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Mar 2013

Volume 23, Issue 1, Articles (01xxxx)

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Chaos 23, 013101 (2013); http://dx.doi.org/10.1063/1.4772970 (6 pages)

Hien Dao, John C. Rodgers, and Thomas E. Murphy
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Referee acknowledgment for 2012

D. Campbell, Editor-in-Chief

Chaos 23, 010201 (2013); http://dx.doi.org/10.1063/1.4795747 (3 pages)

Online Publication Date: 18 March 2013

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Abstract Unavailable
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99.10.Np Editorial note
05.45.-a Nonlinear dynamics and chaos
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Chaotic dynamics of a frequency-modulated microwave oscillator with time-delayed feedback

Hien Dao, John C. Rodgers, and Thomas E. Murphy

Chaos 23, 013101 (2013); http://dx.doi.org/10.1063/1.4772970 (6 pages)

Online Publication Date: 4 January 2013

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We present a chaotic frequency-modulated microwave source that is governed by a simple, first-order nonlinear delay differential equation. When a sinusoidal nonlinearity is incorporated, the dynamical behaviors range from fixed-point to periodic to chaotic, depending on the feedback strength. When the sinusoidal nonlinearity is replaced by a binary nonlinearity, the system exhibits a complex periodic attractor with no fixed-point solution.
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05.45.-a Nonlinear dynamics and chaos
02.30.-f Function theory, analysis

Controlling phase multistability in coupled period-doubling oscillators

A. V. Shabunin

Chaos 23, 013102 (2013); http://dx.doi.org/10.1063/1.4772972 (10 pages)

Online Publication Date: 4 January 2013

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A simple method of switching between coexisting attractors in two coupled period-doubling oscillators is proposed. It is based on “pulling” phases of oscillations into suitable value by means of two periodic forces which simultaneously influence the both sub-systems. The frequency and the phase-shift of the forces are key parameters of the control. Their choice determines the resulted regime. The method is tested on example of coupled Chua's oscillators and exhibits its efficiency both for periodic and for chaotic attractors.
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05.45.Xt Synchronization; coupled oscillators

Influence of chaotic synchronization on mixing in the phase space of interacting systems

Sergey V. Astakhov, Anton Dvorak, and Vadim S. Anishchenko

Chaos 23, 013103 (2013); http://dx.doi.org/10.1063/1.4773824 (6 pages)

Online Publication Date: 4 January 2013

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Using the concept of the relative metric entropy, we study the influence of the synchronization phenomenon on mixing rate in the phase space of deterministic and noisy chaotic systems. We show that transition to both complete and phase synchronization of chaos is accompanied by the decrease of the level of mixing induced by internal nonlinear mechanisms of interacting systems as well as by external noise influence. Therefore, the decrease of the mixing rate in the phase space of interacting systems may indicate transition to synchronization. The obtained results are important for time series analysis in various types of real noisy systems (e.g., biological, social, and financial systems).
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05.45.Xt Synchronization; coupled oscillators
05.70.Ce Thermodynamic functions and equations of state
05.45.Tp Time series analysis
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Spectral coarse graining for random walks in bipartite networks

Yang Wang, An Zeng, Zengru Di, and Ying Fan

Chaos 23, 013104 (2013); http://dx.doi.org/10.1063/1.4773823 (7 pages)

Online Publication Date: 7 January 2013

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Many real-world networks display a natural bipartite structure, yet analyzing and visualizing large bipartite networks is one of the open challenges in complex network research. A practical approach to this problem would be to reduce the complexity of the bipartite system while at the same time preserve its functionality. However, we find that existing coarse graining methods for monopartite networks usually fail for bipartite networks. In this paper, we use spectral analysis to design a coarse graining scheme specific for bipartite networks, which keeps their random walk properties unchanged. Numerical analysis on both artificial and real-world networks indicates that our coarse graining can better preserve most of the relevant spectral properties of the network. We validate our coarse graining method by directly comparing the mean first passage time of the walker in the original network and the reduced one.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
05.90.+m Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems (restricted to new topics in section 05)
02.50.Cw Probability theory
02.60.Cb Numerical simulation; solution of equations

On the existence and multiplicity of one-dimensional solid particle attractors in time-dependent Rayleigh-Bénard convection

Marcello Lappa

Chaos 23, 013105 (2013); http://dx.doi.org/10.1063/1.4773001 (9 pages) | Cited 1 time

Online Publication Date: 9 January 2013

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For the first time evidence is provided that one-dimensional objects formed by the accumulation of tracer particles can emerge in flows of thermogravitational nature (in the region of the space of parameters, in which the so-called OS (oscillatory solution) flow of the Busse balloon represents the dominant secondary mode of convection). Such structures appear as seemingly rigid filaments, rotating without changing their shape. The most interesting (heretofore unseen) feature of such a class of physical attractors is their variety. Indeed, distinct shapes are found for a fixed value of the Rayleigh number depending on parameters accounting for particle inertia and viscous drag. The fascinating “sea” of existing potential paths, their multiplicity and tortuosity are explained according to the granularity of the loci in the physical space where conditions for phase locking between the traveling thermofluid-dynamic disturbance and the “turnover time” of particles in the basic toroidal flow are satisfied. It is shown, in particular, how the observed wealth of geometric objects and related topological features can be linked to a general overarching attractor representing an intrinsic (particle-independent) property of the base velocity field.
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47.52.+j Chaos in fluid dynamics
47.55.Kf Particle-laden flows
47.55.P- Buoyancy-driven flows; convection
47.57.eb Diffusion and aggregation
47.20.Bp Buoyancy-driven instabilities (e.g., Rayleigh-Benard)

Criticality in conserved dynamical systems: Experimental observation vs. exact properties

Dimitrije Marković, Claudius Gros, and André Schuelein

Chaos 23, 013106 (2013); http://dx.doi.org/10.1063/1.4773003 (6 pages)

Online Publication Date: 9 January 2013

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Conserved dynamical systems are generally considered to be critical. We study a class of critical routing models, equivalent to random maps, which can be solved rigorously in the thermodynamic limit. The information flow is conserved for these routing models and governed by cyclic attractors. We consider two classes of information flow, Markovian routing without memory and vertex routing involving a one-step routing memory. Investigating the respective cycle length distributions for complete graphs, we find log corrections to power-law scaling for the mean cycle length, as a function of the number of vertices, and a sub-polynomial growth for the overall number of cycles. When observing experimentally a real-world dynamical system one normally samples stochastically its phase space. The number and the length of the attractors are then weighted by the size of their respective basins of attraction. This situation is equivalent, for theory studies, to “on the fly” generation of the dynamical transition probabilities. For the case of vertex routing models, we find in this case power law scaling for the weighted average length of attractors, for both conserved routing models. These results show that the critical dynamical systems are generically not scale-invariant but may show power-law scaling when sampled stochastically. It is hence important to distinguish between intrinsic properties of a critical dynamical system and its behavior that one would observe when randomly probing its phase space.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.10.De Algebraic structures and number theory
02.10.Ox Combinatorics; graph theory
02.50.Cw Probability theory
02.50.Ga Markov processes

Ray chaos in an architectural acoustic semi-stadium system

Xiaojian Yu and Yu Zhang

Chaos 23, 013107 (2013); http://dx.doi.org/10.1063/1.4772969 (10 pages)

Online Publication Date: 11 January 2013

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The semi-stadium system is composed of a semicircular cap and a rectilinear platform. In this study, a dynamic model of the side, position, and angle variables is applied to investigate the acoustic ray chaos of the architectural semi-stadium system. The Lyapunov exponent is calculated in order to quantitatively describe ray instability. The model can be reduced to the semi-circular and rectilinear platform systems when the rectilinear length is sufficiently small and large. The quasi-rectilinear platform and the semicircular systems both produce regular trajectories with the maximal Lyapunov exponent approaching zero. Ray localizations, such as flutter-echo and sound focusing, are found in these two systems. However, the semi-stadium system produces chaotic ray behaviors with positive Lyapunov exponents and reduces ray localizations. Furthermore, as the rectilinear length increases, the scaling laws of the Lyapunov exponent of the semi-stadium system are revealed and compared with those of the stadium system. The results suggest the potential application of the proposed model to simulate chaotic dynamics of acoustic ray in architectural enclosed systems.
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43.55.Fw Auditorium and enclosure design
05.45.-a Nonlinear dynamics and chaos
43.20.Dk Ray acoustics
43.30.Gv Backscattering, echoes, and reverberation in water due to combinations of boundaries

Topological field theory of dynamical systems. II

Igor V. Ovchinnikov

Chaos 23, 013108 (2013); http://dx.doi.org/10.1063/1.4775755 (8 pages)

Online Publication Date: 15 January 2013

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This paper is a continuation of the study [Chaos.22.033134] of the relation between the stochastic dynamical systems (DS) and the Witten-type topological field theories (TFT). Here, it is discussed that the stochastic expectation values of a DS must be complemented on the TFT side by math, where math is the ghost number operator. The role of this inclusion is to unfold the natural path-integral representation of the TFT, i.e., the Witten index that equals up to a topological constant to the partition function of the stochastic noise, into the physical partition function of TFT/DS. It is also shown that on the DS side, the TFT's wavefunctions are the conditional probability densities.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
03.65.Ta Foundations of quantum mechanics; measurement theory
02.30.Rz Integral equations
02.50.Ey Stochastic processes

The estimation of neurotransmitter release probability in feedforward neuronal network based on adaptive synchronization

Ming Xue, Jiang Wang, Chenhui Jia, Haitao Yu, Bin Deng, Xile Wei, and Yanqiu Che

Chaos 23, 013109 (2013); http://dx.doi.org/10.1063/1.4775757 (7 pages)

Online Publication Date: 15 January 2013

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In this paper, we proposed a new approach to estimate unknown parameters and topology of a neuronal network based on the adaptive synchronization control scheme. A virtual neuronal network is constructed as an observer to track the membrane potential of the corresponding neurons in the original network. When they achieve synchronization, the unknown parameters and topology of the original network are obtained. The method is applied to estimate the real-time status of the connection in the feedforward network and the neurotransmitter release probability of unreliable synapses is obtained by statistic computation. Numerical simulations are also performed to demonstrate the effectiveness of the proposed adaptive controller. The obtained results may have important implications in system identification in neural science.
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87.19.lg Synapses: chemical and electrical (gap junctions)
87.19.lj Neuronal network dynamics
87.18.Sn Neural networks and synaptic communication
87.16.D- Membranes, bilayers, and vesicles

Lévy noise induced switch in the gene transcriptional regulatory system

Yong Xu, Jing Feng, JuanJuan Li, and Huiqing Zhang

Chaos 23, 013110 (2013); http://dx.doi.org/10.1063/1.4775758 (11 pages)

Online Publication Date: 15 January 2013

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The study of random fluctuations in a gene transcriptional regulatory system is extended to the case of non-Gaussian Lévy noise, which can describe unpredictable jump changes of the random environment. The stationary probability densities are given to explore the key roles of Lévy noise in a gene transcriptional regulatory system. The results demonstrate that the parameters of Lévy noise, including noise intensity, stability index, and skewness parameter, can induce switches between distinct gene-expression states. A further concern is the switching time (from the high concentration state to the low concentration one or from the low concentration state to the high concentration one), which is a random variable and often referred to as the mean first passage time. The effects of Lévy noise on expression and degradation time are studied by computing the mean first passage time in two directions and a number of different peculiarities of non-Gaussian Lévy noise compared with Gaussian noise are observed.
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87.18.Cf Genetic switches and networks
02.50.Cw Probability theory

Chaos M-ary modulation and demodulation method based on Hamilton oscillator and its application in communication

Yongqing Fu, Xingyuan Li, Yanan Li, Wei Yang, and Hailiang Song

Chaos 23, 013111 (2013); http://dx.doi.org/10.1063/1.4790831 (12 pages)

Online Publication Date: 7 February 2013

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Chaotic communication has aroused general interests in recent years, but its communication effect is not ideal with the restriction of chaos synchronization. In this paper a new chaos M-ary digital modulation and demodulation method is proposed. By using region controllable characteristics of spatiotemporal chaos Hamilton map in phase plane and chaos unique characteristic, which is sensitive to initial value, zone mapping method is proposed. It establishes the map relationship between M-ary digital information and the region of Hamilton map phase plane, thus the M-ary information chaos modulation is realized. In addition, zone partition demodulation method is proposed based on the structure characteristic of Hamilton modulated information, which separates M-ary information from phase trajectory of chaotic Hamilton map, and the theory analysis of zone partition demodulator's boundary range is given. Finally, the communication system based on the two methods is constructed on the personal computer. The simulation shows that in high speed transmission communications and with no chaos synchronization circumstance, the proposed chaotic M-ary modulation and demodulation method has outperformed some conventional M-ary modulation methods, such as quadrature phase shift keying and M-ary pulse amplitude modulation in bit error rate. Besides, it has performance improvement in bandwidth efficiency, transmission efficiency and anti-noise performance, and the system complexity is low and chaos signal is easy to generate.
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05.45.Vx Communication using chaos
84.40.Ua Telecommunications: signal transmission and processing; communication satellites
89.70.Hj Communication complexity
02.50.-r Probability theory, stochastic processes, and statistics
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Nucleation pathways on complex networks

Chuansheng Shen, Hanshuang Chen, Miaolin Ye, and Zhonghuai Hou

Chaos 23, 013112 (2013); http://dx.doi.org/10.1063/1.4790832 (6 pages)

Online Publication Date: 7 February 2013

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Identifying nucleation pathway is important for understanding the kinetics of first-order phase transitions in natural systems. In the present work, we study nucleation pathway of the Ising model in homogeneous and heterogeneous networks using the forward flux sampling method, and find that the nucleation processes represent distinct features along pathways for different network topologies. For homogeneous networks, there always exists a dominant nucleating cluster to which relatively small clusters are attached gradually to form the critical nucleus. For heterogeneous ones, many small isolated nucleating clusters emerge at the early stage of the nucleation process, until suddenly they form the critical nucleus through a sharp merging process. Moreover, we also compare the nucleation pathways for different degree-mixing networks. By analyzing the properties of the nucleating clusters along the pathway, we show that the main reason behind the different routes is the heterogeneous character of the underlying networks.
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89.75.Hc Networks and genealogical trees
05.45.-a Nonlinear dynamics and chaos
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

Characterizing chaotic dynamics from simulations of large strain behavior of a granular material under biaxial compression

Michael Small, David M. Walker, Antoinette Tordesillas, and Chi K. Tse

Chaos 23, 013113 (2013); http://dx.doi.org/10.1063/1.4790833 (14 pages)

Online Publication Date: 8 February 2013

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For a given observed time series, it is still a rather difficult problem to provide a useful and compelling description of the underlying dynamics. The approach we take here, and the general philosophy adopted elsewhere, is to reconstruct the (assumed) attractor from the observed time series. From this attractor, we then use a black-box modelling algorithm to estimate the underlying evolution operator. We assume that what cannot be modeled by this algorithm is best treated as a combination of dynamic and observational noise. As a final step, we apply an ensemble of techniques to quantify the dynamics described in each model and show that certain types of dynamics provide a better match to the original data. Using this approach, we not only build a model but also verify the performance of that model. The methodology is applied to simulations of a granular assembly under compression. In particular, we choose a single time series recording of bulk measurements of the stress ratio in a biaxial compression test of a densely packed granular assembly—observed during the large strain or so-called critical state regime in the presence of a fully developed shear band. We show that the observed behavior may best be modeled by structures capable of exhibiting (hyper-) chaotic dynamics.
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05.45.Tp Time series analysis

Self avoiding paths routing algorithm in scale-free networks

Abdeljalil Rachadi, Mohamed Jedra, and Noureddine Zahid

Chaos 23, 013114 (2013); http://dx.doi.org/10.1063/1.4790864 (7 pages)

Online Publication Date: 8 February 2013

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In this paper, we present a new routing algorithm called “the self avoiding paths routing algorithm.” Its application to traffic flow in scale-free networks shows a great improvement over the so called “efficient routing” protocol while at the same time maintaining a relatively low average packet travel time. It has the advantage of minimizing path overlapping throughout the network in a self consistent manner with a relatively small number of iterations by maintaining an equilibrated path distribution especially among the hubs. This results in a significant shifting of the critical packet generation rate over which traffic congestion occurs, thus permitting the network to sustain more information packets in the free flow state. The performance of the algorithm is discussed both on a Barábasi-Albert network and real autonomous system network data.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
05.90.+m Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems (restricted to new topics in section 05)
FREE

Bouncing droplets on a billiard table

David Shirokoff

Chaos 23, 013115 (2013); http://dx.doi.org/10.1063/1.4790840 (10 pages) | Cited 1 time

Online Publication Date: 11 February 2013

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In a set of experiments, Couder et al. demonstrate that an oscillating fluid bed may propagate a bouncing droplet through the guidance of the surface waves. I present a dynamical systems model, in the form of an iterative map, for a droplet on an oscillating bath. I examine the droplet bifurcation from bouncing to walking, and prescribe general requirements for the surface wave to support stable walking states. I show that in addition to walking, there is a region of large forcing that may support the chaotic motion of the droplet. Using the map, I then investigate the droplet trajectories in a square (billiard ball) domain. I show that in large domains, the long time trajectories are either non-periodic dense curves or approach a quasiperiodic orbit. In contrast, in small domains, at low forcing, trajectories tend to approach an array of circular attracting sets. As the forcing increases, the attracting sets break down and the droplet travels throughout space.
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47.52.+j Chaos in fluid dynamics
47.55.D- Drops and bubbles
47.55.Lm Fluidized beds
02.30.Oz Bifurcation theory
47.20.Ky Nonlinearity, bifurcation, and symmetry breaking

A unified model for the dynamics of driven ribbon with strain and magnetic order parameters

Ritupan Sarmah and G. Ananthakrishna

Chaos 23, 013116 (2013); http://dx.doi.org/10.1063/1.4790845 (10 pages)

Online Publication Date: 11 February 2013

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We develop a unified model to explain the dynamics of driven one dimensional ribbon for materials with strain and magnetic order parameters. We show that the model equations in their most general form explain several results on driven magnetostrictive metallic glass ribbons such as the period doubling route to chaos as a function of a dc magnetic field in the presence of a sinusoidal field, the quasiperiodic route to chaos as a function of the sinusoidal field for a fixed dc field, and induced and suppressed chaos in the presence of an additional low amplitude near resonant sinusoidal field. We also investigate the influence of a low amplitude near resonant field on the period doubling route. The model equations also exhibit symmetry restoring crisis with an exponent close to unity. The model can be adopted to explain certain results on magnetoelastic beam and martensitic ribbon under sinusoidal driving conditions. In the latter case, we find interesting dynamics of a periodic one orbit switching between two equivalent wells as a function of an ac magnetic field that eventually makes a direct transition to chaos under resonant driving condition. The model is also applicable to magnetomartensites and materials with two order parameters.
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75.40.Gb Dynamic properties (dynamic susceptibility, spin waves, spin diffusion, dynamic scaling, etc.)
75.80.+q Magnetomechanical effects, magnetostriction
61.43.Fs Glasses

Control of a model of DNA division via parametric resonance

Wang Sang Koon, Houman Owhadi, Molei Tao, and Tomohiro Yanao

Chaos 23, 013117 (2013); http://dx.doi.org/10.1063/1.4790835 (18 pages)

Online Publication Date: 14 February 2013

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We study the internal resonance, energy transfer, activation mechanism, and control of a model of DNA division via parametric resonance. While the system is robust to noise, this study shows that it is sensitive to specific fine scale modes and frequencies that could be targeted by low intensity electro-magnetic fields for triggering and controlling the division. The DNA model is a chain of pendula in a Morse potential. While the (possibly parametrically excited) system has a large number of degrees of freedom and a large number of intrinsic time scales, global and slow variables can be identified by (1) first reducing its dynamic to two modes exchanging energy between each other and (2) averaging the dynamic of the reduced system with respect to the phase of the fastest mode. Surprisingly, the global and slow dynamic of the system remains Hamiltonian (despite the parametric excitation) and the study of its associated effective potential shows how parametric excitation can turn the unstable open state into a stable one. Numerical experiments support the accuracy of the time-averaged reduced Hamiltonian in capturing the global and slow dynamic of the full system.
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87.14.gk DNA
87.15.H- Dynamics of biomolecules
02.60.-x Numerical approximation and analysis
36.20.Ey Conformation (statistics and dynamics)

Generalized variable projective synchronization of time delayed systems

Santo Banerjee, S. Jeeva Sathya Theesar, and J. Kurths

Chaos 23, 013118 (2013); http://dx.doi.org/10.1063/1.4791589 (6 pages) | Cited 2 times

Online Publication Date: 14 February 2013

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We study generalized variable projective synchronization between two unified time delayed systems with constant and modulated time delays. A novel Krasovskii-Lyapunov functional is constructed and a generalized sufficient condition for synchronization is derived analytically using the Lyapunov stability theory and adaptive techniques. The proposed scheme is valid for a system of n-numbers of first order delay differential equations. Finally, a new neural oscillator is considered as a numerical example to show the effectiveness of the proposed scheme.
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05.45.Xt Synchronization; coupled oscillators
02.30.Hq Ordinary differential equations

On the absence of analytic integrability of the Bianchi Class B cosmological models

Antoni Ferragut, Jaume Llibre, and Chara Pantazi

Chaos 23, 013119 (2013); http://dx.doi.org/10.1063/1.4790828 (8 pages)

Online Publication Date: 15 February 2013

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We follow Bogoyavlensky's approach to deal with Bianchi class B cosmological models. We characterize the analytic integrability of such systems.
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98.80.Jk Mathematical and relativistic aspects of cosmology

On the geometric formulation of Hamiltonian dynamics

Eran Calderon, Lawrence Horwitz, Raz Kupferman, and Steven Shnider

Chaos 23, 013120 (2013); http://dx.doi.org/10.1063/1.4791588 (12 pages)

Online Publication Date: 15 February 2013

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Under a proper assignment of a metric and a connection, the (classical) dynamical trajectories can be identified as geodesics of the underlying manifold. We show how these geometric structures can be derived; specifically, we construct them explicitly for configuration and phase spaces of Hamiltonian systems. We demonstrate how the correspondence between geometry and dynamics can be applied to study the conserved quantities of a dynamical system. Lastly, we demonstrate how the mean-curvature of the energy level-sets in phase-space might be correlated with strongly chaotic behavior.
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05.45.Gg Control of chaos, applications of chaos
02.30.Xx Calculus of variations
02.40.Hw Classical differential geometry

Reducing the vulnerability of network by inserting modular topologies

Zhiyun Zou, Junyi Lai, and Jianzhi Gao

Chaos 23, 013121 (2013); http://dx.doi.org/10.1063/1.4792267 (8 pages)

Online Publication Date: 15 February 2013

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In this paper, we present a strategy whose purpose is to reduce the vulnerability of a network via inserting modular topologies. The modular topologies are generated as WS small-world random network, which is relatively highly robust. Using betweenness and betweenness centrality as the vulnerability measurement, the strategy searches for remote nodes with low betweenness in the network and sets these nodes to be connected to the modular topologies. We test our strategy on some basis networks and the results show sufficient availability of our strategy. And by comparing with other methods of adding topologies into the network, we show that our strategy is especially efficient in reducing the vulnerability of the critical network components.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.40.Pc General topology
02.50.-r Probability theory, stochastic processes, and statistics

Nonautonomous motion study on accelerated and decelerated solitons for the variable-coefficient Lenells-Fokas model

Xing Lü and Mingshu Peng

Chaos 23, 013122 (2013); http://dx.doi.org/10.1063/1.4790827 (7 pages) | Cited 1 time

Online Publication Date: 19 February 2013

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In this paper, the nonautonomous Lenells-Fokas (LF) model is studied with the bilinear method and symbolic computation. Such analytical solutions of the nonautonomous LF model as one-soliton, two-soliton, and earthwormons are derived. Nonautonomous characteristics are then symbolically and graphically investigated, and it is finally found that the soliton velocity is time-dependent, and there exist soliton accelerating and decelerating motions. Further, two necessary conditions for the occurrence of earthwormon acceleration and deceleration (and their alternation) are pointed out.
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42.65.Tg Optical solitons; nonlinear guided waves
42.79.Sz Optical communication systems, multiplexers, and demultiplexers
42.81.Dp Propagation, scattering, and losses; solitons
02.30.Hq Ordinary differential equations

Two-particle circular billiards versus randomly perturbed one-particle circular billiards

Sandra Ranković and Mason A. Porter

Chaos 23, 013123 (2013); http://dx.doi.org/10.1063/1.4775756 (9 pages) | Cited 1 time

Online Publication Date: 21 February 2013

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We study a two-particle circular billiard containing two finite-size circular particles that collide elastically with the billiard boundary and with each other. Such a two-particle circular billiard provides a clean example of an “intermittent” system. This billiard system behaves chaotically, but the time scale on which chaos manifests can become arbitrarily long as the sizes of the confined particles become smaller. The finite-time dynamics of this system depends on the relative frequencies of (chaotic) particle-particle collisions versus (integrable) particle-boundary collisions, and investigating these dynamics is computationally intensive because of the long time scales involved. To help improve understanding of such two-particle dynamics, we compare the results of diagnostics used to measure chaotic dynamics for a two-particle circular billiard with those computed for two types of one-particle circular billiards in which a confined particle undergoes random perturbations. Importantly, such one-particle approximations are much less computationally demanding than the original two-particle system, and we expect them to yield reasonable estimates of the extent of chaotic behavior in the two-particle system when the sizes of confined particles are small. Our computations of recurrence-rate coefficients, finite-time Lyapunov exponents, and autocorrelation coefficients support this hypothesis and suggest that studying randomly perturbed one-particle billiards has the potential to yield insights into the aggregate properties of two-particle billiards, which are difficult to investigate directly without enormous computation times (especially when the sizes of the confined particles are small).
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05.45.-a Nonlinear dynamics and chaos
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion

Multi-stage complex contagions

Sergey Melnik, Jonathan A. Ward, James P. Gleeson, and Mason A. Porter

Chaos 23, 013124 (2013); http://dx.doi.org/10.1063/1.4790836 (13 pages)

Online Publication Date: 21 February 2013

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The spread of ideas across a social network can be studied using complex contagion models, in which agents are activated by contact with multiple activated neighbors. The investigation of complex contagions can provide crucial insights into social influence and behavior-adoption cascades on networks. In this paper, we introduce a model of a multi-stage complex contagion on networks. Agents at different stages—which could, for example, represent differing levels of support for a social movement or differing levels of commitment to a certain product or idea—exert different amounts of influence on their neighbors. We demonstrate that the presence of even one additional stage introduces novel dynamical behavior, including interplay between multiple cascades, which cannot occur in single-stage contagion models. We find that cascades—and hence collective action—can be driven not only by high-stage influencers but also by low-stage influencers.
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89.75.Hc Networks and genealogical trees
05.90.+m Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems (restricted to new topics in section 05)
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