Volume 22, Issue 2, June 2012

The dynamics in weakly chaotic Hamiltonian systems strongly depends on initial conditions (ICs) and little can be affirmed about generic behaviors. Using two distinct Hamiltonian systems, namely one particle in an open rectangular billiard and four particles globally coupled on a discrete lattice, we show that in these models, the transition from integrable motion to weak chaos emerges via chaotic stripes as the nonlinear parameter is increased. The stripes represent intervals of initial conditions which generate chaotic trajectories and increase with the nonlinear parameter of the system. In the billiard case, the initial conditions are the injection angles. For higherdimensional systems and small nonlinearities, the chaotic stripes are the initial condition inside which Arnold diffusion occurs.
 REGULAR ARTICLES


Components in timevarying graphs
View Description Hide DescriptionReal complex systems are inherently timevarying. Thanks to new communication systems and novel technologies, today it is possible to produce and analyze social and biological networks with detailed information on the time of occurrence and duration of each link. However, standard graph metrics introduced so far in complex network theory are mainly suited for static graphs, i.e., graphs in which the links do not change over time, or graphs built from timevarying systems by aggregating all the links as if they were concurrent in time. In this paper, we extend the notion of connectedness, and the definitions of node and graph components, to the case of timevarying graphs, which are represented as timeordered sequences of graphs defined over a fixed set of nodes. We show that the problem of finding strongly connected components in a timevarying graph can be mapped into the problem of discovering the maximalcliques in an opportunely constructed static graph, which we name the affine graph. It is, therefore, an NPcomplete problem. As a practical example, we have performed a temporal component analysis of timevarying graphs constructed from three data sets of human interactions. The results show that taking time into account in the definition of graph components allows to capture important features of real systems. In particular, we observe a large variability in the size of node temporal in and outcomponents. This is due to intrinsic fluctuations in the activity patterns of individuals, which cannot be detected by static graph analysis.

A general fractionalorder dynamical network: Synchronization behavior and state tuning
View Description Hide DescriptionA general fractionalorder dynamical network model for synchronization behavior is proposed. Different from previous integerorder dynamical networks, the model is made up of coupled units described by fractional differential equations, where the connections between individual units are nondiffusive and nonlinear. We show that the synchronous behavior of such a network cannot only occur, but also be dramatically different from the behavior of its constituent units. In particular, we find that simple behavior can emerge as synchronized dynamics although the isolated units evolve chaotically. Conversely, individually simple units can display chaotic attractors when the network synchronizes. We also present an easily checked criterion for synchronization depending only on the eigenvalues distribution of a decomposition matrix and the fractional orders. The analytic results are complemented with numerical simulations for two networks whose nodes are governed by fractionalorder Lorenz dynamics and fractionalorder Rössler dynamics, respectively.

Iterated function system models in data analysis: Detection and separation
View Description Hide DescriptionWe investigate the use of iterated function system (IFS) models for data analysis. An IFS is a discretetime dynamical system in which each time step corresponds to the application of one of the finite collection of maps. The maps, which represent distinct dynamical regimes, may be selected deterministically or stochastically. Given a time series from an IFS, our algorithm detects the sequence of regime switches under the assumption that each map is continuous. This method is tested on a simple example and an experimental computer performance data set. This methodology has a wide range of potential uses: from changepoint detection in timeseries data to the field of digital communications.

Neuronal avalanches of a selforganized neural network with activeneurondominant structure
View Description Hide DescriptionNeuronal avalanche is a spontaneous neuronal activity which obeys a powerlaw distribution of population event sizes with an exponent of –3/2. It has been observed in the superficial layers of cortex both invivo and invitro. In this paper, we analyze the information transmission of a novel selforganized neural network with activeneurondominant structure. Neuronal avalanches can be observed in this network with appropriate input intensity. We find that the process of network learning via spiketiming dependent plasticity dramatically increases the complexity of network structure, which is finally selforganized to be activeneurondominant connectivity. Both the entropy of activity patterns and the complexity of their resulting postsynaptic inputs are maximized when the network dynamics are propagated as neuronal avalanches. This emergent topology is beneficial for information transmission with high efficiency and also could be responsible for the large information capacity of this network compared with alternative archetypal networks with different neural connectivity.

Experimental evidence of synchronization of timevarying dynamical network
View Description Hide DescriptionWe investigate synchronization of time varying networks and stability conditions. We derive interesting relations between the critical coupling constants for synchronization and switching times for timevarying and time average networks. The relations are based on the additive property of Lyapunov exponents and are verified experimentally in electronic circuit.

Stochastic stability of genetic regulatory networks with a finite set delay characterization
View Description Hide DescriptionIn this paper, the delaydistributiondependent stability is derived for the stochastic genetic regulatory networks (GRNs) with a finite set delay characterization and interval parameter uncertainties. One important feature of the obtained results here is that the timevarying delays are assumed to be random and the sum of the occurrence probabilities of the delays is assumed to be 1. By employing a new LyapunovKrasovskii functional dependent on auxiliary delay parameters which allow the timevarying delays to be not differentiable, less conservative meansquare stochastic stability criteria are obtained. Finally, two examples are given to illustrate the effectiveness and superiority of the derived results.

Using filtering effects to identify objects
View Description Hide DescriptionReflecting signals off of targets is a method widely used to locate objects, but the reflected signal also contains information that can be used to identify the object. In radar or sonar, the signal amplitudes used are small enough that only linear effects are present, so we can consider the effect of the target on the signal as a linear filter. Using the known effects of linear filters on chaotic signals, we can create a reference that allows us to match a particular target to a particular reflected signal. Furthermore, if some parts of this “filter” vary only slowly as the aspect angle of the object changes, we can produce a reference that averages out the parts that are highly angle dependent so that one reference can be used to identify the target over a range of angles.

Partial synchronization in stochastic dynamical networks with switching communication channels
View Description Hide DescriptionIn this paper, the partial synchronization problem of stochastic dynamical networks (SDNs) is investigated. Unlike the existing models, the SDN considered in this paper suffers from a class of communication constraint—only part of nodes’ states can be transmitted. Thus, less nodes’ states can be used to synchronize the SDN, which makes the analysis of the synchronization problem much harder. A set of channel matrices are introduced to reflect such kind of constraint. Furthermore, due to unpredictable environmental changes, the channel matrices can switch among some communication modes. The switching considered here is governed by a Markov process. To overcome the difficulty, a regrouping method is employed to derive our main results. The obtained conditions guarantee that partial synchronization can be achieved for SDNs under switching communication constraint. Finally, numerical examples are given to illustrate the effectiveness of the theoretical results and how the communication constraint influences synchronization result.

Finitetime stochastic combination synchronization of three different chaotic systems and its application in secure communication
View Description Hide DescriptionIn this paper, the finitetime stochastic combination synchronization of three different chaotic systems is investigated. Based on the adaptive technique and the properties of Weiner process, a novel sufficient condition is obtained to ensure combination synchronization under stochastic perturbations. Moreover, a secure communication scheme based on the adaptive combination synchronization of three different systems, i.e., the Lorenz system, Chen system, and Lü system, with uncertainties, unknown parameters, and stochastic perturbation is presented. The simulation results show the feasibility of the proposed method.

Annual variability in a conceptual climate model: Snapshot attractors, hysteresis in extreme events, and climate sensitivity
View Description Hide DescriptionIn a conceptual model of global atmospheric circulation, the effects of annually periodic driving are investigated. The driven system is represented in terms of snapshot attractors, which may remain fractal at all times. This is due to the transiently chaotic behavior in the regular parameter regimes of the undriven system. The driving with annual periodicity is found to be relatively fast: There is a considerable deviation from the undriven case. Accordingly, the existence of a hysteresis loop is identified, namely, the extremal values of a given variable depend not only on the actual strength of the insolation but also on the sign of its temporal change. This hysteresis is due to a kind of internal memory. In the thresholddependence of mean return times of various extreme events, a roughly exponential scaling is found. Climate sensitivity parameters are defined, and the measure of certain types of extremal behavior is found to be strongly susceptible to changes in insolation.

Classical helium atom with radiation reaction
View Description Hide DescriptionWe study a classical model of helium atom in which, in addition to the Coulomb forces, the radiation reaction forces are taken into account. This modification brings in the model a new qualitative feature of a global character. Indeed, as pointed out by Dirac, in any model of classical electrodynamics of point particles involving radiation reaction one has to eliminate, from the a priori conceivable solutions of the problem, those corresponding to the emission of an infinite amount of energy. We show that the Dirac prescription solves a problem of inconsistency plaguing all available models which neglect radiation reaction, namely, the fact that in all such models, most initial data lead to a spontaneous breakdown of the atom. A further modification is that the system thus acquires a peculiar form of dissipation. In particular, this makes attractive an invariant manifold of special physical interest, the zerodipole manifold that corresponds to motions in which no energy is radiated away (in the dipole approximation). We finally study numerically the invariant measure naturally induced by the timeevolution on such a manifold, and this corresponds to studying the formation process of the atom. Indications are given that such a measure may be singular with respect to that of Lebesgue.

Pattern formation in a reactiondiffusionadvection system with wave instability
View Description Hide DescriptionIn this paper, we show by means of numerical simulations how new patterns can emerge in a system with wave instability when a unidirectional advective flow (plug flow) is added to the system. First, we introduce a three variable model with one activator and two inhibitors with similar kinetics to those of the Oregonator model of the BelousovZhabotinsky reaction. For this model, we explore the type of patterns that can be obtained without advection, and then explore the effect of different velocities of the advective flow for different patterns. We observe standing waves, and with flow there is a transition from out of phase oscillations between neighboring units to inphase oscillations with a doubling in frequency. Also mixed and clustered states are generated at higher velocities of the advective flow. There is also a regime of “waving Turing patterns” (quasistationary structures that come close and separate periodically), where low advective flow is able to stabilize the stationary Turing pattern. At higher velocities, superposition and interaction of patterns are observed. For both types of patterns, at high velocities of the advective field, the known flow distributed oscillations are observed.

Exact foldedband chaotic oscillator
View Description Hide DescriptionAn exactly solvable chaotic oscillator with foldedband dynamics is shown. The oscillator is a hybrid dynamical system containing a linear ordinary differential equation and a nonlinear switching condition. Bounded oscillations are provably chaotic, and successive waveform maxima yield a onedimensional piecewiselinear return map with segments of both positive and negative slopes. Continuoustime dynamics exhibit a foldedband topology similar to Rössler’s oscillator. An exact solution is written as a linear convolution of a fixed basis pulse and a discrete binary sequence, from which an equivalent symbolic dynamics is obtained. The foldedband topology is shown to be dependent on the symbol grammar.

Diffusion in a collisional standard map
View Description Hide DescriptionTest particle evaluation of the diffusion coefficient in the presence of magnetic field fluctuations and binary collisions is presented. Chaotic magnetic field lines originate from resonant magnetic perturbations (RMPs). To lowest order, charged particles follow magnetic field lines. Drifts and interaction (collisions) with other particles decorrelate particles from the magnetic field lines. We model the binary collision process by a constant collision frequency. The magnetic field configuration including perturbations on the integrable Hamiltonian part is such that the single particle motion can be followed by a collisional version of a ChirikovTaylor (standard) map. Frequent collisions are allowed for. Scaling of the diffusion beyond the quasilinear and subdiffusive behaviour is investigated in dependence on the strength of the magnetic perturbations and the collision frequency. The appearance of the so called RechesterRosenbluth regime is verified. It is further shown that the so called KadomtsevPogutse diffusion coefficient is the strong collisional limit of the RechesterRosenbluth formula. The theoretical estimates are supplemented by numerical simulations.

On finitesize Lyapunov exponents in multiscale systems
View Description Hide DescriptionWe study the effect of regime switches on finite size Lyapunov exponents (FSLEs) in determining the error growth rates and predictability of multiscale systems. We consider a dynamical system involving slow and fast regimes and switches between them. The surprising result is that due to the presence of regimes, the error growth rate can be a nonmonotonic function of initial error amplitude. In particular, troughs in the large scales of FSLE spectra are shown to be a signature of slow regimes, whereas fast regimes are shown to cause large peaks in the spectra where error growth rates far exceed those estimated from the maximal Lyapunov exponent. We present analytical results explaining these signatures and corroborate them with numerical simulations. We show further that these peaks disappear in stochastic parametrizations of the fast chaotic processes, and the associated FSLE spectra reveal that large scale predictability properties of the full deterministic model are well approximated, whereas small scale features are not properly resolved.

Regular and chaotic dynamics of a fountain in a stratified fluid
View Description Hide DescriptionIn the present paper, we study by direct numerical simulation (DNS) and theoretical analysis, the dynamics of a fountain penetrating a pycnocline (a sharp density interface) in a densitystratified fluid. A circular, laminar jet flow of neutral buoyancy is considered, which propagates vertically upwards towards the pycnocline level, penetrates a distance into the layer of lighter fluid, and further stagnates and flows down under gravity around the upflowing core thus creating a fountain. The DNS results show that if the Froude number (Fr) is small enough, the fountain top remains axisymmetric and steady. However, if Fr is increased, the fountain top becomes unsteady and oscillates in a circular flapping (CF) mode, whereby it retains its shape and moves periodically around the jet central axis. If Fr is increased further, the fountain top rises and collapses chaotically in a bobbing oscillation mode (or Bmode). The development of these two modes is accompanied by the generation of different patterns of internal waves (IW) in the pycnocline. The CFmode generates spiral internal waves, whereas the Bmode generates IW packets with a complex spatial distribution. The dependence of the amplitude of the fountaintop oscillations on Fr is well described by a Landautype twomodecompetition model.

“Quorum sensing” generated multistability and chaos in a synthetic genetic oscillator
View Description Hide DescriptionWe model the dynamics of the synthetic genetic oscillator Repressilator equipped with quorum sensing. In addition to a circuit of 3 genes repressing each other in a unidirectional manner, the model includes a phaserepulsive type of the coupling module implemented as the production of a small diffusive molecule—autoinducer (AI). We show that the autoinducer (which stimulates the transcription of a target gene) is responsible for the disappearance of the limit cycle (LC) through the infinite period bifurcation and the formation of a stable steady state (SSS) for sufficiently large values of the transcription rate. We found conditions for hysteresis between the limit cycle and the stable steady state. The parameters’ region of the hysteresis is determined by the mRNA to protein lifetime ratio and by the level of transcriptionstimulating activity of the AI. In addition to hysteresis, increasing AIdependent stimulation of transcription may lead to the complex dynamic behavior which is characterized by the appearance of several branches on the bifurcation continuation, containing different regular limit cycles, as well as a chaotic regime. The multistability which is manifested as the coexistence between the stable steady state, limit cycles, and chaos seems to be a novel type of the dynamics for the ring oscillator with the added quorum sensing positive feedback.

Generalized complexity measures and chaotic maps
View Description Hide DescriptionThe logistic and Tinkerbell maps are studied with the recently introduced generalized complexity measure. The generalized complexity detects periodic windows. Moreover, it recognizes the intersection of periodic branches of the bifurcation diagram. It also reflects the fractal character of the chaotic dynamics. There are cases where the complexity plot shows changes that cannot be seen in the bifurcation diagram.

Alternative interpretations of powerlaw distributions found in nature
View Description Hide DescriptionWe investigate two inherently different classes of probability density functions (pdfs) that share the common property of power law tails: the αstable Lévy process and the linear Markov diffusion process with additive and multiplicative Gaussian noise. Dynamical processes described by these distributions cannot be uniquely identified as belonging to one or the other class either by diverging variance due to powerlaw tails in the pdf or by the possible existence of skew. However, there are distinguishing features that may be found in sufficiently well sampled time series. We examine these features and discuss how they may guide the development of proper approximations to equations of motion underlying dynamical systems. An additional result of this research was the identification of a variable describing the relative importance of the multiplicative and independent additive noise forcing in our linear Markov process. The distribution of this variable is generally skewed, depending on the level of correlation between the additive and multiplicative noise.

Van der Pol and the history of relaxation oscillations: Toward the emergence of a concept
View Description Hide DescriptionRelaxation oscillations are commonly associated with the name of Balthazar van der Pol via his paper (Philosophical Magazine, 1926) in which he apparently introduced this terminology to describe the nonlinear oscillations produced by selfsustained oscillating systems such as a triode circuit. Our aim is to investigate how relaxation oscillations were actually discovered. Browsing the literature from the late 19th century, we identified four selfoscillating systems in which relaxation oscillations have been observed: (i) the series dynamo machine conducted by GérardLescuyer (1880), (ii) the musical arc discovered by Duddell (1901) and investigated by Blondel (1905), (iii) the triode invented by de Forest (1907), and (iv) the multivibrator elaborated by Abraham and Bloch (1917). The differential equation describing such a selfoscillating system was proposed by Poincaré for the musical arc (1908), by Janet for the series dynamo machine (1919), and by Blondel for the triode (1919). Once Janet (1919) established that these three selfoscillating systems can be described by the same equation, van der Pol proposed (1926) a generic dimensionless equation which captures the relevant dynamical properties shared by these systems. Van der Pol’s contributions during the period of 19261930 were investigated to show how, with Le Corbeiller’s help, he popularized the “relaxation oscillations” using the previous experiments as examples and, turned them into a concept.
