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Dec 2011

Volume 21, Issue 4, Articles (04xxxx)

Issue Cover Spotlight Figure

Chaos 21, 041106 (2011); http://dx.doi.org/10.1063/1.3668192 (1 page)

Joshua E. S. Socolar

Cover image from Joshua E. S. Socolar, Chaos 21, 041106 (2011). A single unit that can tile the plane, but not in any periodic pattern.

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Editorial: Honoring Janis Bennett

David K. Campbell, Editor-in-Chief, for the Editors

Chaos 21, 040401 (2011); http://dx.doi.org/10.1063/1.3670710 (1 page)

Online Publication Date: 12 December 2011

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05.45.-a Nonlinear dynamics and chaos
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Introduction: Eighth Annual Gallery of Nonlinear Images (Dallas, Texas, 2011)

Karin Dahmen, Thomas Halsey, Wolfgang Losert, and Jon Machta

Chaos 21, 041101 (2011); http://dx.doi.org/10.1063/1.3671937 (1 page)

Online Publication Date: 20 December 2011

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01.10.Cr Announcements, news, and awards
05.45.-a Nonlinear dynamics and chaos
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Four-dimensional structural dynamics of sheared collagen networks

Richard C. Arevalo, Jeffrey S. Urbach, and Daniel L. Blair

Chaos 21, 041102 (2011); http://dx.doi.org/10.1063/1.3666225 (1 page)

Online Publication Date: 20 December 2011

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61.25.H- Macromolecular and polymers solutions; polymer melts
82.70.Gg Gels and sols
87.15.B- Structure of biomolecules
62.10.+s Mechanical properties of liquids
83.80.Kn Physical gels and microgels
78.55.Bq Liquids
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Tracking the Brownian diffusion of a colloidal tetrahedral cluster

Kazem V. Edmond, HyunJoo Park, Mark T. Elsesser, Gary L. Hunter, David J. Pine, and Eric R. Weeks

Chaos 21, 041103 (2011); http://dx.doi.org/10.1063/1.3665984 (1 page)

Online Publication Date: 20 December 2011

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05.40.Jc Brownian motion
05.60.-k Transport processes
82.70.Dd Colloids
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Mathematical genealogy and department prestige

Sean A. Myers, Peter J. Mucha, and Mason A. Porter

Chaos 21, 041104 (2011); http://dx.doi.org/10.1063/1.3668043 (1 page)

Online Publication Date: 20 December 2011

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02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)
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Linear shear in a model granular system

Jie Ren, Joshua A. Dijksman, and Robert P. Behringer

Chaos 21, 041105 (2011); http://dx.doi.org/10.1063/1.3664407 (1 page)

Online Publication Date: 20 December 2011

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81.40.Lm Deformation, plasticity, and creep
62.20.F- Deformation and plasticity
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Local action, global impact: Forcing nonperiodicity with a single structural unit

Joshua E. S. Socolar

Chaos 21, 041106 (2011); http://dx.doi.org/10.1063/1.3668192 (1 page)

Online Publication Date: 20 December 2011

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05.45.-a Nonlinear dynamics and chaos
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How granular materials jam in a hopper

J. Tang and R. P. Behringer

Chaos 21, 041107 (2011); http://dx.doi.org/10.1063/1.3669495 (1 page)

Online Publication Date: 20 December 2011

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47.57.Gc Granular flow
47.80.Jk Flow visualization and imaging
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Vibrational resonance in excitable neuronal systems

Haitao Yu, Jiang Wang, Chen Liu, Bin Deng, and Xile Wei

Chaos 21, 043101 (2011); http://dx.doi.org/10.1063/1.3644390 (10 pages)

Online Publication Date: 3 October 2011

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In this paper, we investigate the effect of a high-frequency driving on the dynamical response of excitable neuronal systems to a subthreshold low-frequency signal by numerical simulation. We demonstrate the occurrence of vibrational resonance in spatially extended neuronal networks. Different network topologies from single small-world networks to modular networks of small-world subnetworks are considered. It is shown that an optimal amplitude of high-frequency driving enhances the response of neuron populations to a low-frequency signal. This effect of vibrational resonance of neuronal systems depends extensively on the network structure and parameters, such as the coupling strength between neurons, network size, and rewiring probability of single small-world networks, as well as the number of links between different subnetworks and the number of subnetworks in the modular networks. All these parameters play a key role in determining the ability of the network to enhance the outreach of the localized subthreshold low-frequency signal. Considering that two-frequency signals are ubiquity in brain dynamics, we expect the presented results could have important implications for the weak signal detection and information propagation across neuronal systems.
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87.18.Sn Neural networks and synaptic communication
89.75.Hc Networks and genealogical trees

Dual-lag synchronization between coupled chaotic lasers due to path-delay interference

J. Tiana-Alsina, J. H. Garcia-Lopez, M. C. Torrent, and J. Garcia-Ojalvo

Chaos 21, 043102 (2011); http://dx.doi.org/10.1063/1.3644392 (7 pages)

Online Publication Date: 3 October 2011

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We study experimentally the synchronization dynamics of two semiconductor lasers coupled unidirectionally via two different delayed paths. The emitter laser operates in a chaotic regime characterized by low-frequency fluctuations due to optical feedback and induces a synchronized dynamical activity in the receiver laser, which operates in the continuous-wave regime when uncoupled. Different delays in the two coupling paths lead to the coexistence of two time lags in the synchronized dynamics of the oscillators. This dual-lag synchronization degrades the average synchronization quality of the system of coupled lasers and hinders the transmission of information between them. Numerical simulation results agree with the experimental observations, and allow us to explore this phenomenon in a wide parameter range, and quantify the degree of signal transmission degradation caused by this chaotic path-delay interference.
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05.45.Xt Synchronization; coupled oscillators

Center manifold reduction for large populations of globally coupled phase oscillators

Hayato Chiba and Isao Nishikawa

Chaos 21, 043103 (2011); http://dx.doi.org/10.1063/1.3647317 (10 pages) | Cited 1 time

Online Publication Date: 7 October 2011

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A bifurcation theory for a system of globally coupled phase oscillators is developed based on the theory of rigged Hilbert spaces. It is shown that there exists a finite-dimensional center manifold on a space of generalized functions. The dynamics on the manifold is derived for any coupling functions. When the coupling function is sin θ, a bifurcation diagram conjectured by Kuramoto is rigorously obtained. When it is not sin θ, a new type of bifurcation phenomenon is found due to the discontinuity of the projection operator to the center subspace.
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05.45.-a Nonlinear dynamics and chaos
02.30.Oz Bifurcation theory

Lyapunov exponents for multi-parameter tent and logistic maps

Mark McCartney

Chaos 21, 043104 (2011); http://dx.doi.org/10.1063/1.3645185 (7 pages)

Online Publication Date: 11 October 2011

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The behaviour of logistic and tent maps is studied in cases where the control parameter is dependent on iteration number. Analytic results for global Lyapunov exponent are presented in the case of the tent map and numerical results are presented in the case of the logistic map. In the case of a tent map with N control parameters, the fraction of parameter space for which the global Lyapunov exponent is positive is calculated. The case of bi-parameter maps of period N are investigated.
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05.45.-a Nonlinear dynamics and chaos
02.60.-x Numerical approximation and analysis

Plykin type attractor in electronic device simulated in MULTISIM

Sergey P. Kuznetsov

Chaos 21, 043105 (2011); http://dx.doi.org/10.1063/1.3646903 (8 pages)

Online Publication Date: 13 October 2011

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An electronic device is suggested representing a non-autonomous dynamical system with hyperbolic chaotic attractor of Plykin type in the stroboscopic map, and the results of its simulation with software package NI MULTISIM are considered in comparison with numerical integration of the underlying differential equations. A main practical advantage of electronic devices of this kind is their structural stability that means insensitivity of the chaotic dynamics in respect to variations of functions and parameters of elements constituting the system as well as to interferences and noises.
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05.45.Gg Control of chaos, applications of chaos
05.45.Pq Numerical simulations of chaotic systems

Understanding the complexity of the Lévy-walk nature of human mobility with a multi-scale cost/benefit model

Nicola Scafetta

Chaos 21, 043106 (2011); http://dx.doi.org/10.1063/1.3645184 (10 pages)

Online Publication Date: 14 October 2011

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Probability distributions of human displacements have been fit with exponentially truncated Lévy flights or fat tailed Pareto inverse power law probability distributions. Thus, people usually stay within a given location (for example, the city of residence), but with a non-vanishing frequency they visit nearby or far locations too. Herein, we show that an important empirical distribution of human displacements (range: from 1 to 1000 km) can be well fit by three consecutive Pareto distributions with simple integer exponents equal to 1, 2, and (math) 3. These three exponents correspond to three displacement range zones of about 1 km  ≲Δr ≲10 km, 10 km  ≲Δr ≲300 km, and 300 km  ≲Δr ≲1000 km, respectively. These three zones can be geographically and physically well determined as displacements within a city, visits to nearby cities that may occur within just one-day trips, and visit to far locations that may require multi-days trips. The incremental integer values of the three exponents can be easily explained with a three-scale mobility cost/benefit model for human displacements based on simple geometrical constrains. Essentially, people would divide the space into three major regions (close, medium, and far distances) and would assume that the travel benefits are randomly/uniformly distributed mostly only within specific urban-like areas. The three displacement distribution zones appear to be characterized by an integer (1, 2, or math 3) inverse power exponent because of the specific number (1, 2, or math 3) of cost mechanisms (each of which is proportional to the displacement length). The distributions in the first two zones would be associated to Pareto distributions with exponent β = 1 and β = 2 because of simple geometrical statistical considerations due to the a priori assumption that most benefits are searched in the urban area of the city of residence or in the urban area of specific nearby cities. We also show, by using independent records of human mobility, that the proposed model predicts the statistical properties of human mobility below 1 km ranges, where people just walk. In the latter case, the threshold between zone 1 and zone 2 may be around 100–200 m and, perhaps, may have been evolutionary determined by the natural human high resolution visual range, which characterizes an area of interest where the benefits are assumed to be randomly and uniformly distributed. This rich and suggestive interpretation of human mobility may characterize other complex random walk phenomena that may also be described by a N-piece fit Pareto distributions with increasing integer exponents. This study also suggests that distribution functions used to fit experimental probability distributions must be carefully chosen for not improperly obscuring the physics underlying a phenomenon.
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87.19.ru Locomotion
87.10.Ca Analytical theories

Linear matrix inequality criteria for robust synchronization of uncertain fractional-order chaotic systems

Liping Chen, Yi Chai, and Ranchao Wu

Chaos 21, 043107 (2011); http://dx.doi.org/10.1063/1.3650237 (12 pages)

Online Publication Date: 14 October 2011

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This paper is devoted to synchronization of uncertain fractional-order chaotic systems with fractional-order α: 0 < α < 1 and 1 ≤ α < 2, respectively. On the basis of the stability theory of fractional-order differential system and the observer-based robust control, two sufficient and necessary conditions for synchronizing uncertain fractional-order chaotic systems with parameter perturbations are presented in terms of linear matrix inequality, which is an efficient method and could be easily solved by the toolbox of MATLAB. Finally, fractional-order uncertain chaotic Lü system with fractional-order α = 0.95 and fractional-order uncertain chaotic Lorenz system with fractional-order α = 1.05 are taken as numerical examples to show the validity and feasibility of the proposed method.
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05.45.Xt Synchronization; coupled oscillators
05.45.Pq Numerical simulations of chaotic systems
02.60.Lj Ordinary and partial differential equations; boundary value problems
02.60.Dc Numerical linear algebra

Combinatorial games with a pass: A dynamical systems approach

Rebecca E. Morrison, Eric J. Friedman, and Adam S. Landsberg

Chaos 21, 043108 (2011); http://dx.doi.org/10.1063/1.3650234 (14 pages)

Online Publication Date: 17 October 2011

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By treating combinatorial games as dynamical systems, we are able to address a longstanding open question in combinatorial game theory, namely, how the introduction of a “pass” move into a game affects its behavior. We consider two well known combinatorial games, 3-pile Nim and 3-row Chomp. In the case of Nim, we observe that the introduction of the pass dramatically alters the game’s underlying structure, rendering it considerably more complex, while for Chomp, the pass move is found to have relatively minimal impact. We show how these results can be understood by recasting these games as dynamical systems describable by dynamical recursion relations. From these recursion relations, we are able to identify underlying structural connections between these “games with passes” and a recently introduced class of “generic (perturbed) games.” This connection, together with a (non-rigorous) numerical stability analysis, allows one to understand and predict the effect of a pass on a game.
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05.45.-a Nonlinear dynamics and chaos
02.60.-x Numerical approximation and analysis
02.50.Le Decision theory and game theory
02.10.Ox Combinatorics; graph theory

Multiple coherence resonance induced by time-periodic coupling in stochastic Hodgkin–Huxley neuronal networks

Xiu Lin, Yubing Gong, and Li Wang

Chaos 21, 043109 (2011); http://dx.doi.org/10.1063/1.3652847 (6 pages)

Online Publication Date: 17 October 2011

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In this paper, we study the effect of time-periodic coupling strength (TPCS) on the spiking coherence of Newman–Watts small-world networks of stochastic Hodgkin–Huxley (HH) neurons and investigate the relations between the coupling strength and channel noise when coherence resonance (CR) occurs. It is found that, when the amplitude of TPCS is varied, the spiking induced by channel noise can exhibit CR and coherence bi-resonance (CBR), and the CR moves to a smaller patch area (bigger channel noise) when the amplitude increases; when the frequency of TPCS is varied, the intrinsic spiking can exhibit CBR and multiple CR, and the CR always occurs when the frequency is equal to or multiple of the spiking period, manifesting as the locking between the frequencies of the intrinsic spiking and the coupling strength. These results show that TPCS can greatly enhance and optimize the intrinsic spiking coherence, and favors the spiking with bigger channel noise to exhibit CR. This implies that, compared to constant coupling strength, TPCS may play a more efficient role for improving the time precision of the information processing in stochastic neuronal networks.
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87.18.Sn Neural networks and synaptic communication
87.18.Tt Noise in biological systems
07.05.Mh Neural networks, fuzzy logic, artificial intelligence
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
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Noise reduction by recycling dynamically coupled time series

M. Eugenia Mera and Manuel Morán

Chaos 21, 043110 (2011); http://dx.doi.org/10.1063/1.3645969 (14 pages)

Online Publication Date: 17 October 2011

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We say that several scalar time series are dynamically coupled if they record the values of measurements of the state variables of the same smooth dynamical system. We show that much of the information lost due to measurement noise in a target time series can be recovered with a noise reduction algorithm by crossing the time series with another time series with which it is dynamically coupled. The method is particularly useful for reduction of measurement noise in short length time series with high uncertainties.
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05.45.Tp Time series analysis
02.30.Lt Sequences, series, and summability

Adaptive tuning of feedback gain in time-delayed feedback control

J. Lehnert, P. Hövel, V. Flunkert, P. Yu. Guzenko, A. L. Fradkov, and E. Schöll

Chaos 21, 043111 (2011); http://dx.doi.org/10.1063/1.3647320 (6 pages)

Online Publication Date: 18 October 2011

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We demonstrate that time-delayed feedback control can be improved by adaptively tuning the feedback gain. This adaptive controller is applied to the stabilization of an unstable fixed point and an unstable periodic orbit embedded in a chaotic attractor. The adaptation algorithm is constructed using the speed-gradient method of control theory. Our computer simulations show that the adaptation algorithm can find an appropriate value of the feedback gain for single and multiple delays. Furthermore, we show that our method is robust to noise and different initial conditions.
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89.20.Kk Engineering
02.30.Yy Control theory
02.60.-x Numerical approximation and analysis

Fractal descriptors in the Fourier domain applied to color texture analysis

João Batista Florindo and Odemir Martinez Bruno

Chaos 21, 043112 (2011); http://dx.doi.org/10.1063/1.3650233 (10 pages)

Online Publication Date: 18 October 2011

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The present work proposes the development of a novel method to provide descriptors for colored texture images. The method consists of two steps. First, we apply a linear transform in the color space of the image aiming at highlighting spatial structuring relations among the color of pixels. Second, we apply a multiscale approach to the calculus of fractal dimension based on Fourier transform. From this multiscale operation, we extract the descriptors that are used to discriminate the texture represented in digital images. The accuracy of the method is verified in the classification of two color texture datasets, by comparing the performance of the proposed technique to other classical and state-of-the-art methods for color texture analysis. The results showed an advantage of almost 3% of the proposed technique over the second best approach.
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42.30.Sy Pattern recognition
42.30.Kq Fourier optics
05.45.Df Fractals

Melnikov’s criteria, parametric control of chaos, and stationary chaos occurrence in systems with asymmetric potential subjected to multiscale type excitation

C. A. Kitio Kwuimy, C. Nataraj, and G. Litak

Chaos 21, 043113 (2011); http://dx.doi.org/10.1063/1.3650699 (12 pages)

Online Publication Date: 18 October 2011

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We consider the problems of chaos and parametric control in nonlinear systems under an asymmetric potential subjected to a multiscale type excitation. The lower bound line for horseshoes chaos is analyzed using the Melnikov’s criterion for a transition to permanent or transient nonperiodic motions, complement by the fractal or regular shape of the basin of attraction. Numerical simulations based on the basins of attraction, bifurcation diagrams, Poincaré sections, Lyapunov exponents, and phase portraits are used to show how stationary dissipative chaos occurs in the system. Our attention is focussed on the effects of the asymmetric potential term and the driven frequency. It is shown that the threshold amplitude |γc| of the excitation decreases for small values of the driven frequency ω and increases for large values of ω. This threshold value decreases with the asymmetric parameter α and becomes constant for sufficiently large values of α. γc has its maximum value for asymmetric load in comparison with the symmetric load. Finally, we apply the Melnikov theorem to the controlled system to explore the gain control parameter dependencies.
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05.45.Df Fractals
05.45.Pq Numerical simulations of chaotic systems
02.30.-f Function theory, analysis

Combination synchronization of three classic chaotic systems using active backstepping design

Luo Runzi, Wang Yinglan, and Deng Shucheng

Chaos 21, 043114 (2011); http://dx.doi.org/10.1063/1.3655366 (6 pages) | Cited 1 time

Online Publication Date: 21 October 2011

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In this paper, an active backstepping design is proposed to achieve combination synchronization between three different chaotic systems: Lorenz system, Chen’s system, and Lü system. The proposed method is a systematic design approach and consists in a recursive procedure that interlaces the choice of a Lyapunov function with the design of active control. Numerical simulations are shown to verify the feasibility and effectiveness of the proposed control technique.
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05.45.Xt Synchronization; coupled oscillators
02.60.Cb Numerical simulation; solution of equations

Enhancement and weakening of stochastic resonance for a coupled system

Jing-hui Li

Chaos 21, 043115 (2011); http://dx.doi.org/10.1063/1.3657918 (5 pages)

Online Publication Date: 2 November 2011

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In the paper, we investigate the phenomenon of stochastic resonance of a system with finite locally coupled linear elements driven by multiplicative dichotomous noise and temporal periodic signal. It is shown that, for some suitably selected values of the parameters, with increasing the size of the system or the coupling among the nearest elements, the stochastic resonance phenomenon can be enhanced; while for some other suitably selected parameters’ values, with the increase of the size or the coupling, the phenomenon of stochastic resonance can be weakened. Our results can provide some useful insights for the investigation of the stochastic resonance phenomenon of the systems with locally (or globally) coupled finite (or infinite) elements.
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05.40.Ca Noise
02.50.Ey Stochastic processes
02.50.Fz Stochastic analysis

Finding communities in weighted networks through synchronization

Xuyang Lou and Johan A. K. Suykens

Chaos 21, 043116 (2011); http://dx.doi.org/10.1063/1.3655371 (9 pages)

Online Publication Date: 7 November 2011

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Community detection in weighted networks is an important challenge. In this paper, we introduce a local weight ratio scheme for identifying the community structures of weighted networks within the context of the Kuramoto model by taking into account weights of links. The proposed scheme takes full advantage of the information of the link density among vertices and the closeness of relations between each vertex and its neighbors. By means of this scheme, we explore the connection between community structures and dynamic time scales of synchronization. Moreover, we can also unravel the hierarchical structures of weighted networks with a well-defined connectivity pattern by the synchronization process. The performance of the proposed method is evaluated on both computer-generated benchmark graphs and real-world networks.
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05.45.Xt Synchronization; coupled oscillators
02.10.Ox Combinatorics; graph theory

On stochastic stabilization of the Kelvin-Helmholtz instability by three-wave resonant interaction

S. V. Kostrykin, N. N. Romanova, and I. G. Yakushkin

Chaos 21, 043117 (2011); http://dx.doi.org/10.1063/1.3656800 (11 pages)

Online Publication Date: 7 November 2011

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An analytical investigation of the effect of three-wave resonant interactions with the linearly unstable wave is proposed. We consider the waves in the Kelvin-Helmholtz model, consisting of two fluid layers with different densities and velocities. We suppose that the velocity shear is weakly supercritical, the instability is of the algebraic type, i.e., the amplitude of the unstable wave grows linearly, and the instability occurs within the framework of a single mode. The amplitudes of two other waves taking part in the nonlinear interaction are assumed to be stable. The initial amplitudes of these waves are supposed to be small in comparison with the initial amplitude of the unstable wave. We present an analysis of the system of amplitude equations derived for this case using JWKB-method. As a result, we obtain equations that couple solutions pre- and post-passing the singular point, i.e., the point where the amplitude of the unstable wave has a local minimum. These equations give us the transformation rule of a parameter that characterizes the phase shift between fast and slow waves and defines the behavior of the system. This parameter is constant between two singular points and varies by chance at a singular point. As long as it stays positive, the amplitude of the wave remains limited and performs stochastic oscillations. If this parameter passes over zero, then we leave the region of stabilization and turn out in the region, where the amplitude grows infinitely. Accordingly, the transition to the region of instability happens stochastically. However, if the time interval, when the amplitude remains bounded, is large enough, the proposed scenario can be treated as a partial stabilization of instability.
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47.20.Ft Instability of shear flows (e.g., Kelvin-Helmholtz)
47.35.-i Hydrodynamic waves
02.50.Ey Stochastic processes
47.11.-j Computational methods in fluid dynamics
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