The fractal properties of the clusters corresponding to the regions whose contour is a fractional brownian path have been extensively investigated. The clusters form a stationary sequence, which has been characterized by analyzing the length, the lifetime and the area of the single cluster. The rich fractal structure of the patterns has allowed to determine the time dependent Hurst exponent with great accuracy. We have also demonstrated that the cluster area, length and lifetime exhibit the characteristic scaling behavior of systems evolving through self-organized critical states.

We obtain the Wigner-Ville spectrum of the Wiener process with arbitrary initial conditions. Two different approaches are presented leading to the same result. The solution allows one to study the stationary part, the transient part, and the initial condition dependence.

We study the reaction front for the process A + B → C in which the reagents move subdiffusively. We propose a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. Scaling solutions to these equations are presented and compared with those of a direct numerical integration of the equations. We find that for reactants whose mean square displacement varies sublinearly with time as (r²) ~ t^γ, the scaling behaviors of the reaction front can be recovered from those of the corresponding diffusive problem with the substitution t → t^γ.

We present an analytic relation between the correlation function of dichotomous (taking two values, ± 1) noise and the probability density function (PDF) of the zero crossing interval. The relation is exact if the values of the zero crossing interval τ are uncorrelated. It is proved that when the PDF has an asymptotic form L(τ) = 1/τ^c, the power spectrum density (PSD) of the dichotomous noise becomes S(f) = 1/fβ where β = 3 - c. On the other hand it has recently been found that the PSD of the dichotomous transform of Gaussian 1/f^α noise has the form 1/f^β with the exponent β given by β = α for 0 < α < 1 and β = (α + 1)/2 for 1 < α < 2. Noting that the zero crossing interval of any time series is equal to that of its dichotomous transform, we conclude that the PDF of level-crossing intervals of Gaussian 1/f^α noise should be given by L(τ) = 1/τ^c, where c = 3 - α for 0 < α < 1 and c = (5 - α)/2 for 1 < α < 2. Recent experimental results seem to agree with the present theory when the exponent α is in the range 0.7 ⪅ α < 2 but disagrees for 0 < α ⪅ 0.7. The disagreement between the analytic and the numerical results will be discussed.

We study the distribution of resistance fluctuations of conducting thin films with different levels of internal disorder. The film is modeled as a resistor network in a steady state determined by the competition between two biased processes, breaking and recovery of the elementary resistors. The fluctuations of the film resistance are calculated by Monte Carlo simulations which are performed under different bias conditions, from the linear regime up to the threshold for electrical breakdown. Depending on the value of the external current, on the level of disorder and on the size of the system, the distribution of the resistance fluctuations can exhibit significant deviations from Gaussianity. As a general trend, a size dependent, non universal distribution is found for systems with low and intermediate disorder. However, for strongly disordered systems, close to the critical point of the conductor-insulator transition, the non-Gaussianity persists when the size is increased and the distribution of resistance fluctuations is well described by the universal Bramwell-Holdsworth-Pinton distribution.

Magnetization reversal in a ferromagnetic nanowire which is much narrower than the exchange length is believed to be accomplished through the thermally activated growth of a spatially localized nucleus, which initially occupies a small fraction of the total volume. To date, the most detailed theoretical treatments of reversal as a field-induced but noise-activated process have focused on the case of a very long ferromagnetic nanowire, i.e., a highly elongated cylindrical particle, and have yielded a reversal rate per unit length, due to an underlying assumption that the nucleus may form anywhere along the wire. But in a bounded-length (though long) cylindrical particle with flat ends, it is energetically favored for nucleation to begin at either end. We indicate how to compute analytically the energy of the critical nucleus associated with either end, i.e., the activation barrier to magnetization reversal, which governs the reversal rate in the low-temperature (Kramers) limit. Our treatment employs elliptic functions, and is partly analytic rather than numerical. We also comment on the Kramers prefactor, which for this reversal pathway does not scale linearly as the particle length increases, and tends to a constant in the low-temperature limit.

We study the initial stages of a directional solidification experiment of a mixture by means of a non-variational phase field model with fluctuations. This model does not invoke fluctuation-dissipation theorem to account for the fluctuations statistics. We devote our attention to the transient regime during which concentration gradients are building and fluctuations act to destabilize the interface. To this end we calculate both the temporally dependent growth rate of each mode and the power spectrum of the interface evolving under the effect of fluctuations. Theoretical predictions are compared to phase field simulations.

We present here a study of multiplicative-noise Stochastic Partial Differential Equations (SPDE) and their sensitivity to the stochastic interpretation (Ito or Stratonovich). We analyze both static effects such as noise-induced phase transitions and dynamical ones such as the domain growth of the spatial structures in their way towards the steady state. We discuss in which circumstances a particular choice of stochastic interpretation induces qualitative changes.

Experiments show that the amplitude of turbulent pulsation in submerged jets rises with increasing distance from the nozzle, at first slowly and then, after a certain distance, rapidly. This dependence on distance from the nozzle closely resembles the dependence of an order parameter on temperature in the case of a second-order phase transition. Following an idea introduced by Landa and Zaikin in 1996, it is suggested that the onset of turbulence is a noise-induced phase transition similar to that in a pendulum with a randomly vibrated suspension axis. The Krylov-Bogolyubov asymptotic method is used to provide an approximate description of the transition. Results obtained in this way are shown to coincide closely with experimental data. Such an approach is appropriate because the convective character of the instability means that turbulence in nonclosed flows cannot be a self-oscillatory process, as is often assumed. Rather, it must originate in the external random disturbances that are always present in real flows.

We present a comprehensive study of phase transitions in single-field extended systems that relax to a non-equilibrium global steady state. The mechanism we focus on is not the so-called Stratonovich drift but is instead similar to the one associated with noise-induced transitions a la Horsthemke-Lefever in zero-dimensional systems. As a consequence, the noise interpretation (e.g., Ito vs Stratonovich) merely shifts the phase boundaries. With the help of a mean-field approximation, we present a broad qualitative picture of the various phase diagrams that can be found in these systems.

A noise-induced signal propagation is reported in oscillatory media with FitzHugh-Nagumo dynamics which is based on a noise-induced phase transition to excitability. This transition occurs via a noise-induced suppression of self-excited oscillations, while the overall phase-space structure of the system is maintained. The noise-induced excitability enables the information transport in the originally oscillatory media. We demonstrate this new feature by the propagation of a wave front and the formation of a spiral in a two dimensional lattice. These spatio-temporal structures transport information and can be observed only in the presence of suitable amount of noise and not in the deterministic self-sustained oscillatory system. Thus we extend classes of nonlinear systems with signal transmission properties also to oscillatory systems, which demonstrate a noise-induced phase transition to excitability. Further on, the mechanism of noise-induced excitability provides the opportunity to control the information transport by noise via a triggering mechanism, i.e. the information channel is switched on in the presence of noise and switched off in its absence.

The notion of a strategy in the multi-player version of the Parrondo game is reviewed. We calculate the gain for the greedy strategy as a function of the number N of players, including exact analytic results for N < 4 and in the limit N→∞. We show that the greedy strategy is optimal for N = 1 and N = 2 but not for N = 3. In the limit N→∞ our analysis reveals a very rich behavior including the possibility of phase transitions as a function of the chosen strategy.

We analyze the stability boundaries of the convective Ginzburg-Landau equation which describes the phase transitions in moving systems. The stabiity criteria are different for convective velocities which are constant, random or varying periodically with time or coordinate. For the vortices in superconductors subjected to a magnetic field and bias current, the instability will manifest itself in the drastic change in the distribution of order and disorder state along a sample.

The goal of this paper is to discuss the link between the quantum phenomenon of Anderson localization on the one hand, and the parametric instability of classical linear oscillators with stochastic frequency on the other. We show that these two problems are closely related to each other. On the base of analytical and numerical results we predict under which conditions colored parametric noise suppresses the instability of linear oscillators.

A numerical approach based on dynamic importance sampling (DIMS) is introduced to investigate the activation problem in two-dimensional nonequilibrium systems. DIMS accelerates the simulations and allows the investigation to access noise intensities that were previously forbidden. A shift in the position of the escape path compared to a heteroclinic trajectory calculated in the limit of zero noise intensity is directly observed. A theory to account for such shifts is presented and shown to agree with the simulations for a wide range of noise intensities.

An application of the path-integral approach to an analysis of the fluctuations in complex dynamical systems is discussed. It is shown that essentially the same ideas underly recent progress in the solutions of a number of long-standing problems in complex dynamics. In particular, we consider the problems of prediction, control and inference of chaotic dynamics perturbed by noise in the framework of path-integral approach.

Synchronization is a fundamental problem in natural and artificial coupled multi-component systems. We investigate to what extent small-world couplings (extending the original local relaxational dynamics through the random links) lead to the suppression of extreme fluctuations in such systems. We use the framework of non-equilibrium surface growth to study and characterize the degree of synchronization in the system. In the absence of the random links, the surface in the steady state is "rough" (strongly de-synchronized state) and the average and the extreme height fluctuations diverge in the same power-law fashion with the system size (number of nodes). With small-world links present, the average size of the fluctuations becomes finite (synchronized state) and the extreme heights diverge only logarithmically in the large system-size limit. This latter property ensures synchronization in a practical sense in coupled multi-component autonomous systems with short-tailed noise and effective relaxation through the links. The statistics of the extreme heights is governed by the Fisher-Tippett-Gumbel distribution. We illustrate our findings through an actual synchronization problem in parallel discrete-event simulations.

The problem of estimating periodic properties of periodically non-stationary stochastic processes is studied. A recently introduced measure, the measure of periodicity (MP), of stochastic oscillations is discussed. The MP estimates the "periodicity level" of the oscillations, i.e. the ratio of the periodic to the non-periodic components of the stochastic processes. The introduced measure differs fundamentally from the traditional measure, SNR, because the MP lets us estimate the value of the oscillation period. The MP is particularly useful in systems that display stochastic synchronisation phenomenon where the ratio of the periods of the external force and the response of the studied system is m:n, where m and n are positive integer numbers. The MP is used to study synchronisation in two different systems, a bistable system and a neuronal model driven by noise and a sinusoidal signal. The dependence of MP on parameters is compared with the behaviour of the cross-correlation coefficient and the effective diffusion coefficient. The influence of asymmetry in the bistable system is also studied. In the autonomous neuronal model it is shown that the coherence resonance phenomenon is well described by the MP.

We revisit the issue of directed motion induced by zero average forces in extended systems driven by ac forces. It has been shown recently that a directed energy current appears if the ac external force, f(t), breaks the symmetry f(t) = -f(t+T/2), T being the period, if topological solitons (kinks) existed in the system. In this work, a collective coordinate approach allows us to identify the mechanism through which the width oscillation drives the kink and its relation with the mathematical symmetry conditions. Furthermore, our theory predicts, and numerical simulations confirm, that the direction of motion depends on the initial phase of the driving, while the system behaves in a ratchet-like fashion if averaging over initial conditions. Finally, the presence of noise overimposed to the ac driving does not destroy the directed motion; on the contrary, it gives rise to an activation process that increases the velocity of the motion. We conjecture that this could be a signature of resonant phenomena at larger noises.

Constructive effects of noise have been well studied in spatially extended systems. In most of these studies, the media are static, reaction-diffusion type, and the constructive effects are a consequence of the interplay between local excitation due to noise perturbation and propagation of excitation due to diffusion. Many chemical or biological processes occur in a fluid environment with mixing. In this paper, we investigate the interplay among noise, excitability, diffusion and mixing in excitable media advected by a chaotic flow, in a 2D Fitz Hugh-Nagumo model described by a set of reaction-advection-diffusion equations. Without stirring, noise can only generate non-coherent excited patches of the static media. In the presence of stirring, we observe three dynamical and pattern formation regimes: (1) Non-coherent excitation, when mixing is not strong enough to achieve synchronization of independent excitations developed at different locations; (2) Coherent global excitation, when the noise-induced perturbation propagates by mixing and generates a synchronized excitation of the whole domain; and (3) Homogenization, when strong stirring dilutes quickly those noise-induced local excitations. In the presence of an external sub-threshold periodic forcing, the period of the noise-sustained oscillations can be locked by the forcing period with different ratios. Our results may be verified in experiments and find applications in population dynamics of oceanic ecological systems.

We studied two non-dynamical stochastic resonators, the level-crossing detector (LCD) and the Schmitt trigger, driven by a periodic pulse train plus 1/f^κ-type coloured noises, and we have examined the dependence of the SNR gain maxima on the spectral exponent κ of the random excitation. We have found, in accordance with what previous studies predict for the output SNR in non-dynamical systems, that the correlation only degrades the SNR gain: greater noise amplitudes are required for the gain to peak if we increase the spectral exponent. We have observed that the two different kinds of SNR gains we used, the narrow-band and the wide-band gain, describe the behaviour of these systems rather differently: while the maximum of the wide-band gain decreases monotonically with the spectral exponent κ, the narrow-band gain is optimal at a certain κ. We have also surveyed how the value of the optimal κ depends on the frequency conditions.

Suprathreshold Stochastic Resonance (SSR) is a recently discovered form of stochastic resonance that occurs in populations of neuron-like devices. A key feature of SSR is that all devices in the population possess identical threshold nonlinearities. It has previously been shown that information transmission through such a system is optimized by nonzero internal noise. It is also clear that it is desirable for the brain to transfer information in an energy efficient manner. In this paper we discuss the energy efficient maximization of information transmission for the case of variable thresholds and constraints imposed on the energy available to the system, as well as minimization of energy for the case of a fixed information rate. We aim to demonstrate that under certain conditions, the SSR configuration of all devices having identical thresholds is optimal. The novel feature of this work is that optimization is performed by finding the optimal threshold settings for the population of devices, which is equivalent to solving a noisy optimal quantization problem.

Characterization of noise-induced phase synchronization (NIPS) in a vertical-cavity surface-emitting laser (VCSEL) is reported. An optimal amount of white, gaussian noise induces an effective phase entrainment between the polarized laser emission and the periodic pump modulation. The phenomenon is characterized through suitable indicators as the average frequency of the output signal and the diffusion coefficient of the relative phase. Their values are roughly independent on different waveforms of periodic input, provided that a simple condition for the amplitudes is satisfied. The experimental results are compared with a Langevin model, Monte-Carlo simulations and analytical solutions of a Master equation for the phase dynamics.

We report on stochastic resonance in an array of four unidirectionally coupled Schmitt triggers driven by global noise and a spatiotemporal modulation. By introducing phase shifts in the drives of these bistable electronic triggers we were able to control the amplification of the periodic input signal, which is a characteristic of stochastic resonance. For phase shifts allowing for periodic boundary conditions, we found array-enhanced stochastic resonance. Moreover, evidence for spatiotemporal stochastic multiresonance was obtained, where the array exhibits more than one maximum of amplification. We attribute these phenomena to a competition between the external drive and the coupling of an array element with its neighbor.

We study an extended system that without noise shows a spatially homogeneous state, but when submitted to an adequate multiplicative noise, some "noise-induced patterns" arise. The stochastic resonance between these structures is investigated theoretically, and the knowledge of the exact nonequilibrium potential allows us to obtain the output signal-to-noise ratio. In agreement with previous studies, its maximum is predicted in the symmetric case for which both stable attractors have the same nonequilibrium potential value.

Many diseases, such as childhood diseases, dengue fever, and West Nile virus, appear to oscillate randomly as a function of seasonal environmental or social changes. Such oscillations appear to have a chaotic bursting character, although it is still uncertain how much is due to random fluctuations. Such bursting in the presence of noise is also observed in driven lasers. In this talk, I will show how noise can excite random outbreaks in simple models of seasonally driven outbreaks, as well as lasers. The models for both population dynamics will be shown to share the same class of underlying topology, which plays a major role in the cause of observed stochastic bursting.

In this paper we study the sidebranching development of solidifying dendrites both experimentally and by numerical integration of a phase-field model with a noise term. Our results support the idea that sidebranching is originated through the selective amplification of natural noise at the tip. The initial stages turn to be of crucial importance in the selection of the final noisy shape. However, our results suggest that after the stochastic initial disturbance, a deterministic mechanism dominates the growth and screening-off process of sidebranching.

We analyze the characteristic features of jam formation on a circular one-lane road. We have applied an optimal velocity model including stochastic noise, where cars are treated as moving and interacting particles. The motion of N cars is described by the system of 2N stochastic differential equations with multiplicative white noise. Our system of cars behaves in qualitatively different ways depending on the values of control parameters c (dimensionless density), b (sensitivity parameter characterising the fastness of relaxation), and α (dimensionless noise intensity). In analogy to the gas-liquid phase transition in supersaturated vapour at low enough temperatures, we observe three different regimes of traffic flow at small enough values of b < b_cr. There is the free flow regime (like gaseous phase) at small densities of cars, the coexistence of a jam and free flow (like liquid and gas) at intermediate densities, and homogeneous dense traffic (like liquid phase) at large densities. The transition from free flow to congested traffic occurs when the homogeneous solution becomes unstable and evolves into the limit cycle. The opposite process takes place at a different density, so that we have a hysteresis effect and phase transition of the first order. A phase transition of second order, characterised by critical exponents, takes place at a certain critical density c = c_cr. Inclusion of the stochastic noise allows us to calculate the distribution of headway distances and time headways between the successive cars, as well as the distribution of jam (car cluster) sizes in a congested traffic.

The behaviour of stock markets has been modelled actively during recent years. In some cases the market is modelled as a whole through the time series analysis of some indexes. But the market is made of companies whose time series can be studied independently. In this paper we have paid attention to the characterization of correlations and covariance among different companies in order to extract information about the market. We have used a statistical technique based on the analysis of the covariance matrix between the indexes of companies. When taking into account the sampling uncertainties and high order cumulants of index probability distribution, it is possible to classify automatically trends or clusters of companies in order to identify some independent “submarkets.” The method is applied to some finance data sets coming from the Spanish financial market IBEX35.

This paper explores fluctuations and noise in various facets of cancer development. The three areas of particular focus are the stochastic progression of cells to cancer, fluctuations of the tumor size during treatment, and noise in cancer cell signalling. We explore the stochastic dynamics of tumor growth and response to treatment using a Markov model, and fluctutions in tumor size in response to treatment using partial differential equations. We also explore noise within gene networks in cancer cells, and noise in inter-cell signalling.

We focus on the stochastic description of the stock price dynamics. Thereby we concentrate on the Heston model and the Hull-White model. We derive the stationary probability density distribution of the variance of both models in the case of zero correlation coefficient. These distributions are used to calculate solutions for the logarithmic returns of the stock price for short time lags. Furthermore we compare the received results with numerical simulations. In addition we apply the solutions of both models to the German tick-by-tick Dax data. The data are from May 1996 to December 2001. We use the probability density distributions of the logarithmic returns, calculated out of the data, and fit these distributions to the theoretical distributions.

A new method of inferencing of coupled stochastic nonlinear oscillators is described. The technique does not require extensive global optimization, provides optimal compensation for noise-induced errors and is robust in a broad range of dynamical models. We illustrate the main ideas of the technique by inferencing a model of five globally and locally coupled noisy oscillators. Specific modifications of the technique for inferencing hidden degrees of freedom of coupled nonlinear oscillators is discussed in the context of physiological applications.

We present a heuristic theory describing the recently discovered [PRL 90, 17410 (2003)] drastic facilitation of the onset of global chaos in time periodically perturbed Hamiltonian systems possessing two or more separatrices: the minimal magnitude of the perturbation which provides chaos in the whole energy range between the separatrices possesses deep spikes as a function of the perturbation frequency. The relevant scenario is the combination of the overlap in the phase space between resonances of the same order and their overlap in energy with chaotic layers associated with the separatrices. We discuss a few applications: the increase of the dc conductivity of the 2D electron gas in a magnetic superlattice, the decrease of the activation barrier in the problem of noise-induced escape, the facilitation of the stochastic web formation in time periodically perturbed oscillators.

We introduce a continuum approach to studying the lifetimes of monovalent metal nanowires. By modelling the thermal fluctuations of cylindrical nanowires through the use of stochastic Ginzburg-Landau classical field theories, we construct a self-consistent approach to the fluctuation-induced "necking" of nanowires. Our theory provides quantitative estimates of the lifetimes for alkali metal nanowires in the conductance range 10 < G/G₀ < 100 (where G₀ = 2e/h is the conductance quantum), and allows us to account for qualitative differences in the conductance histograms of alkali vs. noble metal nanowires.

Two specialized algorithms for the numerical integration of the equations of motion of a Brownian walker obeying detailed balance are introduced. The algorithms become symplectic in the appropriate limits, and reproduce the equilibrium distributions to some higher order in the integration time step. Comparisons with other existing integration schemes are carried out both for static and dynamical quantities.

We analyse theoretically and experimentally the residence time distribution of bistable systems in the presence of noise and time-delayed feedback. The feedback provides a memory mechanism for the system which leads to non-Markovian dynamics. We demonstrate and explain various non-exponential features of the residence time distribution using a two-state as well as a continuous model. The experimental results are based on a Schmitt Trigger where the feedback is provided by a computer generated delay loop and on a semiconductor laser with opto-electronic feedback.

We extend a recently developed relation between the master equation describing the Parrondo's games and the formalism of the Fokker-Planck equation to the case in which the games are modified with the introduction of "self-transition probabilities." This accounts for the possibility that the capital can neither increase nor decrease during a game. Using this exact relation, we obtain expressions for the stationary probability and current (games gain) in terms of an effective potential. We also demonstrate that the expressions obtained are nothing but a discretised version of the equivalent expressions in terms of the solution of the Fokker-Planck equation with multiplicative noise.

If a cosinusoidal (harmonic) force acts on a locking dynamic system, the system may be synchronized not only to this force but to its harmonics also. This effect refers to parametric phenomena and has been studied in many systems. If a stationary random process is, for example, a signal phase, the angular vibration of the ring laser, or the phase of a periodic potential, the additive external noise in the systems is green one when its spectral density is zero at zero frequency. In this work we suppose that the green noise is the time derivative of a Ornstein-Uhlenbeck process and the locking system is the ring laser. We use a Krylov-Bogoluibov averaging method to find an effective potential which describes the system response near the locking regions located at the frequencies of the high harmonics of the force. We show that the effective Shapiro steps are well apparent but narrower then in the case of zero noise. The step size is given by the function of the external noise intensity and the harmonic force amplitude. This result is compared with that of numerical simulation accomplished by the predictor-corrector algorithm. The coincidence is excellent even if the green noise is strong enough. We also made the numeric simulation for the case of white noise. This showed that the parametric synchronization regions become ill-defined even for a very small white noise intensity.

In this paper the effect of noise on activity of a neural network of diffusively coupled excitatory-inhibitory cells with time delay is analyzed. We distinguish between global and local noise studying the different types of spatiotemporal behaviors observed when a space-time correlated noise is applied to the system. The results show a synchronization of the network in aperiodic behavior with the help of tiny noisy perturbations on the external signal depending on the spatial correlation of noise and the coupling coefficients.

Random matrix ensembles (RME) of quantum statistical Hamiltonians, Gaussian random matrix ensembles (GRME) and Ginibre random matrix ensembles (Ginibre RME), found applications in study of following quantum statistical systems: molecular systems, nuclear systems, disordered materials, random Ising spin systems, quantum chaotic systems, two-dimensional quantum gravity and two-dimensional electron systems (Wigner-Dyson electrostatic analogy). Quantum statistical information functional is defined as negentropy (minus entropy). Entropy is neginformation (minus information). The distribution functions for the various random matrix ensembles are derived from the maximum entropy principle.

The possibility of measuring weak noise in nonlinear systems on the basis of the phenomenon of prebifurcation noise amplification is proposed. This phenomenon is shortly outlined with special emphasis on the transition from linear regime to the regime of nonlinear saturation of fluctuation amplification. Estimates of the fluctuation variance are obtained both for the linear (away from the bifurcation threshold) and for the nonlinear regime (in the vicinity of the bifurcation threshold). These estimates have proved to be efficient for two simple bifurcation models: period doubling bifurcation and bifurcation of spontaneous symmetry breaking. Theoretical estimates have proved to be in good agreement with the results of numerical simulation. It is shown, that in the saturation regime, fluctuation variance is proportional to the square root of external noise variance, whereas in linear regime, fluctuation variance is proportional to noise variance. The approach to weak noise measuring is based on comparison of maximal fluctuation variance at the bifurcation threshold with variance away from that threshold. The applicability of this approach is limited by the necessity to perform rather long-term observations.

We study the dynamics of a pair of two uncoupled identical chaotic elements driven by common noise. When each element exhibits type-I intermittency, we observe that two uncoupled elements synchronize each other after a finite time of interval for a certain range of the noise intensity. In order to clarify the mechanism of this noise-induced synchronization phenomenon, we focus on the effect of external noise on the fluctuation of the local expansion rate of orbits to perturbations of type-I intermittency. It is found that the probability that the finite-time Liapunov exponent (FTLE) takes a negative value may increase due to the introduction of noise, whereas the Liapunov exponent itself remains to be positive. We show that this noise-induced enhancement of fluctuation may cause the synchronization and also discuss the relation between the statistical properties of relaxation process of the synchronization and the fluctuation properties of FTLE in terms of the thermodynamic formalism.

Noise processes are often modelled as stochastic processes. We have used a multivariate method based on the application of Principal Component Analysis (PCA) in order to classify different spatial-temporal structures taken as noise. When the structures have a correlation in time, a parameter distinguishing between fast and slow dynamics appears naturally. We have found this parameter in previous contributions with a different meaning depending on the context. Especially interesting is the application to the characterization of 1/f noise. In this paper we have extended the method in order to apply it to different kind of systems exhibiting, for example, self-organizing properties or brownian motion. One goal is trying to define a criterion to distinguish between fast and slow dynamics parameters. Finally, a statistical analysis is made in order to find the conditions for the application of the method to a wide range of different systems.

We have found that the width of the chaotic layer in case of a low frequency of perturbation is significantly larger than that predicted by the conventional heuristic criteria. The underlying reason is a randomness of the sign of the energy change if the motion occurs in the vicinity of the separatrix, namely in the energy band of the order of the separatrix split. Moreover, in case when the separatrix is unbounded while the time-periodic perturbation is of a dipole type, we have found the dramatic widening of the chaotic layer as the perturbation frequency decreases. This occurs because the system in the slowly rocked Hamiltonian is accelerated during long periods of time and, therefore manages to gain large energy. The acceleration periods alternate with the braking ones so that the system returns to the vicinity of the separatrix where it may be trapped for some time in one of the regions of the phase space inside the separatrix loops. We have developed the explicit adiabatic theory which nicely agrees with simulations.

We study the energy fluctuations in 3D Ising model near the phase transition point. Specific heat is a relevant quantity which is directly related to the mean squared amplitude of the energy fluctuations in the system. We have made extensive Monte Carlo simulations in 3D Ising model to clarify the character of the singularity of the specific heat C_v based on the finite-size scaling of its maximal values C_v^max depending on the linear size of the lattice L. An original iterative method has been used which automatically finds the pseudocritical temperature corresponding to the maximum of C_v. The simulations made up to L ≤ 128 with application of the Wolff's cluster algorithm allowed us to verify the possible power-like as well as logarithmic singularity of the specific heat predicted by different theoretical treatments. The most challenging and interesting result we have obtained is that the finite-size scaling of C_v^max in 3D Ising model is well described by a logarithmic rather than power-like ansatz, just like in 2D case. Another modification of our iterative method has been considered to estimate the critical coupling of 3D Ising model from the Binder cumulant data within L ε [96; 384]. Furthermore, the critical exponent β has been evaluated from the simulated magnetization data within the range of reduced temperatures t ≥ 0.000086 and system sizes L ≤ 410.

We show that when paced chaotic oscillators, which can be flows or maps, are coupled appropriately, phase slips produced by each oscillator are synchronized. If a periodically driven chaotic oscillator strays from a phase synchronization region, the phase difference between the oscillator and the pacer jumps intermittently by 2π, which is called a phase slip. When two sinusoidally forced Roessler oscillators are coupled appropriately, phase slips produced by the two oscillators occur simultaneously, that is the phase slips are synchronized. We also show that if the coupled oscillator deviates slightly from the slip synchronization region, a portion of the simultaneous phase slips are desynchronized, namely only one of the two oscillators produces a phase slip. Such phenomena as synchronized phase slips and partially synchronized phase slips can be reproduced by a coupled map system. We investigate some statistical properties and dynamical structures of the phenomena by investigating the coupled map system.

We investigate an overdamped Brownian motion in symmetric periodic potential switching by Markovian dichotomous noise between two configurations. Second configuration differs from the first one by the half of spatial period displacement when the maxima of a potential profile become the minima and vice versa. We establish a validity of the formula for effective diffusion constant previously obtained in the case of fixed periodic potentials and, thus, reduce the problem to calculation of mean first-passage times (MFPTs). At the same time the MFPT in this formula should be substituted for the semi-sum of MFPTs regarding two initial configurations of potential profile. A set of equations for MFPTs in flipping potentials can be solved for the sawtooth periodic potential. As a result, after cumbersome calculations we obtain the exact complex expression for effective diffusion coefficient of Brownian particles in such a medium which is valid for arbitrary mean frequency of potential switchings. We detect the acceleration of diffusion in comparison with the case of fixed sawtooth potential profile as it was demonstrated for a symmetric periodic potential modulating by external Gaussian white noise.

We study the relation between the discrete-time version of the flashing ratchet known as Parrondo's games and a compression technique used very recently with thermal ratchets for evaluating the transfer of information -- negentropy -- between the Brownian particle and the source of fluctuations. We present some results concerning different versions of Parrondo's games showing, for each case, a good qualitative agreement between the gain and the inverse of the entropy.

Parrondo's paradox states that there are losing gambling games which, when being combined stochastically or in a suitable deterministic way, give rise to winning games. Here we investigate the probabilistic background. We show how the properties of the equilibrium distributions of the Markov chains under consideration give rise to the paradoxical behavior, and we provide methods how to find the best a priori strategies.

We consider the dynamics of an attractor neural network of a finite size N, trained with two patterns, which is subject to the action of an external stimulus (or field). This field drives the system to one of the patterns or to another for alternating intervals of duration T. It is observed that, for not too strong fields, the response of the network to the evolving field is optimal for some finite size, decreasing for smaller or larger systems. This is the so-called system size resonance, already reported for the Ising model. The explanation of this results is related to the phenomenon of stochastic resonance.

We show that deterministic stochastic resonance (DSR) can be enhanced by coupling of chaotic oscillators. We study periodic-forced chaotic oscillators coupled to each other. One oscillator phase and periodic force phase are synchronized with each other when the force strength is larger than a critical value. When we set the force strength below the critical value, the phase synchronization occasionally fails. We can observe a quick jump in phase difference between one oscillator and periodic force. In this study, we focus on this phase slip and consider one forced chaotic oscillator as a resonator using the phase slips. When we consider coupled resonators, the coupling strength becomes a bifurcation parameter that has a critical point between asynchronous and synchronous phase slip state. Increases in coupling strength leads to a higher degree of phase slip synchronization. The coupling helps to synchronize the slips with a cooperative effect. Therefore, it can enhance the coincident response to the signal. Optimal coupling strength maximizes the resonance response. This enhancement provides some advantages for signal detection applications using DSR. It is considered that intrinsic fluctuations are important for information processing in biological system. This coupled system may be useful for a model study of neural information processing.

There exist a common belief that random sequences are produced from very complicated phenomena, making impossible the construction of accurate mathematical models. It has been recently shown that under specific conditions the exact solutions to some chaotic functions can be generalized to produce truly random sequences. This establishes a transition from chaos to stochastic dynamics. Using this result we can obtain explicit output expressions for stochastic dynamics problems like those posed by stochastic resonant nonlinear systems. We show that in this kind of systems the phenomenon of noise-induced disorder-order can be more efficiently described with an information-theory approach through the determination of a parameter that measures the complexity of the dynamics. The Stochastic Resonance (SR) is just an example of the principal phenomenon wherein the complex stochastic dynamics is converted into a simpler one. Then we show the opposite phenomenon whereby the autonomous (without input noise) transition from chaotic order to stochastic disorder is achieved by a static non-invertible non-linearity. We build electronic systems to simulate and produce experimentally all these phenomena.

The phenomenon of noise enhanced signal transfer, or stochastic resonance, has been observed in many nonlinear systems such as neurons and ion channels. Initial studies of stochastic resonance focused on systems driven by a periodic signal, and hence used a signal to noise ratio based measure for comparison between the input and output of the system. It has been pointed out that for the more general case of aperiodic signals other measures are required, such as cross-correlation or information theoretical tools. In this paper we present simulation results obtained in a model neural system driven by a broadband aperiodic signal, and producing a signal imitating neural spikes. The system is analyzed by using cross-spectral measures.

The relation between diffusion and conductivity is established for a case of diffusing particle moved by means of Levy hops (flights). It is shown that due to of an unusual character of Levy flight a particle velocity depends on electrical field in a nonlinear way in arbitrary weak fields.

We propose a harmonic velocity noise, i.e., time derivative of the harmonic noise, which has a broadband-passing feature and a vanishing Markovian friction. If this noise is regarded as a thermal one, which will lead to nonvanishing mean velocity and ballistic diffusion of a free particle in long time limit. The effective temperature of the system coupled to such a structured heat bath represented by harmonic velocity noise is introduced, and it is shown that the fluctuation-dissipation theorem still holds when the particle at the initial time is in thermal equilibrium. When applied to a titled periodic potential, we find that the Brownian particle does not have a stationary state in the phase space, however, a constant acceleration is obtained.

Random walk process was investigated with PDF of random time intervals similar to fractional exponential law on small times and to regular exponential law on long times. Generalized fractional Kolmogorov-Feller equation was derived for such kind of process. Asymptotics of its PDF in the long time limit and for intermediate times were found. They obey standard diffusion law or fractional diffusion law respectively. Exact solutions of mentioned equations were numerically calculated, demonstrating crossover of fractional diffusion law into the linear one.

Epidemiological processes are studied within a recently proposed social network model using the susceptible-infected-refractory dynamics (SIR) of an epidemic. Within the network model, a population of individuals may be characterized by H independent hierarchies or dimensions, each of which consists of groupings of individuals into layers of subgroups. Detailed numerical simulations reveals that for H > 1, the global spreading results regardless of the degree of homophily α of the individuals forming a social circle. For H = 1, a transition from a global to a local spread occurs as the population becomes decomposed into increasingly homophilous groups. Multiple dimensions in classifying individuals (nodes) thus make a society (computer network) highly susceptible to large scale outbreaks of infectious diseases (viruses). The SIR-model can be extended by the inclusion of waiting times resulting in modified distribution function of the recovered.

The effects of a variable amount of random dilution of the synaptic couplings in Q-Ising multi-state neural networks with Hebbian learning are examined. A fraction of the couplings is explicitly allowed to be anti-Hebbian. Random dilution represents the dying or pruning of synapses and, hence, a static disruption of the learning process which can be considered as a form of multiplicative noise in the learning rule. Both parallel and sequential updating of the neurons can be treated. Symmetric dilution in the statics of the network is studied using the mean-field theory approach of statistical mechanics. General dilution, including asymmetric pruning of the couplings, is examined using the generating functional (path integral) approach of disordered systems. It is shown that random dilution acts as additive gaussian noise in the Hebbian learning rule with a mean zero and a variance depending on the connectivity of the network and on the symmetry. Furthermore, a scaling factor appears that essentially measures the average amount of anti-Hebbian couplings.

In this short note we propose an approach for calculating option prices in financial markets in the framework of path integrals. We review various techniques from engineering and physics applied to the theory of financial risks. We explore how the path integral methods may be used to study financial markets quantitatively and we also suggest a method in calculating transition probabilities for option pricing using real data in that framework.

Models of stock price fluctuations based on simple random walks do not agree with empirical stock price data. We point out an analogy with motion in a one-dimensional random field which generalizes the stock dynamics to include random dependence on the current price in a natural way. Results of an analytically tractable limit are presented, demonstrating that some of the characteristics of real stock data may be reproduced by such models. Shortcomings of the model are noted, and a numerical simulation method for extension beyond the analytically tractable case is presented.

This paper investigates the different effects of chaotic switching on Parrondo's games, as compared to random and periodic switching. The rate of winning of Parrondo's games with chaotic switching depends on coefficient(s) defining the chaotic generator, initial conditions of the chaotic sequence and the proportion of Game A played. Maximum rate of winning can be obtained with all the above mentioned factors properly set, and this occurs when chaotic switching approaches periodic behavior.