability of synchronization in asymmetrically connected networks of limit-cycle or chaotic oscillators. We extend the connection graph stability method to directed graphs with node balance, the property that all nodes in the network have equal input and output weight sums. We obtain the same upper bound for synchronization in asymmetrically connected networks as in the network with a symmetrized matrix, provided that the condition of node balance is satisfied. In terms of graphs, the symmetrization operation amounts to replacing each directed edge by an undirected edge of half the coupling strength. It should be stressed that without node balance this property in general does not hold.
TL;DR: The topology of several music recommendation networks, which arise from relationships between artist, co-occurrence of songs in play lists or experts' recommendation, is studied.
Abstract: We study the topology of several music recommendation networks, which arise from relationships between artist, co-occurrence of songs in play lists or experts’ recommendation. The analysis uncovers the emergence of complex network phenomena in these kinds of recommendation networks, built considering artists as nodes and their resemblance as links. We observe structural properties that provide some hints on navigation and possible optimizations on the design of music recommendation systems. Finally, the analysis derived from existing music knowledge sources provides a deeper understanding of the human music similarity perception.
TL;DR: Findings show that intrinsic noise is not attenuated by the addition of negative or positive feedback to transcriptional cascades, and synchrony in oscillations among a cell population can be improved by increasing network depth and tightening the regulation at one of the repression stages.
Abstract: This paper analyzes how the delay and repression strength of negative feedback in single-gene and multigene transcriptional networks influences intrinsic noise propagation and oscillatory behavior. We simulate a variety of transcriptional networks using a stochastic model and report two main findings. First, intrinsic noise is not attenuated by the addition of negative or positive feedback to transcriptional cascades. Second, for multigene negative feedback networks, synchrony in oscillations among a cell population can be improved by increasing network depth and tightening the regulation at one of the repression stages. Our long term goal is to understand how the noise characteristics of complex networks can be derived from the properties of modules that are used to compose these networks.
TL;DR: The results indicate that random, long-range links among distant modules is the key to synchronization, and a viable strategy to achieve synchronous behavior in social networks is suggested.
Abstract: Networks with a community (or modular) structure arise in social and biological sciences. In such a network individuals tend to form local communities, each having dense internal connections. The linkage among the communities is, however, much more sparse. The dynamics on modular networks, for instance synchronization, may be of great social or biological interest. (Here by synchronization we mean some synchronous behavior among the nodes in the network, not, for example, partially synchronous behavior in the network or the synchronizability of the network with some external dynamics.) By using a recent theoretical framework, the master-stability approach originally introduced by Pecora and Carroll in the context of synchronization in coupled nonlinear oscillators, we address synchronization in complex modular networks. We use a prototype model and develop scaling relations for the network synchronizability with respect to variations of some key network structural parameters. Our results indicate that random, long-range links among distant modules is the key to synchronization. As an application we suggest a viable strategy to achieve synchronous behavior in social networks.
TL;DR: Some basic dynamical behaviors of this general multiscroll Lorenz system family are investigated, including bifurcations, maximum Lyapunov exponents, and parameters regions, which are physically verified by using digital signal processors.
Abstract: This paper proposes a general multiscroll Lorenz system family by introducing a novel parameterized nth-order polynomial transformation. Some basic dynamical behaviors of this general multiscroll Lorenz system family are then investigated, including bifurcations, maximum Lyapunov exponents, and parameters regions. Furthermore, the general multiscroll Lorenz attractors are physically verified by using digital signal processors.
TL;DR: A surprising finding is that, in the common case where multiple destinations exist for scattering trajectories, Wada basin boundaries are common and they appear to be structurally stable under weak dissipation, even when other characteristics of the nonhyperbolic scattering dynamics are not.
Abstract: Chaotic scattering in open Hamiltonian systems under weak dissipation is not only of fundamental interest but also important for problems of current concern such as the advection and transport of inertial particles in fluid flows. Previous work using discrete maps demonstrated that nonhyperbolic chaotic scattering is structurally unstable in the sense that the algebraic decay of scattering particles immediately becomes exponential in the presence of weak dissipation. Here we extend the result to continuous-time Hamiltonian systems by using the Henon-Heiles system as a prototype model. More importantly, we go beyond to investigate the basin structure of scattering dynamics. A surprising finding is that, in the common case where multiple destinations exist for scattering trajectories, Wada basin boundaries are common and they appear to be structurally stable under weak dissipation, even when other characteristics of the nonhyperbolic scattering dynamics are not. We provide numerical evidence and a geometric theory for the structural stability of the complex basin topology.
TL;DR: By employing the Lyapunov method and some inequality techniques, the global point dissipativity of neural networks with mixed time-varying delays is studied and the proposed linear matrix inequality approach is computationally efficient.
Abstract: By employing the Lyapunov method and some inequality techniques, the global point dissipativity is studied for neural networks with both discrete time-varying delays and distributed time-varying delays. Simple sufficient conditions are given for checking the global point dissipativity of neural networks with mixed time-varying delays. The proposed linear matrix inequality approach is computationally efficient as it can be solved numerically using standard commercial software. Illustrated examples are given to show the usefulness of the results in comparison with some existing results.
TL;DR: Patterns of chaotic transitions between synchronization and propagation of bursts are found and it is shown that they consist of transient standing and rotating waves induced by symmetry-breaking bifurcations, and can be viewed as a manifestation of the phenomenon of chaotic itinerancy.
Abstract: The present paper studies regular and complex spatiotemporal behaviors in networks of coupled map-based bursting oscillators. In-phase and antiphase synchronization of bursts are studied, explaining their underlying mechanisms in order to determine how network parameters separate them. Conditions for emergent bursting in the coupled system are derived from our analysis. In the region of emergence, patterns of chaotic transitions between synchronization and propagation of bursts are found. We show that they consist of transient standing and rotating waves induced by symmetry-breaking bifurcations, and can be viewed as a manifestation of the phenomenon of chaotic itinerancy.
TL;DR: It is found that the star coupled and scale-free networks with nonuniform inner couplings can be in the state of synchronization if only a fraction of nodes are controlled.
Abstract: We consider the problem of synchronization in uncertain generic complex networks For generic complex networks with unknown dynamics of nodes and unknown coupling functions including uniform and nonuniform inner couplings, some simple linear feedback controllers with updated strengths are designed using the well-known LaSalle invariance principle The state of an uncertain generic complex network can synchronize an arbitrary assigned state of an isolated node of the network The famous Lorenz system is stimulated as the nodes of the complex networks with different topologies We found that the star coupled and scale-free networks with nonuniform inner couplings can be in the state of synchronization if only a fraction of nodes are controlled
TL;DR: "pseudochaos" is called the case when the randomness exists with zero Lyapunov exponent, i.e., the dispersion of initially close trajectories is subexponential.
Abstract: We consider a nonlinear oscillator of the Duffing type with fractional derivative of the order 1
TL;DR: Studies like these indicate that network theory can be of substantial use in uncovering collaborative structure in political bodies such as the U.S. Congress, without requiring input in the form of political opinions or judgments by the researcher.
Abstract: Studies like these indicate that network theory can be of substantial use in uncovering collaborative structure in political bodies such as the U.S. Congress, without requiring input in the form of political opinions or judgments by the researcher.
TL;DR: It is analytically proved that chaos synchronization could be achieved with probability one merely via white-noise-based coupling and supports the observation of an interesting phenomenon that a certain kind of white noise could enhance chaos synchronization between two chaotic oscillators.
Abstract: In the paper, complete synchronization of two chaotic oscillators via unidirectional coupling determined by white noise distribution is investigated. It is analytically proved that chaos synchronization could be achieved with probability one merely via white-noise-based coupling. The established theoretical result supports the observation of an interesting phenomenon that a certain kind of white noise could enhance chaos synchronization between two chaotic oscillators. Furthermore, numerical examples are provided to illustrate some possible applications of the theoretical result.
TL;DR: It is shown that the system performance in the presence of noise can be improved by including long-range connections between the robots, and a suitable strategy based on this model to control exploration and transport is introduced.
Abstract: In this paper the dynamical network model of the interactions within a group of mobile robots is investigated and proposed as a possible strategy for controlling the robots without central coordination. Motivated by the results of the analysis of our simple model, we show that the system performance in the presence of noise can be improved by including long-range connections between the robots. Finally, a suitable strategy based on this model to control exploration and transport is introduced.
TL;DR: In this article, the synchronization properties of coupled dynamics on time-varying networks and the corresponding time-average networks were investigated and it was shown that for noncommuting Laplacians, the stability of the synchronized state for both the time changing and the time average topologies is approximately the same.
Abstract: We consider synchronization properties of coupled dynamics on time-varying networks and the corresponding time-average network. We find that if the different Laplacians corresponding to the time-varying networks commute with each other then the stability of the synchronized state for both the time-varying and the time-average topologies are approximately the same. On the other hand for noncommuting Laplacians the stability of the synchronized state for the time-varying topology is in general better than the time-average topology.
TL;DR: With this new and effective method, parameters identification and Q-S (lag, anticipated, and complete) time-varying synchronization can be achieved simultaneously.
Abstract: In this paper, a new type of generalized Q-S (lag, anticipated, and complete) time-varying synchronization is defined. Adaptive Q-S (lag, anticipated, and complete) time-varying synchronization and parameters identification of uncertain delayed neural networks have been considered, where the delays are multiple time-varying delays. A novel control method is given by using the Lyapunov functional method. With this new and effective method, parameters identification and Q-S (lag, anticipated, and complete) time-varying synchronization can be achieved simultaneously. Simulation results are given to justify the theoretical analysis in this paper.
TL;DR: It is demonstrated that in a system where the Turing band is slightly below criticality, spatially uniform internal or external oscillations can create oscillating square patterns and the square superlattices are the first examples of time-independent square Turing patterns.
Abstract: The best known Turing patterns are composed of stripes or simple hexagonal arrangements of spots. Until recently, Turing patterns with other geometries have been observed only rarely. Here we present experimental studies and mathematical modeling of the formation and stability of hexagonal and square Turing superlattice patterns in a photosensitive reaction-diffusion system. The superlattices develop from initial conditions created by illuminating the system through a mask consisting of a simple hexagonal or square lattice with a wavelength close to a multiple of the intrinsic Turing pattern’s wavelength. We show that interaction of the photochemical periodic forcing with the Turing instability generates multiple spatial harmonics of the forcing patterns. The harmonics situated within the Turing instability band survive after the illumination is switched off and form superlattices. The square superlattices are the first examples of time-independent square Turing patterns. We also demonstrate that in a system where the Turing band is slightly below criticality, spatially uniform internal or external oscillations can create oscillating square patterns.
TL;DR: The theoretical analysis shows that all scrolls are located around the equilibria corresponding to the saturated plateaus of the saturated function series on a line in the 1-D case, a plane in the 2-D cases, and a three-dimensional space in the 3-Dcase, respectively.
Abstract: This paper proposes a saturated function series approach for generating multiscroll chaotic attractors from the fractional differential systems, including one-directional (1-D) n-scroll, two-directional (2-D) n×m-grid scroll, and three-directional (3-D) n×m×l-grid scroll chaotic attractors. Our theoretical analysis shows that all scrolls are located around the equilibria corresponding to the saturated plateaus of the saturated function series on a line in the 1-D case, a plane in the 2-D case, and a three-dimensional space in the 3-D case, respectively. In particular, each saturated plateau corresponds to a unique equilibrium and its unique scroll of the whole attractor. In addition, the number of scrolls is equal to the number of saturated plateaus in the saturated function series. Finally, some underlying dynamical mechanisms are then further investigated for the fractional differential multiscroll systems.