Volume 22, Issue 1, March 2012

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.
 REGULAR ARTICLES


The impact of awareness on epidemic spreading in networks
View Description Hide DescriptionWe explore the impact of awareness on epidemic spreading through a population represented by a scalefree network. Using a network meanfield 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.

Multiscale dynamics in communities of phase oscillators
View Description Hide DescriptionWe 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 lowerdimensional 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 of neutrally stable equilibria, and we show that all other equilibria are unstable. For M ≥ 3, 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 . We use these equations to study the dynamics of the groups and compare the results with numerical simulations.

Resonance phenomena and longterm chaotic advection in volumepreserving systems
View Description Hide DescriptionCreating 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 longtime resonant mixing in 3D nearintegrable flows. We use the flow between two coaxial elliptic counterrotating 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 diffusiontype 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.

Propagation of spiking regularity and double coherence resonance in feedforward networks
View Description Hide DescriptionWe investigate the propagation of spiking regularity in noisy feedforward networks (FFNs) based on FitzHughNagumo neuron model systematically. It is found that noise could modulate the transmission of firing rate and spiking regularity. Noiseinduced synchronization and synfireenhanced 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.

Transcripts: An algebraic approach to coupled time series
View Description Hide DescriptionOrdinal 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.

Saddlepoint solutions and grazing bifurcations in an impacting system
View Description Hide DescriptionThis paper focuses on the intricate relationship between smooth and nonsmooth phenomena in an impacting system. In particular a boundary saddlepoint solution, that is born in a nonsmooth fold, is analysed. Accessible boundary saddlepoint 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.

Multiscale characterization of recurrencebased phase space networks constructed from time series
View Description Hide DescriptionRecently, 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 recurrencebased 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 quasisymmetric 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.

Fractal variability: An emergent property of complex dissipative systems
View Description Hide DescriptionThe 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 scaleinvariant fractal temporal structures that characterize healthy biologic variability remain unknown. Quite independently, atmospheric and planetary scientists have led breakthroughs in the science of nonequilibrium 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 selforganizing 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.

Analytical properties of horizontal visibility graphs in the Feigenbaum scenario
View Description Hide DescriptionTime 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 perioddoubling and bandsplitting 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 fixedpoint graphs reveal their scaling properties. These fixed points are then rederived 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.

Theoretical analysis of multiplicativenoiseinduced complete synchronization in global coupled dynamical network
View Description Hide DescriptionIn 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 Rösslerlike system are performed as numerical examples.

Using timedelayed mutual information to discover and interpret temporal correlation structure in complex populations
View Description Hide DescriptionThis paper addresses how to calculate and interpret the timedelayed mutual information (TDMI) for a complex, diversely and sparsely measured, possibly nonstationary population of timeseries of unknown composition and origin. The primary vehicle used for this analysis is a comparison between the timedelayed mutual information averaged over the population and the timedelayed 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 timedelayed 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.

Vibrational resonance in Duffing systems with fractionalorder damping
View Description Hide DescriptionThe phenomenon of vibrational resonance (VR) is investigated in over and underdamped Duffing systems with fractionalorder damping. It is found that the factionalorder damping can induce change in the number of the steady stable states and then lead to single or doubleresonance behavior. Compared with vibrational resonance in the ordinary systems, the following new results are found in the fractionalorder systems. (1) In the overdamped system with doublewell potential and ordinary damping, there is only one kind of singleresonance, whereas there are doubleresonance and two kinds of singleresonance for the case of fractionalorder damping. The necessary condition for these new resonance behaviors is the value of the fractionalorder satisfies α > 1. (2) In the overdamped system with singlewell potential and ordinary damping, there is no resonance, whereas there is a singleresonance for the case of fractionalorder damping. The necessary condition for the new result is α > 1. (3) In the underdamped system with doublewell potential and ordinary damping, there are doubleresonance and one kind of singleresonance, whereas there are doubleresonance and two kinds of singleresonance for the case of fractionalorder damping. The necessary condition for the new singleresonance is α < 1. (4) In the underdamped system with singlewell potential, there is at most a singleresonance existing for both the cases of ordinary and fractionalorder damping. In the underdamped systems, varying the value of the fractionalorder is equivalent to change the damping parameter for some cases.

Symmetry chaotic attractors and bursting dynamics of semiconductor lasers subjected to optical injection
View Description Hide DescriptionThis 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.

Multistability of twisted states in nonlocally coupled Kuramototype models
View Description Hide DescriptionA ring of N identical phase oscillators with interactions between Lnearest 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 freerunning oscillators can be set to zero. The resulting system is of gradient type, and therefore, all its solutions converge to an equilibrium point. All socalled qtwisted 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 qtwisted states may become unstable and other qtwisted states may become stable. Finally, the existence of additional equilibrium states, called here multitwisted 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 multitwisted states grows exponentially as N → ∞. It is possible to interpret the equilibrium points of the coupled phase oscillator network as trajectories of a discretetime translational dynamical system where the spacevariable (position on the ring) plays the role of time. The qtwisted states are then fixed points, and the multitwisted 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 → ∞.

Geometric and dynamic perspectives on phasecoherent and noncoherent chaos
View Description Hide DescriptionStatistically distinguishing between phasecoherent 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 phasecoherent 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 MackeyGlass system as an example of a timedelay system with noncoherent chaos. Our results demonstrate that especially geometric measures from recurrence network analysis are well suited for tracing transitions between spiral and screwtype chaos, a common route from phasecoherent to noncoherent chaos also found in other nonlinear oscillators. A detailed explanation of the observed behavior in terms of attractor geometry is given.

Longrange interactions between adjacent and distant bases in a DNA and their impact on the ribonucleic acid polymeraseDNA dynamics
View Description Hide DescriptionWhen an inhomogeneous RNApolymerase (RNAP) binds to an inhomogeneous DNA at the physiological temperature, we propose a spinlike model of DNA nonlinear dynamics with longrange interactions (LRI) between adjacent and distant base pairs to study RNAPDNA dynamics. Using HolsteinPrimakoff’s representation and Glauber’s coherent state representation, we show that the model equation is a completely integrable nonlinear Schrödinger equation whose dispersive coefficient depends on LRI’s parameter. Inhomogeneities have introduced perturbation terms in the equation of motion of RNAPDNA dynamics. Considering the homogeneous part of that equation, a detailed study of the solution shows that the number of base pairs which form the bubble, the height, and the width of that bubble depend on the longrange parameter. The results of the perturbation analysis show that the inhomogeneities due to the DNA and RNAP structures do not alter the velocity and amplitude of the soliton, but introduce some fluctuations in the localized region of the soliton. The events that happen in the present study may represent binding of an RNAP to a promoter site in the DNA during the transcription process.

The structure and resilience of financial market networks
View Description Hide DescriptionFinancial markets can be viewed as a highly complex evolving system that is very sensitive to economic instabilities. The complex organization of the market can be represented in a suitable fashion in terms of complex networks, which can be constructed from stock prices such that each pair of stocks is connected by a weighted edge that encodes the distance between them. In this work, we propose an approach to analyze the topological and dynamic evolution of financial networks based on the stock correlation matrices. An entropyrelated measurement is adopted to quantify the robustness of the evolving financial market organization. It is verified that the network topological organization suffers strong variation during financial instabilities and the networks in such periods become less robust. A statistical robust regression model is proposed to quantity the relationship between the network structure and resilience. The obtained coefficients of such model indicate that the average shortest path length is the measurement most related to network resilience coefficient. This result indicates that a collective behavior is observed between stocks during financial crisis. More specifically, stocks tend to synchronize their price evolution, leading to a high correlation between pair of stock prices, which contributes to the increase in distance between them and, consequently, decrease the network resilience.

Alternation of regular and chaotic dynamics in a simple twodegreeoffreedom system with nonlinear inertial coupling
View Description Hide DescriptionWe show that nonlinear inertial coupling between a linear oscillator and an eccentric rotator can lead to very interesting interchanges between regular and chaotic dynamical behavior. Indeed, we show that this model demonstrates rather unusual behavior from the viewpoint of nonlinear dynamics. Specifically, at a discrete set of values of the total energy, the Hamiltonian system exhibits nonconventional nonlinear normal modes, whose shape is determined by phase locking of rotatory and oscillatory motions of the rotator at integer ratios of characteristic frequencies. Considering the weakly damped system, resonance capture of the dynamics into the vicinity of these modes brings about regular motion of the system. For energy levels far from these discrete values, the motion of the system is chaotic. Thus, the succession of resonance captures and escapes by a discrete set of the normal modes causes a sequence of transitions between regular and chaotic behavior, provided that the damping is sufficiently small. We begin from the Hamiltonian system and present a series of Poincaré sections manifesting the complex structure of the phase space of the considered system with inertial nonlinear coupling. Then an approximate analytical description is presented for the nonconventional nonlinear normal modes. We confirm the analytical results by numerical simulation and demonstrate the alternate transitions between regular and chaotic dynamics mentioned above. The origin of the chaotic behavior is also discussed.

Finger tapping movements of Parkinson’s disease patients automatically rated using nonlinear delay differential equations
View Description Hide DescriptionParkinson’s disease is a degenerative condition whose severity is assessed by clinical observations of motor behaviors. These are performed by a neurological specialist through subjective ratings of a variety of movements including 10s bouts of repetitive fingertapping movements. We present here an algorithmic rating of these movements which may be beneficial for uniformly assessing the progression of the disease. Fingertapping movements were digitally recorded from Parkinson’s patients and controls, obtaining one time series for every 10 s bout. A nonlinear delay differential equation, whose structure was selected using a genetic algorithm, was fitted to each time series and its coefficients were used as a sixdimensional numerical descriptor. The algorithm was applied to timeseries from two different groups of Parkinson’s patients and controls. The algorithmic scores compared favorably with the unified Parkinson’s disease rating scale scores, at least when the latter adequately matched with ratings from the Hoehn and Yahr scale. Moreover, when the two sets of mean scores for all patients are compared, there is a strong (r = 0.785) and significant () correlation between them.

Stabilization of chaos systems described by nonlinear fractionalorder polytopic differential inclusion
View Description Hide DescriptionIn this paper, sliding mode control is utilized for stabilization of a particular class of nonlinear polytopic differential inclusion systems with fractionalorder0 < q < 1. This class of fractional order differential inclusion systems is used to model physical chaotic fractional order Chen and Lu systems. By defining a sliding surface with fractional integral formula, exploiting the concept of the state space norm, and obtaining sufficient conditions for stability of the sliding surface, a special feedback law is presented which enables the system states to reach the sliding surface and consequently creates a sliding mode control. Finally, simulation results are used to illustrate the effectiveness of the proposed method.
