International Journal of Bifurcation and Chaos

Our scientific odyssey through the theory of 1-D cellular automata is enriched by the definition of quasi-ergodicity, a new empirical property discovered by analyzing the time-1 return maps of local rules. Quasi-ergodicity plays a key role in the classification of rules into six groups: in fact, it is an exclusive characteristic of complex and hyper Bernoulli-shift rules. Besides introducing quasi-ergodicity, this paper answers several questions posed in the previous chapters of our quest. To start with, we offer a rigorous explanation of the fractal behavior of the time-1 characteristic functions, finding the equations that describe this phenomenon. Then, we propose a classification of rules according to the presence of Isles of Eden, and prove that only 28 local rules out of 256 do not have any of them; this result sheds light on the importance of Isles of Eden. A section of this paper is devoted to the characterization of Bernoulli basin-tree diagrams through modular arithmetic; the formulas obtained allow us to shorten drastically the number of cases to take into consideration during numerical simulations. Last but not least, we present some theorems about additive rules, including an analytical explanation of their scale-free property.

After a short excursion from the discovery of Brownian motion to the Richardson "law of four thirds" in turbulent diffusion, the article introduces the Lévy flight superdiffusion as a self-similar Lévy process. The condition of self-similarity converts the infinitely divisible characteristic function of the Lévy process into a stable characteristic function of the Lévy motion. The Lévy motion generalizes the Brownian motion on the base of the α-stable distributions theory and fractional order derivatives. Further development on this idea lies on the generalization of the Langevin equation with a non-Gaussian white noise source and the use of functional approach. This leads to the Kolmogorov's equation for arbitrary Markovian processes. As a particular case we obtain the fractional Fokker–Planck equation for Lévy flights. Some results concerning stationary probability distributions of Lévy motion in symmetric smooth monostable potentials, and a general expression to calculate the nonlinear relaxation time in barrier crossing problems are derived. Finally, we discuss the results on the same characteristics and barrier crossing problems with Lévy flights, recently obtained by different approaches.

A spherical growth process controlled by velocity fluctuations of particles of a saturated solution is investigated. Velocity fluctuations are modeled by a Gaussian random field. The interface evolution is determined by a Langevin-type equation with a multiplicative random field, which in the case of the quasi-homogeneous random Gaussian field is equivalent to Fokker–Planck dynamics. We analyze numerically the Fokker–Planck equation and compare growth kinetics in the case of noisy (i.e. space-independent) fluctuations. It is shown that for a large class of spatially correlated velocity fluctuations, the growth kinetics is universal, i.e. it does not depend on the details of statistics of fluctuations.

The statistical properties of sequences of Waiting Times generated by Non-Homogeneous Poisson Processes are investigated by means of Renewal Aging, i.e. a statistical analysis being able to detect the presence of genuine renewal events with non-Poisson statistics. The Renewal Aging features of two non-homogeneous Poisson models with different prescriptions of the rate modulation are compared. Both prescriptions are essentially periodic. In the first model, the rate is a totally deterministic and smooth harmonic function. In the second model, the rate modulation is deterministic and smooth almost everywhere, except in some singular points occurring with a periodicity affected by weak random fluctuations.

The main parameter of both models is the modulation speed, defined as the ratio between the modulation period and the average Waiting Time. This allows to distinguish between slow, intermediate and fast modulation of the rate. As expected, in the very slow modulation case both models have a strongly reduced Renewal Aging, practically zero, because the analysis is affected by local clusters of Waiting Times with the same Poisson rate. In the intermediate range of modulation speeds, both models show non-zero Renewal Aging and the behavior of the two models tends to be more different as the modulation speed decreases.

Going towards the fast modulation regime, the behavior of the two models becomes completely different from one another. Model I does not show a definite global Renewal Aging. On the time scale of the modulation period, local oscillations arise in the Probability Density Function and an apparent oscillating aging is also observed. However, this form of aging is not the manifestation of genuine renewal non-Poisson events, but it is related to the local oscillations. On the contrary, Model II displays Renewal Aging features in agreement with a homogeneous Renewal non-Poisson Process. It is argued that this is related to the singular points in the rate modulation of Model II, which, due to their random occurrence, could be identified with genuine critical non-Poisson events. The approach presented here could have interesting applications in problems involving dichotomous noise and, in particular, in the statistical characterization of ON–OFF fluorescence intermittency experimentally observed in complex systems, such as nano-crystals and single bio-molecules stimulated by laser fields.

The chaotic motion of a ray path in a deep water acoustic waveguide with internal-wave-induced fluctuations of the sound speed is investigated. A statistical approach for the description of chaotic rays is discussed. The behavior of ray trajectories is studied using Hamiltonian formalism expressed in terms of action-angle variables. It is shown that the range dependence of the action variable of chaotic ray can be approximated by a random Wiener process. On the basis of this result, analytical expressions for probability density functions of ray parameters are derived. Distributions of coordinates, momenta (grazing angles), and actions of sound rays are evaluated. Numerical simulation shows that statistical characteristics of ray parameters weakly depend on a particular realization of random perturbation giving rise to ray chaos.

The paper is related to studies of a turbulent medium by the sounding method with the interferometric reception. We performed a theoretical analysis of a multiplicative interferometer response to noise quasi-monochromatic emission, disturbed by a turbulent water stream. The effect of turbulent pulsations of the stream velocity on output interferometer signal was considered. Spectral analysis of the output interferometer signal allowed to perform the diagnostics of the propagation media. Particularly, it was shown, that the interferometer power spectrum carries information on the spatial spectrum of fluctuations of cavity bubbles concentration and on the average stream velocity. The theoretical evidence has been compared with the results of interferometric experiments on ultrasonic sounding of the water stream inside a pipe.

We illustrate two distinct approaches to the Mittag–Leffler relaxation, as a mathematical expression suitable for the interpretation of real data produced by complex systems, and especially those of physiological interest. The first approach is based on interpreting the fluctuation–dissipation process under study as obtained via Subordination to the Ordinary Fluctuation–Dissipation (SOFD) process. The second approach rests on the Generalized Langevin Equation (GLE). We prove that in the real cases of truncated time series the two theories generate a survival probability in the form of a stretched exponential, and that this property makes it hard to assess if a given time series obeys the GLE or the SOFD prescription. Some conjectures are made on the possibility of distinguishing the GLE from the SOFD predictions through the analysis of a single time series.

Fractional generalizations of the Poisson process and branching Furry process are considered. The link between characteristics of the processes, fractional differential equations and Lèvy stable densities are discussed and used for the construction of the Monte Carlo algorithm for simulation of random waiting times in fractional processes. Numerical calculations are performed and limit distributions of the normalized variable Z = N/〈N〉 are found for both processes.

We study passive admixture spreading in the atmosphere considering regular (with the wind) and stochastic motion of the particles. The probability distribution of velocities on a boundary of a given spatial region (detector) is found for particles of a passive admixture. The connection between this probability distribution and the solution of the classical problem on the probability distribution of the coordinates and velocity of a Brownian particle at a fixed moment is determined. The dependence of the obtained probability density on the relation among the parameters of the problem under consideration is discussed. Some specific examples are presented. Potential applications of the results are in ecological and meteorological problems.

We propose a new autonomous dynamical system of dimension N = 4 that demonstrates the regime of stable two-frequency motions. It is shown that the system of two generators of quasiperiodic oscillations with symmetric coupling can realize motions on four-dimensional torus with resonant structures on it in the form of three- and two-dimensional torus. We show that with increase of noise intensity when the dimension of torus is higher it is destroyed faster.

A model of working memory system in the form of a network of neuron-like units is considered. The units are stimulated by a periodic subthreshold stimulus and interact via inhibitory feedback. The information storage in the system occurs in the form of periodic clusters of neural activity. Several sequentially firing clusters can coexist in the network simultaneously corresponding to several stored images. The influence of noise on cluster dynamics and noise resistance of the system are studied. Optimal parameters corresponding to maximal noise resistance of the system were found.

The problem regarding shapes of bispectral peak of the triplet representing the triharmonic signal is discussed. This triplet can be used as both standard diagnostic signal for a bispectral analyzers and a simplest element of complex informational patterns of bispectrally organized signals. The problem of analyzing and measuring the parameters of bispectral peak is equally fundamental as the well-known problem concerning the shape and width of spectral line of quasi-monochromatic oscillations. The sectional area of bispectral peak restricts the limiting informational volume of complex bispectral patterns. Two models of an actual triplet with frequency fluctuations are analyzed. Universal bispectral-peak shapes are found for a triplet with proportional frequencies for the limiting cases of extremely slow and extremely fast frequency fluctuations. The phenomenon of bispectral-peak superlocalization is discovered for quasi-static fluctuations and analyzed for the most realistic model of 1/f frequency fluctuations. The experiment investigating the bispectral peak of the real high-frequency synthesizer is presented.

In this article we derive rough formulas for the probability distribution functions of sampled higher-order spectra of poly-harmonic signals corrupted by Gaussian noise. These formulas provide a basis for analyzing the performance detection algorithms for finite-sample length.

The diffusion process on the surface with equivalent potential wells is investigated. These wells are considered deep enough to keep the particle for longer than the correlation time. The quasi-equilibrium Boltzmann distribution is applied inside the well to calculate the escape time. This time characterizes jumps from a well to another one. The coefficient of such a diffusion is calculated as a function of temperature, shape, depth and localization of potential wells and the mass of the particles.

We briefly review the statistical properties of the escape times, or hitting times, for stock price returns by using different models which describe the stock market evolution. We compare the probability function (PF) of these escape times with that obtained from real market data. Afterwards we analyze in detail the effect both of noise and different initial conditions on the escape time in a market model with stochastic volatility and a cubic nonlinearity. For this model, we compare the PF of the stock price returns, the PF of the volatility and the return correlation with the same statistical characteristics obtained from real market data.

Stochastic quantization provides a connection between quantum field theory and statistical mechanics, with applications especially in gauge field theories. Euclidean quantum field theory is viewed as the equilibrium limit of a statistical system coupled to a thermal reservoir. Nonlinear phenomena in stochastic quantization arise when employing nonlinear Brownian motion as an underlying stochastic process. We discuss a novel formulation of the Higgs mechanism in QED.

The problem of information extraction from discrete stochastic time series, produced with some finite sampling frequency, using flicker-noise spectroscopy, a general framework for information extraction based on the analysis of the correlation links between signal irregularities and formulated for continuous signals, is discussed. It is shown that the mathematical notions of Dirac δ- and Heaviside θ-functions used in the analysis of continuous signals may be interpreted as high-frequency and low-frequency stochastic components, respectively, in the case of discrete series. The analysis of electroencephalogram measurements for a teenager with schizophrenic symptoms at two different sampling frequencies demonstrates that the "power spectrum" and difference moment contain different information in the case of discrete signals, which was formally proven for continuous signals. The sampling interval itself is suggested as an additional parameter that should be included in general parameterization procedures for real signals.

Here we discuss the remarkable role of the statistical memory effects in the human brain functioning at photosensitive epilepsy (PSE). We have analyzed three independent statistical memory quantifiers for the magnetoencephalographic (MEG) signals. These quantifiers reflect the dynamical characteristics of neuromagnetic brain responses to a flickering stimulus of different color combinations. Results for a group of control subjects are contrasted with those from a patient with PSE. The emergence of the strong memory and the transition to a regular and robust regime of chaotic behavior of the signals in separate areas is characteristic for a patient with PSE versus a healthy brain.

The role of noise for a single neuron and for an ensemble of mutually coupled neurons is investigated. For a single element we show that an increase in noise intensity in the regime of irregular firing enhances the coherence of the neuronal response. For this regime of spiking a study of the collective behavior for a chain composed of n mutually coupled Hodgkin–Huxley elements that are subjected simultaneously to sinusoidal drive and to a common random forcing is performed. The conditions for complete synchronization for this case are examined both within the framework of a theoretical approach based on the connection graph stability method and through numerical simulation.

Quantum space-time nonlocality, i.e. retardation of the interaction between an electron and its own radiation field at distances about the Compton wavelength, is established. By taking into account a finite variance of electron-coordinate increment in the intrinsic coordinate system, the radiative damping coefficient is obtained as a divergence-free function of frequency that is not subject to the well-known paradoxes of the classical theory of radiative damping. A relation between radiative damping and the electromagnetic mass of the electron is found.

We present the analysis of the mean switching time and its standard deviation of short overdamped Josephson junctions, driven by a direct current and a periodic signal. The effect of noise enhanced stability is investigated. It is shown that fluctuations may both decrease and increase the switching time.

The phenomena of stochastic resonance is studied in overdamped nonlinear monostable systems driven by a periodic signal and Gaussian white noise. It is shown that the signal power amplification as a function of input noise intensity can be different depending on nonlinearity: it can monotonically grow, decrease and it can reach a maximum at certain value of the noise intensity. Nevertheless, the output signal to noise ratio is shown to be always a decreasing function of input noise intensity.

It is shown that Berry model may be successfully applied to study noise wave field properties. The correlation function of the waves stream was obtained and its probability density was analyzed in the case of simple distributions of the wave vector parameters.

In this paper we have analyzed the dynamics of redistribution of the radiative point defects in gallium arsenide during relaxation of the local heating in a semiconductor sample, generated by a radiative particle in the initial stage of stabilization of clusters of defects. We calculated: (i) the maximum heating temperature of a semiconductor sample, and (ii) the point defect diffusion coefficient corresponding to the maximum temperature.

The mixing property is characterized by the metric entropy that is introduced by Kolmogorov for dynamical systems. The Kolmogorov entropy is infinite for a stochastic system. In this work, a relative metric entropy is considered. The relative metric entropy allows to estimate the level of mixing in noisy dynamical systems. An algorithm for calculating the relative metric entropy is described and examples of the metric entropy estimation are provided for certain chaotic systems with various noise intensities. The results are compared to the entropy estimation given by the positive Lyapunov exponents.

The transient dynamics of long overlap Josephson junctions in the frame of the sine-Gordon model with a white noise source is investigated. The effect of noise delayed decay is observed for the case of overdamped sine-Gordon equation. It is shown that this noise induced effect, in the range of small noise intensities, vanishes for junction lengths greater than several Josephson penetration lengths.

By means of a thermodynamic approach we analyze billiards in the form of the Lorentz gas with the open horizon. For periodic and stochastic oscillations of the scatterers, the average velocity of the particle ensemble as a function of time is analytically obtained. It is shown that the consequence of such oscillations is Fermi acceleration which is larger for periodic oscillations. The described results do not depend on the size of scatterers and their position. Only the property of the horizon openness is necessary. It is found that the developed thermodynamic approach is in a very good agreement with the results of the direct numerical simulations at which the corresponding billiard map is used.

The study of the noise induced effects on the dynamics of a chain molecule crossing a potential barrier, in the presence of a metastable state, is presented. A two-dimensional stochastic version of the Rouse model for a flexible polymer has been adopted to mimic the molecular dynamics and to take into account the interactions between adjacent monomers. We obtain a nonmonotonic behavior of the mean first passage time and its standard deviation, of the polymer center of inertia, with the noise intensity. These findings reveal a noise induced effect on the mean crossing time. The role of the polymer length is also investigated.

Spectra and autocorrelation functions are studied in a system of two coupled self-sustained chaotic oscillators. We show that spectral and correlation characteristics are connected with partial effective phase diffusion coefficients. The evolution of spectra and diffusion coefficients are considered from the asynchronous regime to the regime of synchronous chaos. We draw the analogy between spectral characteristics of coupled chaotic oscillators and noisy coupled periodic oscillators.