Show all abstracts

View Session

- Stochastic Ratchets
- Noise in Dynamical Systems: General Aspects
- Path Integrals Method
- Self-Organizing Systems
- Diffusion-Limited Reactions
- Quantum Stochastic Processes
- Periodic Driven Systems and Stochastic Resonance
- Noise-Induced Phase Transitions
- Thresholds, Signals, and Synchronization
- Noise in Systems with Delay
- Poster Session

Stochastic Ratchets

Microscopic models of Brownian ratchets

Show abstract

A hard disk microscopic ratchet is introduced and studied with molecular dynamics. The properties of the systematic motion that appears when its two compartments are at different temperatures is documented.

Applications of Brownian motors

Show abstract

Brownian motors combine asymmetry (such as a “ratchet” potential) and stochastic (thermal) motion with non-equilibrium processes to generate directed particle flow. A brief general introduction to Brownian motors is given, and the relevance of the ratchet model for biological motor proteins is highlighted. However, the impact of research on Brownian motors goes far beyond biophysics. A wealth of novel phenomena has been predicted, and some of these phenomena have been observed in areas as diverse as synthetic chemistry, bio-molecular colloids, self-organizing systems, quantum electronics, micro-fluidics, and materials science. Applications, such as novel actuators and molecular separation techniques, are evolving quickly. In the oral presentation, I will attempt to give an overview on applications of ratchets and Brownian motors. In the present paper, I give a short overview and review then a recent experimental realization of a tunneling ratchet for electrons. Such electron tunnelling ratchets can not only be used to generate particle currents, but also to pump heat. Using a realistic model, the heat pumping properties of the experimental electron ratchet are analysed.

Walking on ratchets: a model of two Brownian motors with bistable coupling

Show abstract

We propose a model for a walker moving on an asymmetric periodic ratchet potential. The walker has two 'feet' represented as two finite-size particles coupled nonlinearly through a double-well potential. In contrast to linear coupling, the bistable potential admits a richer dynamics where the ordering of the particles can alternate. The transitions between the two stable points on the bistable potential, correspond to a walking with alternating particles. In our model, each particle is acted upon by independent white noises, modeling thermal noise, and additionally we have an external time-dependent force that drives the system out of equilibrium, allowing directed transport. This force can be common colored noise, periodic deterministic driving or fluctuations on the bistable potential. In the equilibrium case, where only white noise is present, we perform a bifurcation analysis which reveals different walking patterns available for various parameter settings. Numerical simulations showed the existence of current reversals and significant changes in the effective diffusion constant and in the synchronization index. We obtained an optimal coherent transport, characterized by a maximum dimensionless ratio of the current and the effective diffusion (Peclet number), when the periodicity of the ratchet potential coincides with the equilibrium distance between the two particles.

Noise in Dynamical Systems: General Aspects

Fluctuation-dissipation-dispersion relation for slowly varying processes

Show abstract

The famous Callen-Welton formula [1] is generalized to the systems with slowly varying parameters. Using the momentum method and the time multiscale technique, developed for a nonlocal plasma in [2] it is shown that not only the dissipation but also the time derivatives of the dispersion determine the amplitude and the width of the spectrum lines of the fluctuations. In the general case, the contribution of the second one may be comparable with the first one. This contribution is affected by a new nonlocal dispersive term which is not related to Joule dissipation and appears because of an additional phase shift between the force and response of the system. The general formalism is illustrated by applications to several particular types of system. The influence of the dispersion contributions on the quality factor of the system is discussed.

Solvability of dichotomous flows, dichotomous diffusion, and generalizations

Show abstract

We first consider the one-dimensional stochastic flow dx/dt = f(

*x*) + g(*x*) xi(*t*), where xi(*t*) is a dichotomous Markov noise. A procedure involving the algebra of the relevant differential operators is used to identify the conditions under which the integro-differential equation satisfied by the total probability density*P*(*x*,*t*) of the driven variable can be reduced to a differential equation of finite order. This systematizes the enumeration of the "solvable" cases, of which the case of linear drift and additive noise is a notable one. We then revisit the known formula for the stationary density that exists under suitable conditions in dichotomous flow, and indicate how this expression may be derived and interpreted on direct physical grounds. Finally, we consider a diffusion process driven by an*N*-level extension of dichotomous noise, and explicitly derive the higher-order partial differential equation satisfied by*P*(*x*,*t*) in this case. This multi-level noise driven diffusion is a process that interpolates between the usual extremes of dichotomous diffusion and Brownian motion. We comment on the possible use of certain algebraic techniques to solve the master equation for this generalized diffusion.
Escape times and diffusion coefficients in fluctuating potentials

Show abstract

We investigate an overdamped Brownian particle moving in: (a) a dichotomously fluctuating metastable potential; (b) a random fluctuating periodic potential. For piece-wise linear potential we obtain for case (a) the exact average lifetime and the mean first passage time as a function of the potential parameters, the noise intensity and the mean frequency of switchings of the dichotomous noise. We find noise enhanced stability (NES) in the system investigated. The parameter regions of the fluctuating potential where NES effect can be observed are analytically derived. For case (b) we consider a symmetric periodic potential modulated by white noise. We obtain for such a potential the same relationship between effective diffusion coefficient of Brownian particles and the mean first-passage time, discovered previously for fixed periodic potential (see ref. 3). The phenomenon of diffusion acceleration in comparison with free particle case has been found for arbitrary potential profile. The effective diffusion coefficients for sawtooth, sinusoidal and piecewise parabolic potentials are calculated in closed analytical form.

Different ways of stabilization of metastable states

Show abstract

Different ways to stabilize a classical particle located at metastable or unstable state include an application of a periodic field or noise. We consider discrete and space-extended double-well potentials. Exact calculations are performed for the piecewise potentials.

Effects of weak spatiotemporal noise on a bistable one-dimensional system

Show abstract

We treat analytically a model that captures several features of the
phenomenon of spatially inhomogeneous reversal of an order parameter. The model is a classical Ginzburg-Landau field theory restricted to a bounded one-dimensional spatial domain, perturbed by weak spatiotemporal noise having a flat power spectrum in time and space. Our analysis extends the Kramers theory of noise-induced transitions to the case when the system acted on by the noise has nonzero spatial extent, and the noise itself is spatially dependent. By extending the Langer-Coleman theory of the noise-induced decay of a metastable state, we determine the dependence of the activation barrier and the Kramers reversal rate prefactor on the size
of the spatial domain. As this is increased from zero and passes through a certain critical value, a transition between activation regimes occurs, at which the rate prefactor diverges. Beyond the transition, reversal preferentially takes place in a spatially inhomogeneous rather than in a homogeneous way. Transitions of this sort were not discovered by Langer or Coleman, since they treated only the infinite-volume limit. Our analysis uses higher ranscendental functions to handle the case of finite volume.
Similar transitions between activation regimes should occur in other models of metastable systems with nonzero spatial extent, perturbed by weak noise, as the size of the spatial domain is varied.

Path Integrals Method

Throwing ropes in the dark: the case of oscillating barriers

Show abstract

We present a novel path-integral method for the determination of time-dependent and time-averaged reaction rates in multidimensional, periodically driven escape problems at weak thermal noise. The so obtained general expressions are evaluated explicitly for the situation of a sinusoidally driven, damped particle with inertia moving in a metastable, piecewise parabolic potential. A comparison with data from Monte-Carlo simulations yields a very good agreement with analytic results over a wide parameter range.

Nonequilibrium distribution at finite noise intensity

Show abstract

The non-equilibrium distribution in dissipative dynamical systems with unstable limit cycle is analyzed in the next-to-leading order of the small-noise approximation of the Fokker-Planck equation. The noise-induced variations of the non-equilibrium distribution are described in terms of topological changes in the pattern of optimal paths. It is predicted that singularities in the pattern of optimal paths are shifted and cross the basin boundary in the presence of finite noise. As a result the probability distribution oscillates at the basin boundary. Theoretical predictions are in good agreement with the results of numerical solution of the Fokker-Planck equation and Monte Carlo simulations.

Escape from a chaotic attractor with fractal basin boundaries

Show abstract

We study fluctuational transitions in a discrete dynamical system
between two co-existing chaotic attractors separated by a
fractal basin boundary. It is shown that there is a generic
mechanism of fluctuational transition through a fractal boundary
determined by a hierarchy of homoclinic original saddles. The most
probable escape path from a chaotic attractors to the fractal boundary is found using both statistical analysis of fluctuational trajectories and Hamiltonian theory of fluctuations.

Self-Organizing Systems

Quantifying self-organization in cyclic cellular automata

Show abstract

Cyclic cellular automata (CCA) are models of excitable media. Started from random initial conditions, they produce several different kinds of spatial structure, depending on their control parameters. We introduce new tools from information theory that let us calculate the dynamical information content of spatial random processes. This complexity measure allows us to quantitatively determine the rate of self-organization of these cellular automata, and establish the relationship between parameter values and self-organization in CCA. The method is very general and can easily be applied to other cellular automata or even digitized experimental data.

Diffusion-Limited Reactions

Reaction-diffusion processes in scale-free networks

Show abstract

In this work we investigate the dynamics of reaction-diffusion processes on scale-free networks. Particles of two types, A and B, are randomly distributed on such a network and diffuse using random walk models by hopping to nearest neighbor nodes only. Here we treat the case where one species is immobile and the other is mobile. The immobile species acts as a trap, i.e. when particles of the other species encounter a trap node they are immediately annihilated. We numerically compute Φ(

*n*,*c*), the survival probability of mobile species at time*n*, as a function of the concentration of trap nodes,*c*. We compare our results to the mean-field result (Rosenstock approximation), and the exact result for lattices of Donsker-Varadhan. We find that for high connectivity networks and high trap concentrations the mean-field result of a simple exponential decay is also valid here. But for low connectivity networks and low c the behavior is much more complicated. We explain these trends in terms of the number of sites visited,*S*(_{n}), the system size, and the concentration of traps.
Stochastic description of traffic breakdown

Show abstract

We present a comparison of nucleation in an isothermal-isochoric container with traffic congestion on a one-lane freeway. The analysis is based, in both cases, on the probabilistic description by stochastic master equations. Further we analyze the characteristic features of traffic breakdowns. To describe this phenomenon we apply the stochastic model regarding the jam emergence to the formation of a large car cluster on the highway.

Quantum Stochastic Processes

Numerical solution methods for quantum stochastic processes

Show abstract

The study of quantum stochastic processes presents severe difficulties, both on the theory level as well as on technical grounds. The numerically exact solution remains prohibitive even today. In this paper we review and present new results for three different methods used for the modelling of quantum stochastic processes. These include a mixed quantum classical approach, semiclassical initial value representations of the quantum propagator and the reduced density matrix approach as typified by the quantum Wigner-Fokker-Planck equation. A new semiclassical initial value representation that does away with cumbersome prefactors which depend on the monodromy matrix elements but is exact for a harmonic oscillator is presented and its properties analysed. A recently proposed systematic method for improving semiclassical initial value representations is reviewed. The generalization of the Wigner-Fokker-Planck equation to stochastic processes with memory is obtained by using a novel integral equation representation.

Periodic Driven Systems and Stochastic Resonance

Critical exponents for escape of a driven particle near a bifurcation point

Show abstract

We study the rate of activated escape

*W*in periodically modulated systems close to the saddle-node bifurcation point where the metastable state disappears. The escape rate displays scaling behavior versus modulation amplitude*A*as*A*approaches the bifurcational value*A*_{c}, with 1n*W*∝(*A*_{c}-*A*)μ. For adiabatic modulation, the critical exponent is μ=3/2. Even if the modulation is slow far from the bifurcation point, the adiabatic approximation breaks down close to*A*_{c}. In the weakly nonadiabatic regime we predict a crossover to μ = 2 scaling. For higher driving frequencies, as*A*_{c}is approached there occurs another crossover, from Αμ=2 to μ=3/2. The general results are illustrated using a simple model system.
On stochastic resonance-like effect in detection

Show abstract

One of the most common characteristic of a system exhibiting stochastic resonance is the existence of a maximum in the output signal-to-noise ratio when plotted against the power of the input noise. This property is at the root of the use of stochastic resonance in detection, since it is generally admitted that performance of detection increases with the signal-to-noise ratio. We show in this paper that this statement is not always true by examining the key index of performance in detection: the probability of detection. Furthermore, when the probability of detection can be increased by an increase of the power of the noise, we address the practical problem of adding noise. We consider in particular the alpha-stable case for which addition does not change the probability density function of the noise.

Noise-Induced Phase Transitions

Macroscopic limit cycle via a noise-induced phase transition

Show abstract

Bistability generated via a noise-induced phase transition is reexamined from the view of macroscopic dynamical systems, which clarifies the role of fluctuation better than the conventional Fokker-Plank or Langevin equation approach. Using this approach, we investigated the spatially-extended systems with two degrees of freedom per site. The model systems undergo a noise-induced phase transition through a Hopf bifurcation, leading to a macroscopic limit cycle motion similar to the deterministic relaxation oscillation.

Twofold role of noise in doubly stochastic effects

Show abstract

We study nonlinear systems under two noisy sources to demonstrate the concept of doubly stochastic effects. In such effects noise plays a twofold role: first it induces a special feature in the system, and second it interplays with this feature leading to noise-induced order. For this effect one needs to optimize both noisy sources, hence we call these phenomena doubly stochastic effects. To show the generality of this approach we apply this concept to several basic noise-induced phenomena: stochastic resonance, noise-induced propagation and coherence resonance. Additionally, we discuss an application of this concept to noise-induced transitions and ratchets. In all these noise-induced effects ordering occurs due to the joint action of two noisy sources.

Coupled Brownian motors: anomalous-to-normal hysteresis transition and noise-induced limit cycle

Show abstract

We study a model consisting of

*N*nonlinear oscillators with*global periodic*coupling and*local multiplicative*and additive noises. The model was shown to undergo a nonequilibrium phase transition towards a broken-symmetry phase exhibiting noise-induced ``ratchet" behavior. Here we review some aspects leading to an "anomalous--to--normal" transition in the ratchet's hysteretic behavior and also show -as suggested by the absence of stable solutions when the load force is beyond a critical value- the existence of a limit cycle induced by both: multiplicative noise and*global periodic*coupling.Thresholds, Signals, and Synchronization

Thresholds and noise

Show abstract

Random processes acting through dynamical systems with thresholds lie at the heart of many natural and man-made phenomena. The thresholds here considered are general including not only sharp or “hard” boundaries but also a class of dynamical, nonlinear system functions some of which are themselves mediated by the noise. Processes include noise-induced transitions, postponed and advanced bifurcations, noise enhanced propagation of coherent structures, and stochastic resonance and synchronization. Examples of these processes are found in a wide range of disciplines from physics and chemistry to neuroscience and even human and animal behavior and perception. I will discuss some of these examples connecting them with their fundamental dynamical origins.

Nonrenewal spike trains generated by stochastic neuron models

Show abstract

Many of the stochastic neuron models employed in the neurobiological
literature generate renewal point processes, i.e., successive
intervals between spikes are statistically uncorrelated. Recently,
however, much experimental evidence for positive and negative
correlations in the interspike interval (ISI) sequence of real neurons
has been accumulated. It has been shown that these correlations can
have implications for neuronal functions. We study a leaky
integrate-and-fire (LIF) model with a dynamical threshold or an
adaptation current both of which lead to negative correlations. Conditions are identified where these models are equivalent. The ISI statistics, the serial correlation coefficient, and the power spectrum of the spike train, are numerically investigated for various parameter sets.

Catastrophes in locking systems driven by green noise

Show abstract

We consider a phase-locked loop for the case of an external signal with a stationary fluctuating phase. The problem reduces to the problem of a Brownian particle in a periodic potential driven by “green” noises. We numerically simulate the case in which the random phase is the Ornstain-Uhlenbeck process. The rapid irreversible transition from stationary random motion (a locked state) to a nonstationary one at a high near-constant rate (a running state) is shown to be possible for the case of the massive particle. We found that transition moments change suddenly for small variations of external parameters. We call this phenomenon the “catastrophe”. The numerical results are compared with those obtained by the Krylov-Bogoliubov averaging method. The first approximation of the method is found to be sufficiently accurate if the states coexist and the direct and backward transitions occur frequently enough.

Noise-assisted propagation of signals through the chain of level-crossing detectors

Show abstract

Noise-assisted propagation of periodic signals is investigated for one-dimensional arrays composed of one-way coupled level-crossing detectors (LCD). Analytical expressions are obtained for the signal decay length through chains and the signal decay time through rings, where noise is uncorrelated so that the signal transmission from a LCD to the neighboring one is Markovian. Recent numerical simulations for one-dimensional arrays of one-way coupled bistable oscillators are discussed in comparison to the present analytical results.

Stochastic synchronization: applications to oscillatory electroreceptors

Show abstract

Classical notion of synchronization, introduced originally for periodical self-sustained oscillators, can be extended to stochastic systems. This can be done even in the case when the characteristic times of a system are fully controlled by noise. Stochastic synchronization is then defined by imposing certain conditions to various statistical measures of the process. We review various approaches to stochastic synchronization and apply them to study synchronization in the electrosensory system of paddlefish.

The data processing inequality and stochastic resonance

Show abstract

The data processing inequality of information theory states that given random variables

*X*,*Y*and*Z*which form a Markov chain in the order*X*-->*Y*-->*Z,*then the mutual information between*X*and*Y*is greater than or equal to the mutual information between*X*and*Z*. That is*I*(*X*) >=*I*(*X*;*Z)*. In practice, this means that no more information can be obtained out of a set of data then was there to begin with, or in other words, there is a bound on how much can be accomplished with signal processing. However, in the field of stochastic resonance, it has been reported that a signal to noise ratio gain can occur in some nonlinear systems due to the addition of noise. Such an observation appears to contradict the data processing inequality. In this paper, we investigate this question by using an example model system.Noise in Systems with Delay

Anticipated synchronization in neuronal systems subject to noise

Show abstract

We report the observation of synchrony in two unidirectionally coupled (master-slave) model neurons (implemented by electronic circuits) in a noisy environment. Both neurons are subjected to the same random stimulus, and there is a recurrent inhibitory delayed connection in the slave neuron. We observe that synchrony occurs shifted tin time, such that the slave neuron anticipates, i.e., forecasts, the response of the master neuron. By incorporating the effects of unidirectional coupling, delayed feedback and common noise into models of two spiking neurons, we are able to simulate successfully the experimental observations.

Feedback coupling in dynamical systems

Show abstract

Different evolution models are considered with feedback-couplings. In particular, we study the Lotka-Volterra system under the influence of a cumulative term, the Ginzburg-Landau model with a convolution memory term and chemical rate equations with time delay. The memory leads to a modified dynamical behavior. In case of a positive coupling the generalized Lotka-Volterra system exhibits a maximum gain achieved after a finite time, but the population will die out in the long time limit. In the opposite case, the time evolution is terminated in a crash. Due to the nonlinear feedback coupling the two branches of a bistable model are controlled by the the strength and the sign of the memory. For a negative coupling the system is able to switch over between both branches of the stationary solution. The
dynamics of the system is further controlled by the initial condition. The diffusion-limited reaction is likewise studied in case the reacting entities are not available simultaneously. Whereas for an external feedback the dynamics is altered, but the stationary solution remain unchanged, a self-organized internal feedback leads to a time persistent solution.

Poster Session

Phase synchronization in noisy oscillators with nonisochronicity

Show abstract

We study the synchronization of two nonidentical oscillators with nonvanishing nonisochronicity under the presence of uncorrelated Gaussian noise. To measure the amount of synchronization we calculate the evolution of the phase difference. Without coupling both oscillators rotate with different natural frequencies. Due to the action of coupling this frequency difference is reduced until finally, at a critical coupling strength, synchronization sets in. Under the presence of uncorrelated noise the observed frequency differences is still a monotonically decreasing function of coupling strength but can never become zero due to noise induced phase slips. Here, we show that this usual picture of the transition to synchronization is strongly modified when the oscillators are nonisochronous. In this case the onset of coupling can have different effects and may enlarge or even invert the natural frequency difference of the uncoupled oscillators. Our results can be explained in terms of a noisy particle in a tilted potential.

Noise-induced transitions in overdamped systems: short times

Slanislav M. Soskin,
Valentin I. Sheka,
Tatiana L. Linnik,
et al.

Show abstract

In the problem of the activation energy for a noise-induced transition over a finite given time in an

*arbitrary*overdamped one-dimensional potential system, we find and classify*all*extremal paths and provide a simple algorithm to explicitly select which is the most probable transition path (MPTP). The activation energy is explicitly expressed in quadratures. For the transition beyond the top of the barrier, the MPTP does not possess turning points and the activation energy is a monotonously decreasing function of the transition time. For transitions between points lying on one and the same slope of the potential well, which may be relevant e.g. for the problem of the tails of the prehistory probability density, the situation is more complicated: the activation energy is a non-monotonous function of time and, most important, may possess bends corresponding to jump-wise switches in the topology of the MPTP; it can be proved also that the number of turning points in the MPTP is necessarily less than two. The prefactor is calculated numerically using the scheme suggested by Lehmann, Reimann and Hanggi, PRE 55, 419 (1998). The theory is compared with simulations.
Stochastic excitation and synchronization in coupled FitzHugh-Nagumo elements

Show abstract

We investigate theoretically and numerically the activation process in a single-out and coupled FitzHugh-Nagumo elements. Two qualitatively different types of the dependence of the mean activation time and of the mean cycling time on the coupling strength monotonic and non-monotonic have been found for identical elements. The influence of coupling strength, noise intensity and firing threshold on the synchronization regimes and its characteristics is analyzed

A Fokker-Planck description for Parrondo's games

Show abstract

We discuss in detail two recently proposed relations between the Parrondo's games and the Fokker-Planck equation describing the flashing ratchet as the overdamped motion of a particle in a potential landscape. In both cases it is possible to relate exactly the probabilities of the games to the potential in which the overdamped particle moves. We will discuss under which conditions current-less potentials correspond to fair games and vie versa.

Experimental study of a noisy dissipative-driven ring lattice with Morse interactions

Show abstract

An experimental study has been carried out on a noisy dissipative-driven ring lattice of units coupled via Morse potentials. An electronic circuit mimicking the lattice dynamics and noise sources is used. We show that inclusion of long range attractive forces facilitates clustering (at variance with the repulsive Toda ring) and van der Waals-like transition phenomena.

Mechanism of signal-to-noise ratio gain in a monostable threshold stochastic resonator

Show abstract

In the last few years, several papers have been published that reported high signal-to-noise ratio (SNR) gains in systems showing stochastic resonance. In the present work, we consider a level-crossing detector driven by a periodic pulse train plus Gaussian band-limited white noise, and provide analytical formulae for the dependence of the SNR gain on the relevant parameters of the input (the amplitude and the cut-off frequency of noise, the duty cycle of the deterministic signal and the distance between the threshold and the amplitude of the signal). Our results are valid in the input parameter range wherein high gains are expected, that is, wherein the probabilities of missing and, especially, extra output peaks are very low. We also include numerical simulation results that support the theory, along with illustrations of cases which are outside the validity of our theory.

From theory of infinitely divisible distributions to derivation of generalized master equation for Markov process

Show abstract

We show that the increment of generalized Wiener process (random process with stationary and independent increments) has the properties of a random value with infinitely divisible distribution. This enables us to write the characteristic function of increments and then to obtain the new formula for correlation of the derivative of generalized Wiener process (non-Gaussian white noise) and its arbitrary functional. IN the context of well-known functional approach to analysis of nonlinear dynamical systems based on a correlation formulae for nonlinear stochastic functionals, we apply this result for derivation of generalized Fokker-Planck equation for probability density. We demonstrate that the equation obtained takes the form of ordinary Fokker-Planck equation for Gaussian white noise and, at the same time, transforms in the fractional diffusion equation in the case of non-Gaussian white noise with stable distribution.

Drastic facilitation of the onset of global chaos due to an extremum in the dependence of eigenfrequency on energy

Show abstract

The Chirikov resonance-overlap criterion predicts the onset of global chaos if nonlinear resonances overlap in energy, which is conventionally assumed to require a non-small magnitude of perturbation. We show that, for a time-periodic perturbation, the onset of global chaos may occur at unusually

*small*magnitudes of perturbation if the unperturbed system possesses more than one separatrix. 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 separatrices of the unperturbed system. One of the most important manifestations of this effect is a drastic increase of the energy range involved into the unbounded chaotic transport in spatially periodic system driven by a rather*weak*time-periodic force, which results in turn in the drastic increase either of the dc conductivity, if the system carries an electric charge, or the escape rate, if the system is subject to noise. We develop the asymptotic theory and verify it in simulations. Various generalizations are delineated, in particular for the case of a time-independent perturbation.
Pulse propagation in a model for the photosensitive Belousov-Zhabotinsky reaction with external noise

Show abstract

We study the dynamics of excitation pulses in a modified Oregonator
model for the light-sensitive Belousov-Zhabotinsky (BZ)reaction assuming that the intensity of the applied illumination is a
spatiotemporal stochastic field with finite correlation time and correlation length. For a two-component version of the model we discuss the dependence of the pulse speed on the characteristic parameters of the noise in the framework of a small noise approximation up to the first order in the correlation time. In the full three-component model we find enhancement of coherence resonance for suitable chosen correlation time. Based on this observation, we
propose a mechanism for noise-enhanced propagation of pulse trains
in excitable media subjected to external fluctuations.

Discrete games of chance as models for continuous stochastic transport processes

Show abstract

Discrete games of chance can be used to illustrate principles of stochastic processes. For example, most readers are familiar with the use of discrete random walks to model the microscopic phenomenon of Brownian motion. We show that discrete games of chance, such as those of Parrondo and Astumian, can be used to quantitatively model stochastic transport processes. Discrete games can be used as “toy” models for pedagogic purposes but they can be much more than “toys”. In principle we could perform accurate simulations and we could reduce the errors of approximation to any desired level, provided that we were prepared to pay the computational cost. We consider some different approaches to discrete games, in the literature, and we use partial differential equations to model the particle densities inside a Brownian Ratchet. We apply a finite difference approach and obtain finite difference equations, which are equivalent to the games of Parrondo. The new games generalize Parrondo's original games, in the context of stochastic transport problems. We provide a practical method for constructing sets of discrete games, which can be used to simulate stochastic transport processes. We also attempt to place discrete games, such as those of Parrondo and Astumian, on a more sound philosophical basis.

Solution of the boundary value problem for nonlinear flows and maps

Show abstract

Fluctuational escape via an unstable limit cycle is investigated
in stochastic flows and maps. A new topological method is
suggested for analysis of the corresponding boundary value
problems when the action functional has multiple local minima
along the escape trajectories and the search for the global
minimum is otherwise impossible. The method is applied to the
analysis of the escape problem in the inverted Van der Pol
oscillator and in the Henon map. An application of this technique
to solution of the escape problem in chaotic maps with fractal
boundaries, and in maps with chaotic saddles embedded within the
basin of attraction, is discussed.

Method for detecting the signature of noise-induced structures in spatiotemporal data sets: an application to excitable media

Show abstract

We formulate mathematical tools for analyzing spatiotemporal data sets. The tools are based on nearest-neighbor considerations similar to cellular automata. One of the analysis tools allows for reconstructing the noise intensity in a data set and is an appropriate method for detecting a variety of noise-induced phenomena in spatiotemporal data. The functioning of these methods is illustrated on sample data generated with the forest fire model and with networks of nonlinear oscillators. It is seen that these methods allow the characterization of spatiotemporal stochastic resonance (STSR) in experimental data. Application of these tools to biological spatiotemporal patterns is discussed. For one specific example, the slime mold

*Dictyostelium discoideum*, it is seen, how transitions between different patterns are clearly marked by changes in the spatiotemporal observables.
Oscillatory electrochemical reactions at corroding silicon surface

Show abstract

The paper analyses the nature of chaotic and well-ordered oscillations of the anodic potential and open circuit potential of silicon immersed in aqueous electrolytes. These oscillations are observed when experimental conditions are fine tuned in what corresponds to the current flowing through the system, composition of electrolyte, its viscosity, etc. It is assumed that the oscillations are due to the accumulation of mechanical stress in the thin (50-80 nm) oxide film formed at the surface of silicon as a result of electrochemical anodic reaction. The stress is released by local etching of the oxide and its lifting-on from the Si surface. The process repeats again and again yielding long-lasting oscillations of the anodic potential value (amplitude around 1-15 V, period 20-150 s) or of the open circuit potential (several hundreds milli-volts). Along with temporal ordering of the process (oscillations of potential) there occurs a spatial ordering in the system - the surface of corroding Si sample is covered with hexagonally ordered semi-spherical cells (diameter about 700 nm). The effect is well-fit by the general phenomenology of chaos-order transitions in che4mical systems (bifurcations), strange attractors are the intrinsic features of these oscillations) and its kinetics is very similar to that of the Belousov-Zabotinsky reaction. However, oscillatory processes on the corroding Si surface are caused by quite specific physical and chemical mechanisms, which are not well understood presently. We present the microscopic model for the oscillatory behavior which involves, generation of local mechanical stress at the Si/electrolyte interface, non-linear electrochemical etching of Si, localization of the electric field at the etched surface, etc.

Scaling properties of long-range correlated noisy signals: application to financial markets

Show abstract

Long-range correlation properties of financial stochastic time
series

*y*have been investigated with the main aim to demonstrate the ability of a recently proposed method to extract the scaling parameters of a stochastic series. According to this technique, the Hurst coefficient*H*is calculated by means of the following function: EQUATION where*y*_{n}(*i*)is the moving average of*y*(*i*), defined as EQUATION the moving average window and*N*_{max}is the dimension of the stochastic series. The method is called*Detrending Moving Average Analysis*(DMA) on account of the several analogies with the well-known*Detrended Fluctuation Analysis*(DFA). The DMA technique has been widely tested on stochastic series with assigned*H*generated by suitable algorithms. It has been demonstrated that the ability of the proposed technique relies on very general grounds: the function EQUATION generates indeed a sequence of clusters with power-law distribution of amplitudes and lifetimes. In particular the exponent of the distribution of cluster lifetime varies as the fractal dimension 2 -*H*of the series, as expected on the basis of the box-counting method. In the present paper we will report on the scaling coefficients of real data series (the BOBL and DAX German future) calculated by the DMA technique.
Derivation of 1/f noise from resonances

Show abstract

We study non-interacting particles in a small subsystem which is weakly coupled to a reservoir. We show that this class of systems can be mapped into an extended form of the Friedrichs model. We derive from the Hamiltonian dynamics that the number fluctuation in a subsystem is 1/

*f*or 1/*f*^{β }noise. We show that this effect comes from the sum of resonances.
Dynamical formulation of Gaussian white noise

Show abstract

We study the connection between Hamiltonian dynamics and irreversible, stochastic equations, such as the Langevin equation. We consider a simple model of a harmonic oscillator (Brownian particle) coupled to a field (heat bath). We introduce an invertible transformation operator Λ that brings us to a new representation where dynamics is decomposed into independent Markovian components, including Brownian motion. The effects of Gaussian white noise are obtained by the non-distributive property of Λ with respect to products of dynamical variables. In this way we obtain an exact formulation of white noise effects. Our method leads to a direct link between dynamics of Poincaré nonintegrable systems, probability and stochasticity.

Dynamical model of foreign exchange markets leading to Tsallis distribution

Show abstract

We present a model of financial markets originally proposed for a turbulent flow, as a dynamic basis of its intermittent behavior. Time evolution of the price change is assumed to be described by Brownian motion in a power-law potential, where the 'temperature' fluctuates slowly. The model generally yields a fat-tailed distribution of the price change. Specifically a Tsallis distribution is obtained if the inverse temperature is χ

^{2}-distributed, which qualitatively agrees with intraday data of foreign exchange market. The so-called 'volatility', a quantity indicating the risk or activity in financial markets, corresponds to the temperature of markets and its fluctuation leads to intermittency.
Directed current in ratchets by pseudo-Gaussian white noise: a seeming paradox from a Langevin equation and an alternative master equation

Show abstract

Symmetric white Poissonian shot noise with Gaussian distributed amplitudes is shown from an overdamped Langevin equation to induce directed motion of ratchets. The noise becomes white Gaussian in the limit λ→∞, where λ is the average number of the delta pulses per unit time. The current tends to zero as 1/λ→0, which agrees with the fact that the directed motion cannot be induced by the thermal noise alone. However, a finite value of 1/λ yields a finite value of the current no matter how small the former may be. Since 1/λ cannot be zero physically, this is a seeming contradiction of the second law of thermodynamics, which originates from the Langevin equation. We discuss this point from an alternative master equation without assuming the Langevin equation.

Information essence of chaotic surface structures

Show abstract

A general phenomenological approach - a Flicker Noise Spectroscopy (FNS)- to revelation of information valuable parameters characterizing the arbitrary chaotic surfaces was develop to distinguish their patterns and describe quantitatively their functional properties. The consideration was carried out in terms of correlation lengths and additional parameters characterizing the rate of correlation links lost in the sequences of surface irregularities. The parameters are obtained by fitting the Fourier spectra and structural functions (difference moments of different orders) calculated for the digitized surface profiles using the approximations derived on the base of model representation of the profiles as the sequences of irregularities of different types (“bursts”, “jumps”, etc.). The method developed was applied to revelation of effects of a shungit filling agent in polypropylen matrix on the composite properties, revelation of hydrogen treatment effects on the cleavage surfaces of LiF monocrystals after their dissolution in water with quantitative evaluations of their anisotropy, analysis of activity of vacuum deposited porphyrins layers in a photosensibilized gnenration of singlet oxygen into gaseous phase. The approach elaborated can be used for developing the new control tools in nano-technologies, microelectronics, production of polymeric material with the specific surface properties, and others.

Influence of spatiotemporal 1/f alpha-noise on structure formation in excitable media

Show abstract

The influence of spatiotemporally correlated power-law, i.e. 1/f^\alpha$, noise on pattern formation in a two dimensional excitable medium consisting of coupled FitzHugh-Nagumo (FHN) oscillators is discussed. The signature of Spatiotemporal Stochastic Resonance (STSR) is investigated using the mutual information.
It is found that optimal noise variance for STSR is minimal, if both the spatial and temporal power spectral densities of the noise decay with a characteristic exponent of \alpha$=1. This effect is related to the band-pass frequency filtering characteristic of the FHN oscillator.

Inhomogeneity-enhanced coherence resonance in assemblies of uncoupled chaotic elements

Show abstract

We study the dynamics of assemblies of “uncoupled” identical chaotic elements under the influence of external noisy filed. It is numerically demonstrated that in the case where each chaotic element exhibits type-I intermittency, the degree of the temporal regularity of the mean-field dynamics of the system reaches a maximum at a certain optimal noise intensity. Moreover, we also report that inhomogeneous noise which drives each element partly independently enhances the coherence of the mean-field more than that of the case where all elements of the system receive a completely identical noisy input, and the degree of the coherence as a function against the degree of inhomogeneity of noise shows a convex curve. In noisy uncoupled systems, the common part of noise which drives each element can be regarded as the interaction among elements which corresponds to the coupling term in the case of coupled systems, so our finding that some degree of inhomogeneity enhances the coherence of the dynamics is not trivial.

Deterministic stochastic resonance in chaotic diffusion

Show abstract

We show deterministic stochastic resonance (DSR) in chaotic diffusion when the diffusion map is modulated by a sinusoid. In chaotic diffusion, the map parameter determines the state transition rate and the diffusion coefficient. The transition rate shows the diffusion intensity. Therefore, the parameter represents the intensity of the internal fluctuation. By this fact, increase of the parameter maximizes the response of DSR as in standard stochastic resonance (SR) where the external noise intensity optimizes the response. Sinusoidally modulated diffusion is regarded as a stochastic process whose transition rate is modulated by the sinusoid. Therefore, the transition dynamics can be approximated by a time-dependent random walk process. Using the mean transition rate function against the map parameter, we can derive the DSR response depending on the parameter. Our approach is based on the rate modulation theory for SR. Even when the diffusion map is modulated by the sinusoid and noise from an external environment, the increasing parameter can also maximize the DSR response. We can calculate the DSR response depending on the external noise intensity and the map parameter. DSR takes advantage of applications to signal detection because the system has the control parameter corresponding to the internal fluctuation intensity.

Synchronization patterns in cerebral blood flow and peripheral blood pressure under minor stroke

Show abstract

Stroke is a leading cause of death and disability in the United States. The autoregulation of cerebral blood flow that adapts to changes in systemic blood pressure is impaired after stroke. We
investigate blood flow velocities (BFV) from right and left middle cerebral arteries (MCA) and beat-to-beat blood pressure (BP) simultaneously measured from the finger, in 13 stroke and 11 healthy subjects using the mean value statistics and phase synchronization
method. We find an increase in the vascular resistance and a much stronger cross-correlation with a time lag up to 20 seconds with the instantaneous phase increment of the BFV and BP signals for the subjects with stroke compared to healthy subjects.

Stochastic processes with finite size scale invariance

Show abstract

We present a theory of stochastic processes that are finite size scale invariant. Such processes are invariant under generalized dilations that operate on bounded ranges of scales and amplitudes. We recall here the theory of deterministic finite size scale invariance, and introduce an operator called Lamperti transform that makes equivalent generalized dilations and translations. This operator is then used to defined finite size scale invariant processes as image of stationary processes. The example of the Brownian motion is presented is some details to illustrate the definitions. We further extend the theory to the case of finite size scale invariant processes with stationary increments.

Data modeling of 1/f noise sets

Show abstract

A novel method is presented for solving the inverse fractal problem for 1/f noise sets. The performance of this method is compared with classical data modeling methods. Applicability to different distributions of noise is presented, along with an overview of important applications including data and image compression.