Penta-hepta defect chaos in a model for rotating hexagonal convection

In a model for rotating non-Boussinesq convection with mean flow, we identify a regime of spatiotemporal chaos that is based on a hexagonal planform and is sustained by the induced nucleation of dislocations by penta-hepta defects. The probability distribution function for the number of defects deviates substantially from the usually observed Poisson-type distribution. It implies strong correlations between the defects in the form of density-dependent creation and annihilation rates of defects. We extract these rates from the distribution function and also directly from the defect dynamics.     

Karhunen-Loeve local characterization of spatiotemporal chaos in a reaction-diffusion system

By computing the Karhunen-Loeve decomposition (KLD) correlation length xi(KLD) of a reaction-diffusion system in the extensive chaos regime, we show that it is a sensitive measure of spatial dynamical inhomogeneities. It reveals substantial spatial nonuniformity of the dynamics at the boundaries and can also detect slow spatial variations in system parameters. The intensive length xi(KLD) can be easily computed from small local subsystems and is found to have a similar parametric dependence as the two-point correlation length computed over the full system size.     

Effect of transmission fiber and amplifier noise on optical chaos synchronization

Optical chaos propagation has few constraints peculiar to itself which do not become as significant in conventional nonchaotic optical communication. We have investigated the effects of transmission fiber nonlinearities, dispersion and noise of erbium doped fiber amplifier (EDFA) on chaotic signal synchronization in lumped and distributed configuration. It is found that the effects of fiber dispersion can be easily compensated; however, the effects of fiber nonlinearity on chaos cannot be overdone and must be avoided. Three distinct configurations with different combinations of standard telecommunication fiber, dispersion compensation fiber and lumped and distributed EDF for amplification are analysed. The results are compared in terms of sync diagrams and noise figure. The chaos after propagation through distributed amplification performs better as compared to lumped amplification. Also, a new quantitative measure for the calculation of deviation in sync diagram of chaos is introduced.     

Effect of noise on generalized synchronization of chaos: theory and experiment

The influence of noise on the generalized synchronization regime in the chaotic systems with dissipative coupling is considered. If attractors of the drive and response systems have an infinitely large basin of attraction, generalized synchronization is shown to possess a great stability with respect to noise. The reasons of the revealed particularity are explained by means of the modified system approach [A.E. Hramov, A.A. Koronovskii, Phys. Rev. E 71, 067201 (2005)] and confirmed by the results of numerical calculations and experimental studies. The main results are illustrated using the examples of unidirectionally coupled chaotic oscillators and discrete maps as well as spatially extended dynamical systems. Different types of the model noise are analyzed. Possible applications of the revealed particularity are briefly discussed.     

Spatiotemporal chaos in an electric current driven ionic reaction-diffusion system

Two types of transitions from the time-periodic spatiotemporal patterns to chaotic ones in the spatially one-dimensional ionic reaction-diffusion system forced either with direct or alternating electric field are described and analyzed by numerical techniques. An ionic version of the Brusselator kinetic scheme is considered. The Karhunen-Loeve decomposition technique is shown to be a possible tool for the global representation of dynamic behavior, but fails as a tool in the identification of the route of transition to chaos in the case of direct current forcing. Higher dimensional chaos with two positive Lyapunov exponents has been identified for the case of alternating current forcing. Results of the Karhunen-Loeve analysis are compared to results of classical analysis of local time series (attractor dimensions, Lyapunov exponents).     

Effect of noise on chaos synchronization in time-delayed systems: Numerical and experimental observations

We discuss the constructive role of noise (white and colored) in chaos synchronization in time-delayed systems. We first numerically investigate noise-induced synchronization (NIS) between two identical uncoupled Ikeda and Mackey–Glass systems. We find that synchronization occurs above a critical noise intensity that differs for different colors of noise. Synchronization onset is characterized by the value of the maximum transverse Lyapunov exponent. We then discuss the enhancement of chaos synchronization between two time-delayed systems when they are coupled unidirectionally. The effect of parameter mismatch for NIS is described in detail. We provide experimental evidence of NIS for a Mackey–Glass-like system in an electronic circuit using different colors of noise. An integration scheme for time-delayed systems in the presence of additive white and colored noise is discussed.     

The linear noise approximation for reaction-diffusion systems on networks

Stochastic reaction-diffusion models can be analytically studied on complex networks using the linear noise approximation. This is illustrated through the use of a specific stochastic model, which displays travelling waves in its deterministic limit. The role of stochastic fluctuations is investigated and shown to drive the emergence of stochastic waves beyond the region of the instability predicted from the deterministic theory. Simulations are performed to test the theoretical results and are analyzed via a generalized Fourier transform algorithm. This transform is defined using the eigenvectors of the discrete Laplacian defined on the network. A peak in the numerical power spectrum of the fluctuations is observed in quantitative agreement with the theoretical predictions.     

Turing pattern selection in a reaction-diffusion epidemic model

We present Turing pattern selection in a reaction-diffusion epidemic model under zero-flux boundary conditions.The value of this study is twofold.First,it establishes the amplitude equations for the excited modes,which determines the stability of amplitudes towards uniform and inhomogeneous perturbations.Second,it illustrates all five categories of Turing patterns close to the onset of Turing bifurcation via numerical simulations which indicates that the model dynamics exhibits complex pattern replication:on increasing the control parameter ν,the sequence "H 0 hexagons → H 0-hexagon-stripe mixtures → stripes → H π-hexagon-stripe mixtures → H π hexagons" is observed.This may enrich the pattern dynamics in a diffusive epidemic model.     

Importance of packing in spiral defect chaos

We develop two measures to characterize the geometry of patterns exhibited by the state of spiral defect chaos, a weakly turbulent regime of Rayleigh-Bénard convection. These describe the packing of contiguous stripes within the pattern by quantifying their length and nearest-neighbor distributions. The distributions evolve towards a unique distribution with increasing Rayleigh number that suggests power-law scaling for the dynamics in the limit of infinite system size. The techniques are generally applicable to patterns that are reducible to a binary representation.      

Running pulses of complex shape in a reaction-diffusion model

In a one-dimensional reaction-diffusion model of an active medium, stable steady-state wave pulses of a new type are described. They are called multihumped because their waveforms contain several maxima of similar size. Presumably, the multihumped pulses arise via a bifurcation at which an unstable trigger wave disappears. The parameter governing this bifurcation is the diffusion coefficient for the model inhibitor. The model is analyzed by varying this parameter to determine the conditions for the emergence of multihumped pulses. The results of this analysis show how their waveform and dynamics of excitation depend on the inhibitor diffusion coefficient.     

Spatio-temporal chaos in a chemotaxis model

In this paper we explore the dynamics of a one-dimensional Keller–Segel type model for chemotaxis incorporating a logistic cell growth term. We demonstrate the capacity of the model to self-organise into multiple cellular aggregations which, according to position in parameter space, either form a stationary pattern or undergo a sustained spatio-temporal sequence of merging (two aggregations coalesce) and emerging (a new aggregation appears). This spatio-temporal patterning can be further subdivided into either a time-periodic or time-irregular fashion. Numerical explorations into the latter indicate a positive Lyapunov exponent (sensitive dependence to initial conditions) together with a rich bifurcation structure. In particular, we find stationary patterns that bifurcate onto a path of periodic patterns which, prior to the onset of spatio-temporal irregularity, undergo a “periodic-doubling” sequence. Based on these results and comparisons with other systems, we argue that the spatio-temporal irregularity observed here describes a form of spatio-temporal chaos. We discuss briefly our results in the context of previous applications of chemotaxis models, including tumour invasion, embryonic development and ecology.     

Time-space white noise eliminates global solutions in reaction-diffusion equations

We prove that perturbing the reaction-diffusion equation u_t=u_xx+(u₊)^p (p>1), with time-space white noise produces that solutions explodes with probability one for every initial datum, opposite to the deterministic model where a positive stationary solution exists.     

Deterministic chaos and noise in optical bistability

The time-dependent solutions of the mean-field Maxwell-Bloch equations for optical bistability are studied numerically for the deterministic equations and the stochastic equations with additional noise sources. From the solutions of the deterministic equations, a discrete map is constructed showing that the periodic and chaotic solutions form a Feigenbaum scenarium. Inclusion of noise sources leads to a finite lifetime of the states in the upper bistable branch and to destabilization of higher periodic solutions.     

Numerical study of chaos based on a shell model

A shell model is introduced to study a turbulence driven by the thermal instability (Rayleigh-Benard convection). This model equation describes cascade and chaos in the strong turbulence with high Rayleigh number. The chaos is numerically studied based on this model. The characteristics of the turbulence are analyzed and compared with those of the Gledzer-Ohkitani-Yamada (GOY) model. Quantities such as a mean value of total fluctuation energy, it's standard deviation, time averaged wave spectrum, probability distribution function, frequency spectrum, the maximum instantaneous Lyapunov exponent, distribution of instantaneous Lyapunov exponents, are evaluated. The dependences of these quantities on the error of numerical integration are also examined. There is not a clear correlation between the numerical accuracy and the accuracy of these quantities, since the interaction between a truncation error and an intrinsic nonlinearity of the system exists. A finding is that the maximum Lyapunov exponent is insensitive to a truncation error. (c) 1999 American Institute of Physics.     

The order of chaos on a Bianchi-IX cosmological model

The purpose of this paper is to analyze the chaotic behavior that can arise on a type-IX cosmological model using methods from dynamic systems theory and symbolic dynamics. Specifically, instead of the Belinski-Khalatnikov-Lifschitz model, we use the iterates of a monotonously increasing map of the circle with a discontinuity, and for the Hamiltonean dynamics of Misner's Mixmaster model we introduce the iterates of a noninvertible map. An equivalence between these two models can easily be brought upon by translating them in symbolic-dynamical terms. The resulting symbolic orbits can be inserted in an ordered tree structure set, and so we can present an effective counting and referentation of all period orbits.     

Control of spatiotemporal chaos: A study with an autocatalytic reaction-diffusion system

The characterization of chaotic spatiotemporal dynamics has been studied for a representative nonlinear autocatalytic reaction mechanism coupled with diffusion. This has been carried out by an analysis of the Lyapunov spectrum in spatiallylocalised regions. The linear scaling relationships observed in the invariant measures as a function of thesub-system size have been utilized to assess the controllability, stability and synchronization properties of the chaotic dynamics. The dynamical synchronization properties of this high-dimensional system has been analyzed using suitable Lyapunov functionals. The possibility of controlling spatiotemporal chaos for relevant objectives using available noisy scalar time-series data with simultaneous self-adaptation of the control parameter(s) has also been discussed.     

Unstable periodic orbits and noise in chaos computing

Different methods to utilize the rich library of patterns and behaviors of a chaotic system have been proposed for doing computation or communication. Since a chaotic system is intrinsically unstable and its nearby orbits diverge exponentially from each other, special attention needs to be paid to the robustness against noise of chaos-based approaches to computation. In this paper unstable periodic orbits, which form the skeleton of any chaotic system, are employed to build a model for the chaotic system to measure the sensitivity of each orbit to noise, and to select the orbits whose symbolic representations are relatively robust against the existence of noise. Furthermore, since unstable periodic orbits are extractable from time series, periodic orbit-based models can be extracted from time series too. Chaos computing can be and has been implemented on different platforms, including biological systems. In biology noise is always present; as a result having a clear model for the effects of noise on any given biological implementation has profound importance. Also, since in biology it is hard to obtain exact dynamical equations of the system under study, the time series techniques we introduce here are of critical importance.