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Mar 2012

Volume 22, Issue 1 (partial)

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The impact of awareness on epidemic spreading in networks

Qingchu Wu, Xinchu Fu, Michael Small, and Xin-Jian Xu

Chaos 22, 013101 (2012); doi:10.1063/1.3673573 (8 pages)

Online Publication Date: 3 January 2012

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We explore the impact of awareness on epidemic spreading through a population represented by a scale-free network. Using a network mean-field approach, a mathematical model for epidemic spreading with awareness reactions is proposed and analyzed. We focus on the role of three forms of awareness including local, global, and contact awareness. By theoretical analysis and simulation, we show that the global awareness cannot decrease the likelihood of an epidemic outbreak while both the local awareness and the contact awareness can. Also, the influence degree of the local awareness on disease dynamics is closely related with the contact awareness.
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05.45.-a Nonlinear dynamics and chaos
87.23.Cc Population dynamics and ecological pattern formation
89.75.-k Complex systems

Multiscale dynamics in communities of phase oscillators

Dustin Anderson, Ari Tenzer, Gilad Barlev, Michelle Girvan, Thomas M. Antonsen, and Edward Ott

Chaos 22, 013102 (2012); doi:10.1063/1.3672513 (12 pages)

Online Publication Date: 3 January 2012

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We investigate the dynamics of systems of many coupled phase oscillators with heterogeneous frequencies. We suppose that the oscillators occur in M groups. Each oscillator is connected to other oscillators in its group with “attractive” coupling, such that the coupling promotes synchronization within the group. The coupling between oscillators in different groups is “repulsive,” i.e., their oscillation phases repel. To address this problem, we reduce the governing equations to a lower-dimensional form via the ansatz of Ott and Antonsen, Chaos 18, 037113 (2008). We first consider the symmetric case where all group parameters are the same, and the attractive and repulsive coupling are also the same for each of the M groups. We find a manifold math of neutrally stable equilibria, and we show that all other equilibria are unstable. For M ≥ 3, math has dimension M − 2, and for M = 2, it has dimension 1. To address the general asymmetric case, we then introduce small deviations from symmetry in the group and coupling parameters. Doing a slow/fast timescale analysis, we obtain slow time evolution equations for the motion of the M groups on the manifold math. We use these equations to study the dynamics of the groups and compare the results with numerical simulations.
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05.45.Xt Synchronization; coupled oscillators

Resonance phenomena and long-term chaotic advection in volume-preserving systems

Dmitri L. Vainchtein and Alimu Abudu

Chaos 22, 013103 (2012); doi:10.1063/1.3672510 (8 pages)

Online Publication Date: 3 January 2012

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Creating chaotic advection is the most efficient strategy to achieve mixing on microscale or in very viscous fluids. In this paper, we present a quantitative theory of the long-time resonant mixing in 3D near-integrable flows. We use the flow between two coaxial elliptic counter-rotating cylinders as a demonstrative model, where multiple scatterings on resonance result in mixing by causing the jumps of adiabatic invariants. We improve the existing estimates of the width of the mixing domain. We show that the resulting mixing both on short and long time scales can be described in terms of a single diffusion-type equation with a diffusion coefficient depending on the averaged effect of multiple passages through resonances. We discuss the exact location of the boundaries of the chaotic domain and show how it affects the properties of mixing.
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47.52.+j Chaos in fluid dynamics
47.32.Ef Rotating and swirling flows
47.51.+a Mixing

Propagation of spiking regularity and double coherence resonance in feedforward networks

Cong Men, Jiang Wang, Ying-Mei Qin, Bin Deng, Kai-Ming Tsang, and Wai-Lok Chan

Chaos 22, 013104 (2012); doi:10.1063/1.3676067 (7 pages)

Online Publication Date: 10 January 2012

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We investigate the propagation of spiking regularity in noisy feedforward networks (FFNs) based on FitzHugh-Nagumo neuron model systematically. It is found that noise could modulate the transmission of firing rate and spiking regularity. Noise-induced synchronization and synfire-enhanced coherence resonance are also observed when signals propagate in noisy multilayer networks. It is interesting that double coherence resonance (DCR) with the combination of synaptic input correlation and noise intensity is finally attained after the processing layer by layer in FFNs. Furthermore, inhibitory connections also play essential roles in shaping DCR phenomena. Several properties of the neuronal network such as noise intensity, correlation of synaptic inputs, and inhibitory connections can serve as control parameters in modulating both rate coding and the order of temporal coding.
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05.45.Xt Synchronization; coupled oscillators

Transcripts: An algebraic approach to coupled time series

José M. Amigó, Roberto Monetti, Thomas Aschenbrenner, and Wolfram Bunk

Chaos 22, 013105 (2012); doi:10.1063/1.3673238 (13 pages)

Online Publication Date: 13 January 2012

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Ordinal symbolic dynamics is based on ordinal patterns. Its tools include permutation entropy (in metric and topological versions), forbidden patterns, and a number of mathematical results that make this sort of symbolic dynamics appealing both for theoreticians and practitioners. In particular, ordinal symbolic dynamics is robust against observational noise and can be implemented with low computational cost, which explains its increasing popularity in time series analysis. In this paper, we study the perhaps less exploited aspect so far of ordinal patterns: their algebraic structure. In a first part, we revisit the concept of transcript between two symbolic representations, generalize it to N representations, and derive some general properties. In a second part, we use transcripts to define two complexity indicators of coupled dynamics. Their performance is tested with numerical and real world data.
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05.70.Ce Thermodynamic functions and equations of state
05.45.Tp Time series analysis
02.40.Re Algebraic topology
02.60.-x Numerical approximation and analysis

Saddle-point solutions and grazing bifurcations in an impacting system

Joanna F. Mason and Petri T. Piiroinen

Chaos 22, 013106 (2012); doi:10.1063/1.3673786 (6 pages)

Online Publication Date: 13 January 2012

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This paper focuses on the intricate relationship between smooth and nonsmooth phenomena in an impacting system. In particular a boundary saddle-point solution, that is born in a nonsmooth fold, is analysed. Accessible boundary saddle-point solutions play a key role in determining the global dynamics of a system and here we will show how grazing bifurcations can affect their existence.
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05.45.-a Nonlinear dynamics and chaos
45.40.-f Dynamics and kinematics of rigid bodies

Multiscale characterization of recurrence-based phase space networks constructed from time series

Ruoxi Xiang, Jie Zhang, Xiao-Ke Xu, and Michael Small

Chaos 22, 013107 (2012); doi:10.1063/1.3673789 (10 pages)

Online Publication Date: 17 January 2012

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Recently, a framework for analyzing time series by constructing an associated complex network has attracted significant research interest. One of the advantages of the complex network method for studying time series is that complex network theory provides a tool to describe either important nodes, or structures that exist in the networks, at different topological scale. This can then provide distinct information for time series of different dynamical systems. In this paper, we systematically investigate the recurrence-based phase space network of order k that has previously been used to specify different types of dynamics in terms of the motif ranking from a different perspective. Globally, we find that the network size scales with different scale exponents and the degree distribution follows a quasi-symmetric bell shape around the value of 2k with different values of degree variance from periodic to chaotic Rössler systems. Local network properties such as the vertex degree, the clustering coefficients and betweenness centrality are found to be sensitive to the local stability of the orbits and hence contain complementary information.
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89.75.Hc Networks and genealogical trees
05.45.Tp Time series analysis
02.50.-r Probability theory, stochastic processes, and statistics

Fractal variability: An emergent property of complex dissipative systems

Andrew J. E. Seely and Peter Macklem

Chaos 22, 013108 (2012); doi:10.1063/1.3675622 (7 pages)

Online Publication Date: 19 January 2012

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The patterns of variation of physiologic parameters, such as heart and respiratory rate, and their alteration with age and illness have long been under investigation; however, the origin and significance of scale-invariant fractal temporal structures that characterize healthy biologic variability remain unknown. Quite independently, atmospheric and planetary scientists have led breakthroughs in the science of non-equilibrium thermodynamics. In this paper, we aim to provide two novel hypotheses regarding the origin and etiology of both the degree of variability and its fractal properties. In a complex dissipative system, we hypothesize that the degree of variability reflects the adaptability of the system and is proportional to maximum work output possible divided by resting work output. Reductions in maximal work output (and oxygen consumption) or elevation in resting work output (or oxygen consumption) will thus reduce overall degree of variability. Second, we hypothesize that the fractal nature of variability is a self-organizing emergent property of complex dissipative systems, precisely because it enables the system’s ability to optimally dissipate energy gradients and maximize entropy production. In physiologic terms, fractal patterns in space (e.g., fractal vasculature) or time (e.g., cardiopulmonary variability) optimize the ability to deliver oxygen and clear carbon dioxide and waste. Examples of falsifiability are discussed, along with the need to further define necessary boundary conditions. Last, as our focus is bedside utility, potential clinical applications of this understanding are briefly discussed. The hypotheses are clinically relevant and have potential widespread scientific relevance.
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87.19.Wx Pneumodyamics, respiration

Analytical properties of horizontal visibility graphs in the Feigenbaum scenario

Bartolo Luque, Lucas Lacasa, Fernando J. Ballesteros, and Alberto Robledo

Chaos 22, 013109 (2012); doi:10.1063/1.3676686 (14 pages)

Online Publication Date: 24 January 2012

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Time series are proficiently converted into graphs via the horizontal visibility (HV) algorithm, which prompts interest in its capability for capturing the nature of different classes of series in a network context. We have recently shown [B. Luque et al., PLoS ONE 6, 9 (2011)] that dynamical systems can be studied from a novel perspective via the use of this method. Specifically, the period-doubling and band-splitting attractor cascades that characterize unimodal maps transform into families of graphs that turn out to be independent of map nonlinearity or other particulars. Here, we provide an in depth description of the HV treatment of the Feigenbaum scenario, together with analytical derivations that relate to the degree distributions, mean distances, clustering coefficients, etc., associated to the bifurcation cascades and their accumulation points. We describe how the resultant families of graphs can be framed into a renormalization group scheme in which fixed-point graphs reveal their scaling properties. These fixed points are then re-derived from an entropy optimization process defined for the graph sets, confirming a suggested connection between renormalization group and entropy optimization. Finally, we provide analytical and numerical results for the graph entropy and show that it emulates the Lyapunov exponent of the map independently of its sign.
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05.45.Tp Time series analysis
05.70.Ce Thermodynamic functions and equations of state
02.20.Bb General structures of groups
02.30.Oz Bifurcation theory
02.60.Pn Numerical optimization

Theoretical analysis of multiplicative-noise-induced complete synchronization in global coupled dynamical network

Yuzhu Xiao, Sufang Tang, and Yong Xu

Chaos 22, 013110 (2012); doi:10.1063/1.3677253 (7 pages)

Online Publication Date: 24 January 2012

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In this paper, based on the theory of stochastic differential equation, we study the effect of noise on the synchronization of global coupled dynamical network, when noise presents in coupling term. The theoretical result shows that noise can really induce synchronization. To verify the theoretical result, Cellular Neural Network neural model and Rössler-like system are performed as numerical examples.
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05.45.Xt Synchronization; coupled oscillators
05.90.+m Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems (restricted to new topics in section 05)
02.30.Hq Ordinary differential equations
02.50.Ey Stochastic processes
02.60.Lj Ordinary and partial differential equations; boundary value problems
05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)
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Using time-delayed mutual information to discover and interpret temporal correlation structure in complex populations

D. J. Albers and George Hripcsak

Chaos 22, 013111 (2012); doi:10.1063/1.3675621 (25 pages)

Online Publication Date: 24 January 2012

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This paper addresses how to calculate and interpret the time-delayed mutual information (TDMI) for a complex, diversely and sparsely measured, possibly non-stationary population of time-series of unknown composition and origin. The primary vehicle used for this analysis is a comparison between the time-delayed mutual information averaged over the population and the time-delayed mutual information of an aggregated population (here, aggregation implies the population is conjoined before any statistical estimates are implemented). Through the use of information theoretic tools, a sequence of practically implementable calculations are detailed that allow for the average and aggregate time-delayed mutual information to be interpreted. Moreover, these calculations can also be used to understand the degree of homo or heterogeneity present in the population. To demonstrate that the proposed methods can be used in nearly any situation, the methods are applied and demonstrated on the time series of glucose measurements from two different subpopulations of individuals from the Columbia University Medical Center electronic health record repository, revealing a picture of the composition of the population as well as physiological features.
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05.45.Tp Time series analysis
02.50.-r Probability theory, stochastic processes, and statistics

Vibrational resonance in Duffing systems with fractional-order damping

J. H. Yang and H. Zhu

Chaos 22, 013112 (2012); doi:10.1063/1.3678788 (9 pages)

Online Publication Date: 27 January 2012

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The phenomenon of vibrational resonance (VR) is investigated in over- and under-damped Duffing systems with fractional-order damping. It is found that the factional-order damping can induce change in the number of the steady stable states and then lead to single- or double-resonance behavior. Compared with vibrational resonance in the ordinary systems, the following new results are found in the fractional-order systems. (1) In the overdamped system with double-well potential and ordinary damping, there is only one kind of single-resonance, whereas there are double-resonance and two kinds of single-resonance for the case of fractional-order damping. The necessary condition for these new resonance behaviors is the value of the fractional-order satisfies α > 1. (2) In the overdamped system with single-well potential and ordinary damping, there is no resonance, whereas there is a single-resonance for the case of fractional-order damping. The necessary condition for the new result is α > 1. (3) In the underdamped system with double-well potential and ordinary damping, there are double-resonance and one kind of single-resonance, whereas there are double-resonance and two kinds of single-resonance for the case of fractional-order damping. The necessary condition for the new single-resonance is α < 1. (4) In the underdamped system with single-well potential, there is at most a single-resonance existing for both the cases of ordinary and fractional-order damping. In the underdamped systems, varying the value of the fractional-order is equivalent to change the damping parameter for some cases.
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05.45.-a Nonlinear dynamics and chaos

Symmetry chaotic attractors and bursting dynamics of semiconductor lasers subjected to optical injection

A. D. Mengue and B. Z. Essimbi

Chaos 22, 013113 (2012); doi:10.1063/1.3675623 (10 pages)

Online Publication Date: 1 February 2012

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This paper presents the nonlinear dynamics and bifurcations of optically injected semiconductor lasers in the frame of relative high injection strength. The behavior of the system is explored by means of bifurcation diagrams; however, the exact nature of the involved dynamics is well described by a detailed study of the dynamics evolutions as a function of the effective gain coefficient. As results, we notice the different types of symmetry chaotic attractors with the riddled basins, supercritical pitchfork and Hopf bifurcations, crisis of attractors, instability of chaos, symmetry breaking and restoring bifurcations, and the phenomena of the bursting behavior as well as two connected parts of the same chaotic attractor which merge in a periodic orbit.
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05.45.-a Nonlinear dynamics and chaos

Multistability of twisted states in non-locally coupled Kuramoto-type models

Taras Girnyk, Martin Hasler, and Yuriy Maistrenko

Chaos 22, 013114 (2012); doi:10.1063/1.3677365 (10 pages)

Online Publication Date: 2 February 2012

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A ring of N identical phase oscillators with interactions between L-nearest neighbors is considered, where L ranges from 1 (local coupling) to N/2 (global coupling). The coupling function is a simple sinusoid, as in the Kuramoto model, but with a minus sign which has a profound influence on its behavior. Without the limitation of the generality, the frequency of the free-running oscillators can be set to zero. The resulting system is of gradient type, and therefore, all its solutions converge to an equilibrium point. All so-called q-twisted states, where the phase difference between neighboring oscillators on the ring is 2πq/N, are equilibrium points, where q is an integer. Their stability in the limit N → ∞ is discussed along the line of Wiley et al. [Chaos 16, 015103 (2006)] In addition, we prove that when a twisted state is asymptotically stable for the infinite system, it is also asymptotically stable for sufficiently large N. Note that for smaller N, the same q-twisted states may become unstable and other q-twisted states may become stable. Finally, the existence of additional equilibrium states, called here multi-twisted states, is shown by numerical simulation. The phase difference between neighboring oscillators is approximately 2πq/N in one sector of the ring, −2πq/N in another sector, and it has intermediate values between the two sectors. Our numerical investigation suggests that the number of different stable multi-twisted states grows exponentially as N → ∞. It is possible to interpret the equilibrium points of the coupled phase oscillator network as trajectories of a discrete-time translational dynamical system where the space-variable (position on the ring) plays the role of time. The q-twisted states are then fixed points, and the multi-twisted states are periodic solutions of period N that are close to a heteroclinic cycle. Due to the apparently exponentially fast growing number of such stable periodic solutions, the system shows spatial chaos as N → ∞.
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05.45.Xt Synchronization; coupled oscillators
02.60.-x Numerical approximation and analysis

Geometric and dynamic perspectives on phase-coherent and noncoherent chaos

Yong Zou, Reik V. Donner, and Jürgen Kurths

Chaos 22, 013115 (2012); doi:10.1063/1.3677367 (12 pages)

Online Publication Date: 2 February 2012

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Statistically distinguishing between phase-coherent and noncoherent chaotic dynamics from time series is a contemporary problem in nonlinear sciences. In this work, we propose different measures based on recurrence properties of recorded trajectories, which characterize the underlying systems from both geometric and dynamic viewpoints. The potentials of the individual measures for discriminating phase-coherent and noncoherent chaotic oscillations are discussed. A detailed numerical analysis is performed for the chaotic Rössler system, which displays both types of chaos as one control parameter is varied, and the Mackey-Glass system as an example of a time-delay system with noncoherent chaos. Our results demonstrate that especially geometric measures from recurrence network analysis are well suited for tracing transitions between spiral- and screw-type chaos, a common route from phase-coherent to noncoherent chaos also found in other nonlinear oscillators. A detailed explanation of the observed behavior in terms of attractor geometry is given.
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05.45.Tp Time series analysis
02.50.-r Probability theory, stochastic processes, and statistics
02.60.-x Numerical approximation and analysis
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