Nonmarkovian dynamics of stochastic differential equations with quadratic noise

We consider a stochastic differential equation with a quadratic nonlinearity in the noise. We derive equations for the steady state probability density and joint probability distribution valid beyond a markovian approximation. We do not assume that the strength of the random term is small. The equations are derived for the case of an Ornstein-Uhlenbeck noise and also for a dichotomic noise. A comparison is made. We discuss some examples for which correlation functions and the associated relaxation times are calculated.     

Optimal signal-to-noise ratio in stochastic time-delayed bistable systems

We study the optimal signal-to-noise ratio in a stochastic time-delayed bistable system.By using the small delay approximation, we transform the time-delayed system intostochastic nondelayed differential equations to obtain the analytic expressions of thesignal-to-noise ratio in different mechanisms. In the valid range of small delayapproximation, we compare the peak values of signal-to-noise ratio curves and obtain theoptimal signal-to-noise ratio. From the results, we find that the interplay of time delayand noise has a great influence on time-delayed bistable systems.     

Exponential decay and scaling laws in noisy chaotic scattering

In this Letter we present a numerical study of the effect of noise on a chaotic scattering problem in open Hamiltonian systems. We use the second order Heun method for stochastic differential equations in order to integrate the equations of motion of a two-dimensional flow with additive white Gaussian noise. We use as a prototype model the paradigmatic Hénon-Heiles Hamiltonian with weak dissipation which is a well-known example of a system with escapes. We study the behavior of the scattering particles in the scattering region, finding an abrupt change of the decay law from algebraic to exponential due to the effects of noise. Moreover, we find a linear scaling law between the coefficient of the exponential law and the intensity of noise. These results are of a general nature in the sense that the same behavior appears when we choose as a model a two-dimensional discrete map with uniform noise (bounded in a particular interval and zero otherwise), showing the validity of the algorithm used. We believe the results of this work be useful for a better understanding of chaotic scattering in more realistic situations, where noise is presented.     

Langevin description of markovian integro-differential master equations

For a given master equation of a discontinuous irreversible Markov process, we present the derivation of stochastically equivalent Langevin equations in which the noise is either multiplicative white generalized Poisson noise or a spectrum of multiplicative white Poisson noise. In order to achieve this goal, we introduce two new stochastic integrals of the Ito type, which provide the corresponding interpretation of the Langevin equations. The relationship with other definitions for stochastic integrals is discussed. The results are elucidated by two examples of integro-master equations describing nonlinear relaxation.     

Describing intrinsic noise in Chua's circuit

In this Letter we demonstrate that intrinsic inevitable noise effects, existing in realistic experiments with electronic circuits, are properly described theoretically using a Gaussian noise. For this we integrate numerically the equations of motion from the Chua circuit using a fourth-order stochastic Runge–Kutta integrator. Periodic structures in parameter space, related to periodic motion, start to be destroyed when the noise intensity is increased and vanish at a critical intensity value, for which only chaotic motion remains. We find the appropriate noise intensity interval which satisfactorily reproduces the parameter space from the corresponding experiment and it is in remarkable agreement with the estimated experimental noise. Present achievements should be applicable to describe noise effects in a wide number of electronic circuits.     

Analysis of noise-induced bimodality in a Michaelis–Menten single-step enzymatic cycle

In this paper we study noise-induced bimodality in a specific circuit with many biological implications, namely a single-step enzymatic cycle described by Michaelis–Menten equations. We study the biological feasibility of this phenomenon, which allows for switch-like behavior in response to graded stimuli, considering a small and discrete number of molecules involved in the circuit, and we characterize the conditions necessary for it. We show that intrinsic noise (due to the stochastic character of the Master Equation approach) of a one-dimensional substrate reaction is not sufficient to achieve bimodality, then we characterize analytically the necessary conditions on enzyme number fluctuations. We implement numerically two model circuits that show bimodality over different parameter windows, that depend critically on system size as predicted by our results, providing hints about how such a phenomenon could be exploited in real biological systems.     

Probability Density and Joint Probability Density of SDE with General Nonlinear Gaussian Noise

We consider a stochastic differential equation with a general nonlinearity in Gaussian noise; With both D (fluctuation intensity) and γ (correlation time) small quantities with D/γ < 1/10, approximate equations for the probability density p(q, t) and the joint probability density p(q, t, q^t, t^p) are derived. As applications of our general equations, quadratic noise, exponential noise and triangle function noise are studied.     

3D Stochastic Primitive Equations of the Large-Scale Ocean: Global Well-Posedness and Attractors

In this paper, we consider the global well-posedness and long-time dynamics for the three-dimensional viscous primitive equations describing the large-scale oceanic motion under a random forcing, which is an additive white in time noise. We firstly prove the existence and uniqueness of global strong solutions to the initial boundary value problem for the stochastic primitive equations. Subsequently, by studying the asymptotic behavior of strong solutions, we obtain the existence of random attractors for the corresponding random dynamical system.     

Some new results of IR systems'' sensitivity equations

Typical IR systems' sensitivity equations have been established for decades, these include the NETD equation for IR imaging systems, and the acquisition and tracking range equations for IR searching and tracking systems, but they only fit the specific or simplified conditions. For example, the NETD equation can only be used for systems tested under the standard condition. In the field application, we must deal with the noise equivalent radiance difference NELD, not the NETD, and give out its explicit equation for concrete calculation. For the range equation, when the small target, often called a point source, is not strong enouph, i. e., when the target contrast is seriously decreased, we must consider the unneglectable background effect. In this paper, we summarrize some more general and valuable IR systems' sensitivity equations, which can be used for the practical engineering analysis and projection effectively.     

Shot noise in electron transport through a double quantum dot: a master equation approach

We study shot noise in tunneling current through a double quantum dot connected to two electric leads.We derive two master equations in the occupation-state basis and the eigenstate basis to describe the electron dynamics.The approach based on the occupation-state basis,despite being widely used in many previous studies,is valid only when the interdot coupling strength is much smaller than the energy difference between the two dots.In contrast,the calculations using the eigenstate basis are valid for an arbitrary interdot coupling.Using realistic model parameters,we demonstrate that the predicted currents and shot-noise properties from the two approaches are significantly different when the interdot coupling is not small.Furthermore,properties of the shot noise predicted using the eigenstate basis successfully reproduce qualitative features found in a recent experiment.     

Monte Carlo and numerical studies of coloured noise and stochastic resonance problems

We suggest two algorithms for evaluating dynamical systems described as first order differential equations under the influence of external noise represented by an Ornstein-Uhlenbeck process: a direct Monte Carlo simulation of the equation of motion and a numerical integration of the associated composite marcov equation. The two algorithms complement one another with respect to small and large noise correlation times and produce results which agree within any desired accuracy. We apply our algorithms to the problem of stochastic resonance and present the numerical results of first passage time densities, transition rates und phase histograms as functions of the system parameters frequency of the periodic force, noise correlation time and noise strength.     

Coherence and saturation properties of second-harmonic generation with a nonlinear crystal inside the laser cavity

The basic equations for second-harmonic generation including noise are derived for the case that the nonlinear crystal is put inside the laser cavity. A realistic model of a (detuned) laser with two-level atoms in single-mode operation is taken using the nonlinear theory of laser noise which describes the laser saturation effects, the phase diffusion and the intensity fluctuations. The reaction of the second-harmonic field on the fundamental field is taken into account as well as the reaction of the fundamental field on the laser. The nonlinear crystal is described by microscopic anharmonic oscillator equations (without introducing nonlinear susceptibilities by perturbation theory). The saturation of the polarization of the nonlinear medium is taken into account exactly with the only assumption that the influence of third and higher harmonics should be small. The electromagnetic field is described semiclassically by stochastic equations. In all equations, the damping is introduced simultaneously with Markoffian fluctuating forces by coupling to heatbaths. The equations are solved exactly in the stationary state without noise (the time dependent solution including noise will be presented in a subsequent paper). The most important saturation effect is a frequency shift which depends on the laser intensity.     

Can colored noise improve stochastic resonance?

The phenomenon of stochastic resonance is studied in the presence of colored noise. Several sources of colored noise are introduced and the consequences for the asymptotic time-periodic probability and the (phase-averaged) power spectrum are discussed. Based on space-time symmetry considerations, selection rules for the occurrence of-spikes in the power spectrum are derived. The effect of colored noise on the amplification of small periodic signals is studied in terms of effective, time-periodic Fokker-Planck equations: In overdamped systems driven by colored noise, we find that SR is suppressed with increasing noise color. In contrast, for colored noise induced by inertia (as well as for asymmetric dichotomic noise), one obtains an enhancement of SR. This latter result is obtained by studying the Kramers equation perturbed by a small periodic force.     

Dynamics of individuals and swarms with shot noise induced by stochastic food supply

We study the Brownian dynamics of individual particles with energy depot in two dimensions and extend the model to swarms of such particles. We assume that the elements (energy depots) are provided at discrete times with packets of chemical energy which is subsequently converted into acceleration of motion. In contrast to the mechanical white noise which is incorporated in the equations of mechanical motion and has no preferred direction, the energetic noise, as discussed in this study, is directed and it does not reverse the direction of mechanical motion. We characterize the effective noise acting on the particles and show that the stochastic energy supply may be modeled as a shot-noise driven Ornstein-Uhlenbeck process in energy which finally results in fluctuations of the velocity. We study the energy and velocity distributions for different regimes and estimate the crossover time from ballistic to diffusion motion. Further we investigate the dynamics of swarms and find a transition from translational to rotational motion depending on the rate of the shot noise.     

"Green" noise in quasistationary stochastic systems

We consider the behavior of stochastic systems driven by noise with a zero value of spectral density at zero frequency ("green" noise). For this purpose we propose the version of the Krylov-Bogoliubov averaging method to study the systems which are not stationary in the case of an external white noise. We use the ergodicity of a nonlinear random function in the method, and obtain equations for any approximation of the theory. In particular, it is shown in the first approximation that there is an effective potential to describe the averaged motion of the system. We consider a phase-locked loop as an example and show that metastable states are possible. The lifetime of these states essentially increases if the form of a green noise spectrum becomes sharper in the low-frequency region. The high stability of the system driven by green noise is confirmed by numerical simulation. It is important that the theoretical result obtained by the averaging method and the one obtained in the simulation coincide with sufficient accuracy. In conclusion, we discuss some of the unsolved green noise problems. (c) 2001 American Institute of Physics.     

Analysis of Equilibrium States of Markov Solutions to the 3D Navier-Stokes Equations Driven by Additive Noise

We prove that every Markov solution to the three dimensional Navier-Stokes equations with periodic boundary conditions driven by additive Gaussian noise is uniquely ergodic. The convergence to the (unique) invariant measure is exponentially fast. Moreover, we give a well-posedness criterion for the equations in terms of invariant measures. We also analyse the energy balance and identify the term which ensures equality in the balance.     

Moment analysis of the Hodgkin-Huxley system with additive noise

We consider a classical space-clamped Hodgkin-Huxley (HH) model neuron stimulated by a current which has a mean μ together with additive Gaussian white noise of amplitude σ. A system of 14 deterministic first-order nonlinear differential equations is derived for the first- and second-order moments (means, variances and covariances) of the voltage, V, and the subsidiary variables n, m and h. The system of equations is integrated numerically with a fourth-order Runge-Kutta method. As long as the variances as determined by these deterministic equations remain small, the latter accurately approximate the first- and second-order moments of the stochastic Hodgkin-Huxley system describing spiking neurons. On the other hand, for certain values of μ, when rhythmic spiking is inhibited by larger amplitude noise, the solutions of the moment equation strongly overestimate the moments of the voltage. A more refined analysis of the nature of such irregularities leads to precise insights about the effects of noise on the Hodgkin-Huxley system. For suitable values of μ which enable rhythmic spiking, we analyze, by numerical examples from both simulation and solutions of the moment equations, the three factors which tend to promote its cessation, namely, the increasing variance, the nature and shape of the basins of attraction of the limit cycle and stable equilibrium point and the speed of the process.     

Noise-induced bifurcations and chaos in the average motion of globally-coupled oscillators

A system of coupled master equations simplified from a model of noise-driven globally coupled bistable oscillators under periodic forcing is investigated. In the thermodynamic limit, the system is reduced to a set of two coupled differential equations. Rich bifurcations to subharmonics and chaotic motions are found. This behavior can be found only for certain intermediate noise intensities. Noise with intensities which are too small or too large will certainly spoil the bifurcations. In a system with large though finite size, the bifurcations to chaos induced by noise can still be detected to a certain degree. Received 6 April 1999 and Received in final form 1 November 1999