on the stochastic pendulum with ornstein uhlenbeck noise we study a frictionless pendulum subject to multiplicative random noise because of destructive interference between the angular displacement of the system and the noise term the energy fluctuations are reduced when the noise has a non zero correlation time we derive the long time behaviour of the pendulum in the case of ornstein uhlenbeck noise by a recursive adiabatic elimination procedure an analytical expression for the asymptotic probability distribution function of the energy is obtained and the results agree with numerical simulations lastly we compare our method with other approximation schemes

generalized two step maruyama methods for stochastic differential equations abstract in this paper we propose generalized two step maruyama methods for solving ito stochastic differential equations numerical analysis concerning consistency convergence and numerical stability in the mean square sense is presented we derive sufficient and necessary conditions for linear mean square stability of the generalized two step maruyama methods we compare the stability region of the generalized two step maruyama methods of adams type with that of the corresponding two step maruyama methods of adams type and show that our proposed methods have better linear mean square stability a numerical example is given to confirm our theoretical results

anharmonic oscillator driven by additive ornstein uhlenbeck noise we present an analytical study of a nonlinear oscillator subject to an additive ornstein uhlenbeck noise known results are mainly perturbative and are restricted to the large dissipation limit obtained by neglecting the inertial term or to a quasi white noise i e a noise with vanishingly small correlation time here in contrast we study the small dissipation case we retain the inertial term and consider a noise with finite correlation time our analysis is non perturbative and based on a recursive adiabatic elimination scheme a reduced effective langevin dynamics for the slow action variable is obtained after averaging out the fast angular variable in the conservative case we show that the physical observables grow algebraically with time and calculate the associated anomalous scaling exponents and generalized diffusion constants in the case of small dissipation we derive an analytic expression of the stationary probability distribution function pdf which differs from the canonical boltzmann gibbs distribution our results are in excellent agreement with numerical simulations

stochastic averaging of strongly non linear oscillators under combined harmonic and white noise excitations abstract a stochastic averaging procedure of strongly non linear oscillators subject to external and or parametric excitations of both harmonic and white noise forces is developed by using the so called generalized harmonic functions the procedure is applied to a duffing oscillator with hardening stiffness under both external harmonic excitation and external and parametric excitations of white noises the averaged fokker planck kolmogrov equation is solved by using the path integration method based on the stationary joint probability density of amplitude and phase obtained by using the stochastic averaging and the path integration the stochastic jump of the duffing oscillator under combined harmonic and white noise excitations and its bifurcation as the system parameters frequency ratio strength of the non linearity amplitude of harmonic excitation and intensity of white noise change are examined for the first time

on a complex beam beam interaction model with random forcing abstract in recent years many studies have been devoted to complex differential equations cde which appear in very important applications in physics and engineering this paper aims to investigate one such cde containing a random forcing term z o 2 z e 2 g z e 2 f z z p o t n t where z t x t iy t a complex function i 1 n t is a broad band process with zero mean and e2 and small real parameters in particular we use eq to model the interaction between two colliding beams in particle accelerators setting g z z f z z z z 2 and p o t cos o t and extend the work we had started in an earlier publication mahmoud physica a 216 1995 445 we apply the stochastic averaging method to derive a fokker planck kolmogorov equation for this equation and obtain analytically the exact stationary probability density function and the first and second moments in the amplitude of the solutions numerical simulations are carried out to compare with the theoretical ones and excellent agreement is found

responses of strongly non linear oscillator parametrically excited by random narrow band noise the principal response of a strongly van der pol duffing oscillator subjected to parametric random narrow band excitation is investigated the technique of modified lindstedt poincare mlp method is used to transform the strongly non linear system to a weak one by introducing a new expansion parameter and then the multiple scales method is applied to determine the modulation equations for amplitude and phase of the response of the system the effect of damping detuning and bandwidth on the dynamic behaviors such as stability bifurcation are examined by computing the maximum lyapunov exponent analytically also the numerical simulation is carried out to verify the analytical results and random jump phenomenon may be observed in the region of the parameters of the system the excellent agreement between theoretical results and numerical ones can be found immediately and so the present method in this paper is applicable to solve strongly non linear problems

nonlinear response with dichotomous noise dichotomous noise appears in a wide variety of physical and mathematical models it has escaped attention that the standard results for the long time properties cannot be applied when unstable fixed points are crossed in the asymptotic regime we show how calculations have to be modified to deal with these cases and present as a first application full analytic results for hypersensitive transport

random generation of stochastic area integrals the authors describe a method of random generation of the integrals a 1 2 t t h int t t h int t s dw 1 r dw 2 s int t t h int t s dw 2 r dw 1 s together with the increments w 1 t h w 1 t and w 2 t h w 2 t of a two dimensional brownian path w 1 t w 2 t the method chosen is based on marsaglia s rectangle wedge tail method generalised to higher dimensions the motivation is the need for a numerical scheme for simulation of strong solutions of general multidimensional stochastic differential equations with an order of convergence o h where h is the stepsize previously no method has obtained an order of convergence better than o sqrt h in the general case

response of nonlinear systems in probability domain using stochastic averaging determination of probability density function pdf of the response for strongly nonlinear single degree of freedom system subjected to both multiplicative and additive random excitations using stochastic averaging technique is faced with few difficulties firstly the size of excitations should be small such that the response of the system converges weakly to a markov process secondly the excitations should preferably be broad banded so that the analytical results are reasonably accurate and finally the nonlinear functions should be integrable if closed formed expressions for the result are desired the above issues are examined in the paper with a view to show a limiting values of the size parameter of excitations for which stochastic averaging technique can be applied to obtain reasonable estimates of pdf and mean square value of response b the effect of the nature of the frequency contents of excitation on the response and c the use of the stochastic averaging method employing generalized harmonic functions for nonintegrable functions representing the dynamic system it is shown that the stochastic averaging procedure provides results which compare very well with those obtained from simulation analysis not only for wide band excitations but also for narrow band excitations for a wide range of the size parameter of excitation further the procedure can be easily implemented for highly irregular functions by employing a numerical scheme with fft

influence of the noise spectrum on the anomalous diffusion in a stochastic system we use an effective markovian description to study the long time behavior of a nonlinear second order langevin equation with a gaussian noise when dissipation is neglected the energy of the system grows as with time a power law with an anomalous scaling exponent that depends both on the confining potential and on the high frequency distribution of the noise the asymptotic expression of the probability distribution function in phase space is calculated analytically the results are extended to the case where small dissipative effects are taken into account

on response of nonlinear oscillators with random frequency of excitation abstract some types of nonlinear oscillators for which the frequency of excitation is stochastic are investigated the paper consists of two parts in the first part equations of motion are linearized with the aid of stochastic averaging differential equations for the mean and variance of the process are obtained and integrated numerically this approach is applicable for weakly nonlinear oscillators the case of strong nonlinearity is considered in the second part making use of computer simulation a number of stochastic realizations of the process are computed the stochastic process is characterized by the mean and standard deviation of these realizations calculations have been carried out for the duffing ueda and van der pol equations and for forced vibrations of a pendulum these calculations show that if attractors exist then the deterministic vibrations which may be chaotic turn regular by adding noise and the motion terminates in a stable fixed point or on a limit cycle

numerical analysis of stochastic schemes in geophysics we present and study the stability convergence and order of convergence of a numerical scheme used in geophysics namely the stochastic version of a deterministic implicit leapfrog scheme which has been developed for the approximation of the so called barotropic vorticity model two other schemes which might be useful in the context of geophysical applications are also introduced and discussed

on diffusion equations for dynamical systems driven by noise a unified formalism is presented to study hamiltonian linear systems driven by noise with this formalism the phase averaging approximation valid at weak noise is easily performed already known results are straightforwardly recovered and new ones are obtained after introducing this formalism on the exactly solvable one degree of freedom problem with uncorrelated noise one studies the corresponding exponentially correlated case the validity of the approximate results thus obtained is considered by investigating the systematic weak disorder expansion beyond the quasilinear approximation in particular it is argued that this expansion behaves uniformly for weak and large correlation time the two degrees of freedom problem is completely solved at the low disorder approximation and this result is applied to the two channel anderson localization problem the invariant measure and the two positive lyapunov exponents are computed at all coupling between the channels for systems withn degree of freedom the phase averaging leads to a fokker planck equation for the measure in action space describing the system however it is argued that it is not solvable except in a special case which is explicitly displayed and solved nevertheless in the large n limit it is possible to compute the largest lyapunov exponent moreover generalized lyapunov exponents are calculated in this limit and they do not exhibit a dispersion in particular log log 1 where is the energy of the system and where the brackets denote averaging over the noise on the other hand it is possible to compute at weak noise the sum of all the positive lyapunov exponents taking into account all these results allows more insight on the whole spectrum of lyapunov exponents

statistical linearization of the duffing oscillator under non gaussian external excitation the duffing oscillator under external non gaussian excitations is investigated by means of statistical linearization the input process is modelled as a polynomial of a gaussian process or as a renewal driven impulse process four criteria of statistical linearization are considered the interarrival times of the renewal process are distributed according to a pearson type iii law predictions of the stationary variance are compared with monte carlo simulations

maxwellian gas undergoing a stationary poiseuille flow in a pipe abstract the hierarchy of moment equations derived from the nonlinear boltzmann equation is solved for a gas of maxwell molecules undergoing a stationary poiseuille flow induced by an external force in a pipe the solution is obtained as a perturbation expansion in powers of the force through third order a critical comparison is done between the navier stokes theory and the predictions obtained from the boltzmann equation for the profiles of the hydrodynamic quantities and their fluxes the navier stokes description fails to first order and especially to second order in the force thus the hydrostatic pressure is not uniform the temperature profile exhibits a non monotonic behavior a longitudinal component of the flux exists in the absence of longitudinal thermal gradient and normal stress differences are present on the other hand comparison with the bhatnagar gross krook model kinetic equation shows that the latter is able to capture the correct functional dependence of the fields although the numerical values of the coefficients are in general between 0 38 and 1 38 times the boltzmann values a short comparison with the results corresponding to the planar poiseuille flow is also carried out

stochastic averaging of strongly non linear oscillators under bounded noise excitation abstract a stochastic averaging method for strongly non linear oscillators under external and or parametric excitation of bounded noise is proposed by using the so called generalized harmonics functions the method is then applied to study the primary resonance of duffing oscillator with hardening spring under external excitation of bounded noise the stochastic jump and its bifurcation of the system are observed and explained by using the stationary probability density of amplitude and phase subsequently the method is applied to study the dynamical instability and parametric resonance of duffing oscillator with hardening spring under parametric excitation of bounded noise the primary unstable region is delineated by evaluating the lyapunov exponent of linearized system and the response and jump of non linear system around the unstable region are examined by using the sample functions and stationary probability density of amplitude and phase

anomalous diffusion in nonlinear oscillators with multiplicative noise the time asymptotic behavior of undamped nonlinear oscillators with a random frequency is investigated analytically and numerically we find that averaged quantities of physical interest such as the oscillator s mechanical energy root mean square position and velocity grow algebraically with time the scaling exponents and associated generalized diffusion constants are calculated when the oscillator s potential energy grows as a power of its position u x approximately x 2n for x infinity correlated noise yields anomalous diffusion exponents equal to half the value found for white noise

the influence of noise on the logistic model the behavior of the logistic system which is generated by the functionf x ax 1 x changes in an interesting way if it is perturbed by external noise it turns out that the chaotic behavior which was predicted by li and yorke for orbits of period 3 becomes visible and that a sequence of mergence transitions occurs at the critical parameter the change of the invariant probability density and the lyapunov exponents are examined numerically the power spectrum for the period 3 orbit for different fluctuations is calculated and a recursion formula for the time evolution of the probability density is presented as a discrete time analog of a chapman kolmogorov equation

moment lyapunov exponents of a two dimensional system under both harmonic and white noise parametric excitations the moment lyapunov exponents and the lyapunov exponents of a 2d system under both harmonic and white noise excitations are studied the moment lyapunov exponents and the lyapunov exponents are important characteristics determining the moment and almost sure stability of a stochastic dynamical system the eigenvalue problem governing the moment lyapunov exponent is established a singular perturbation method is applied to solve the eigenvalue problem to obtain second order weak noise expansions of the moment lyapunov exponents the influence of the white noise excitation on the parametric resonance due to the harmonic excitation is investigated

principal response of duffing oscillator to combined deterministic and narrow band random parametric excitation abstract the principal resonance of a duffing oscillator to combined deterministic and narrow band random parametric excitations is investigated in particular the case in which the parametric terms share close frequencies is examined the method of multiple scales is used to determine the equations of modulation of amplitude and phase the behavior stability and bifurcation of steady state response are studied by means of qualitative analyses jumps are shown to occur if the random excitation is small the effects of damping detuning and magnitudes of deterministic and narrow band parametric excitations are analyzed the theoretical analyses are verified by numerical results

stability of a stochastic two dimensional non hamiltonian system we study the top lyapunov exponent of the response of a two dimensional non hamiltonian system driven by additive white noise the origin is not a fixed point for the system however there is an invariant measure for the one point motion of the system in this paper we consider the stability of the two point motion by khasminskii s method of linearization along trajectories the specific system we consider is the third order truncated normal form of the unfolding of a hopf bifurcation we show that in the small noise limit the top lyapunov exponent always approaches zero from below and is thus negative for noise sufficiently small we also show that there exist open sets of parameters for which the top lyapunov exponent is positive thus the two point motion can be either stable or unstable while the stationary density that describes the one point motion always exists

numerical methods for stochastic differential equations stochastic differential equations sde s play an important role in physics but existing numerical methods for solving such equations are of low accuracy and poor stability a general strategy for developing accurate and efficient schemes for solving stochastic equations is outlined here high order numerical methods are developed for the integration of stochastic differential equations with strong solutions we demonstrate the accuracy of the resulting integration schemes by computing the errors in approximate solutions for sde s which have known exact solutions

non newtonian poiseuille flow of a gas in a pipe abstract the bhatnagar gross krook kinetic model of the boltzmann equation is solved for the steady cylindrical poiseuille flow fed by a constant gravity field the solution is obtained as a perturbation expansion in powers of the field through fourth order and for a general class of repulsive potentials the results which are hardly sensitive to the interaction potential suggest that the expansion is only asymptotic a critical comparison with the profiles predicted by the navier stokes equations shows that the latter fail over distances comparable to the mean free path in particular while the navier stokes description predicts a monotonically decreasing temperature as one moves apart from the cylinder axis the kinetic theory description shows that the temperature has a local minimum at the axis and reaches a maximum value at a distance of the order of the mean free path within that distance the radial heat flows from the colder to the hotter points in contrast to what is expected from the fourier law furthermore a longitudinal component of the heat flux exists in the absence of gradients along the longitudinal direction non newtonian effects such as a non uniform hydrostatic pressure and normal stress differences are also present

resonant enhancement of anderson localization analytical approach we study localization properties of the eigenstates and wave transport in a one dimensional system consisting of a set of barriers and or wells of fixed thickness and random heights the inherent peculiarity of the system resulting in the enhanced anderson localization is the presence of the resonances emerging due to the coherent interaction of the waves reflected from the interfaces between the wells and or barriers our theoretical approach allows to derive the localization length in infinite samples both out of the resonances and close to them we examine how the transport properties of finite samples can be described in terms of this length it is shown that the analytical expressions obtained by standard methods for continuous random potentials can be used in our discrete model in spite of the presence of resonances that cannot be described by conventional theories all our results are illustrated with numerical data manifesting an excellent agreement with the theory

averaging method using generalized harmonic functions for strongly non linear oscillators abstract an averaging method in which generalized harmonic functions are used is applied to study the approximate solutions of the strongly non linear oscillators g x e x and g x ef x t where g x is an arbitrary non linear function the method gives the approximate solutions in terms of generalized harmonic functions these functions are also periodic and are exact solutions of strongly non linear differential equations some phenomena considered include limit cycles of strongly non linear autonomous oscillators and steady state response of strongly non linear oscillators subject to weak harmonic excitation the procedure is simple and easy to apply

stochastic jump and bifurcation of a duffing oscillator under narrow band excitation abstract the variance of the displacement the probability densities of the displacement velocity and amplitude and the joint probability density of the displacement and velocity of the stationary response of a duffing oscillator to narrow band gaussian random excitation for a number of combinations of the parameters of the oscillator and excitation obtained from digital simulation are presented based on these results the stochastic jump and bifurcation of the system are examined it is shown that for each combination of the parameters all the statistics of the stationary response are unique and independent of initial conditions in a certain domain of the parameter space there are two more probable motions in the stationary response and jumps may occur the jump is a transition from one more probable motion to another or vice versa outside of this domain the stationary response is either nearly gaussian or like a diffused limit cycle as the parameters cross the boundary of the domain the qualitative behavior of the stationary response changes and it is a special kind of bifurcation it is pointed out that some previous results and explanations on this problem are incorrect

a moment approach to non gaussian colored noise abstract the langevin system subjected to non gaussian colored noise has been discussed by using the second order moment approach with two kinds of models for generating the noise we have derived the effective differential equation de for a variable x from which the stationary probability distribution p x has been calculated with the use of the fokker planck equation the result of p x calculated by the moment method is compared to several expressions obtained by different methods such as the universal colored noise approximation ucna jung and hanggi phys rev a 35 1987 4464 and the functional integral method it has been shown that our p x is in good agreement with that of direct simulations dss we have also discussed dynamical properties of the model with an external input solving des in the moment method

on numerical approximations of forward backward stochastic differential equations a numerical method for a class of forward backward stochastic differential equations fbsdes is proposed and analyzed the method is designed around the four step scheme j douglas jr j ma and p protter ann appl probab 6 1996 pp 940 968 but with a hermite spectral method to approximate the solution to the decoupling quasi linear pde on the whole space a rigorous synthetic error analysis is carried out for a fully discretized scheme namely a first order scheme in time and a hermite spectral scheme in space of the fbsdes equally important a systematical numerical comparison is made between several schemes for the resulting decoupled forward sde including a stochastic version of the adams bashforth scheme it is shown that the stochastic version of the adams bashforth scheme coupled with the hermite spectral method leads to a convergence rate of frac 32 in time which is better than those in previously published work for the fbsdes

active disturbance rejection control approach to output feedback stabilization of a class of uncertain nonlinear systems subject to stochastic disturbance the active disturbance rejection control adrc is now considered as a powerful control strategy in dealing with large uncertainty covering unknown dynamics external disturbance and unknown part in coefficient of the control however all theoretical works up to present are limited to deterministic uncertainty in this technical note we generalize the adrc to uncertain nonlinear systems subject to external bounded stochastic disturbance described by an uncertain stochastic differential equation driven by white noise we first design an extended state observer eso that is used to estimate both state and total disturbance which includes the internal uncertain nonlinear part and the external uncertain stochastic disturbance it is shown that the resulting closed loop system is practically stable in the mean square topology the numerical experiments are carried out to illustrate effectiveness of the proposed approach

multiscaling of noise induced parametric instability we describe the statistical properties of growth rates of a linear oscillator driven by a parametric noise we show that in general the fluctuations of local lyapunov exponents are non gaussian and demonstrate multiscaling analytical calculations of the generalized lyapunov exponents are complemented with approximative and numerical results this allows us to identify the parameter range where the deviations from the gaussian statistics become important

multistep methods for sdes and their application to problems with small noise in this article the numerical approximation of solutions of ito stochastic differential equations is considered in particular for equations with a small parameter epsilon in the noise coefficient we construct stochastic linear multistep methods and develop the fundamental numerical analysis concerning their mean square consistency numerical stability in the mean square sense and mean square convergence for the special case of two step maruyama schemes we derive conditions guaranteeing their mean square consistency further for the small noise case we obtain expansions of the local error in terms of the step size and the small parameter epsilon simulation results using several explicit and implicit stochastic linear k step schemes k 1 2 illustrate the theoretical findings

on convergence of active disturbance rejection control for a class of uncertain stochastic nonlinear systems in this paper the practical mean square convergence of active disturbance rejection control for a class of uncertain stochastic nonlinear systems modelled by the ito type stochastic differential e

stability measures in metastable states with gaussian colored noise we present a study of the escape time from a metastable state of an overdamped brownian particle in the presence of colored noise generated by ornstein uhlenbeck process we analyze the role of the correlation time on the enhancement of the mean first passage time through a potential barrier and on the behavior of the mean growth rate coefficient as a function of the noise intensity we observe the noise enhanced stability effect for all the initial unstable states used and for all values of the correlation time tau c investigated we can distinguish two dynamical regimes characterized by weak and strong correlated noises depending on the value of tau c with respect to the relaxation time of the system

modified kramers formulas for the decay rate in agreement with dynamical modeling abstract accuracy of the kramers approximate formulas for the thermal decay rate of the metastable state is studied for the anharmonic shapes of the potential pocket and the barrier this is done by the comparison with the quasistationary rate resulting from the dynamical modeling disagreement between the kramers rate and the dynamical one is shown to reach 15 in the cases when much better agreement is expected corrections to the kramers formulas accounting for the higher derivatives of the potential are obtained the small parameters are the ratios of the thermal energy to the stiffnesses at the extremes of the potential the distance between the potential barrier and the absorptive border is accounted for as well this corrected kramers rate is demonstrated to agree with the dynamical rate typically within 2 probably the most interesting result is that despite the corrections are derived in the case of the overdamped brownian motion the above 2 agreement holds even in the case of medium friction

largest lyapunov exponent for second order linear systems under combined harmonic and random parametric excitations the principal resonance of a second order linear stochastic oscillator to combined harmonic and random parametric excitations is investigated the method of multiple scales is used to determine the equations of modulation of amplitude and phase the effects of damping detuning bandwidth and magnitudes of random excitation are analyzed the method of path integration is used to obtain the steady state probability density function of the system and then the largest lyapunov exponent is calculated the almost sure stability or instability of the stochastic system depends on the sign of the largest lyapunov exponent the theoretical analyses are verified by numerical results

the van kampen expansion for the fokker planck equation of a duffing oscillator excited by colored noise in rodriguez and van kampen s 1976 paper a method of extracting information from the fokker planck equation without having to solve the equation is outlined the fokker planck equation for a duffing oscillator excited by white noise is expanded about the intensity of the forcing function in weinstein and benaroya the effect of the order of expansion is investigated by carrying the expansion to a higher order the effect of varying the system parameters is also investigated all results are verified by comparison to monte carlo experiments in this paper the van kampen expansion is modified and applied to the case of a duffing oscillator excited by colored noise the effect of the correlation time is investigated again the results are compared to those of monte carlo experiments it is found that the expansion compares closely with those of the monte carlo experiments as the correlation time c is varied from 0 001 to 10 sec examination of the results reveals that the colored noise can be categorized in one of four ways 1 for tau c mathcal o 0 01 text sec the noise can be considered as white for all intents and purposes 2 for tau c mathcal o 0 1 text sec the noise can be considered white for some purposes 3 for tau c mathcal o 1 0 text sec the correlated nature of the noise must be considered in an analysis and 4 for mathcal o 1 0 text sec tau c the noise can be considered as deterministic

on exponential mean square stability of two step maruyama methods for stochastic delay differential equations we are concerned with the exponential mean square stability of two step maruyama methods for stochastic differential equations with time delay we propose a family of schemes and prove that it can maintain the exponential mean square stability of the linear stochastic delay differential equation for every step size of integral fraction of the delay in the equation numerical results for linear and nonlinear equations show that this family of two step maruyama methods exhibits better stability than previous two step maruyama methods

averaging method for quasi integrable hamiltonian systems abstract a deterministic averaging method for quasi integrable hamiltonian systems is proposed to predict the approximate response of many degree of freedom autonomous or non autonomous strongly nonlinear systems the form and dimension of the averaged equations depend on the number of degree of freedom and the number of resonant relations of the associated hamiltonian systems in non resonant case the averaged equations for n action variables or n independent integrals of motion are derived in resonant case the averaged equations for n action variables and combinations of angle variables are derived two examples are given to illustrate the application of the proposed method it is shown that the analytical results agree well with those from numerical solution even for systems with very strong nonlinearity

the asymptotic stability of a noisy non linear oscillator abstract the purpose of this work is to obtain an approximation for the top lyapunov exponent the exponential growth rate of the response of a single well kramers oscillator driven by either a multiplicative or an additive white noise process to this end we consider the equations of motion as dissipative and noisy perturbations of a two dimensional hamiltonian system a perturbation approach is used to obtain explicit expressions for the exponent in the presence of small intensity noise and small dissipation we show analytically that the top lyapunov exponent is positive and for small values of noise intensity e and dissipation e the exponent grows in proportion with e1 3

the law of the euler scheme for stochastic differential equations i convergence rate of the distribution function we study the approximation problem ofef x t byef x t n where x t is the solution of a stochastic differential equation x t n is defined by the euler discretization scheme with stept n andf is a given function for smoothf s talay and tubaro have shown that the erroref x t f x t n can be expanded in powers of 1 n which permits to construct romberg extrapolation precedures to accelerate the convergence rate here we prove that the expansion exists also whenf is only supposed measurable and bounded under an additional nondegeneracy condition of hormander type for the infinitesimal generator of x t to obtain this result we use the stochastic variations calculus in the second part of this work we will consider the density of the law ofx t n and compare it to the density of the law ofx t

finite correlation time effects in nonequilibrium phase transitions i dynamic equation and steady state properties abstract we determine an approximate renormalized equation of evolution for an arbitrary nonlinear single degree of freedom system externally driven by gaussian parametric fluctuations of finite correlation time the renormalization scheme used here gives a second order equation with a time and state dependent diffusion coefficient we are able to calculate the diffusion coefficient in closed form the steady state distribution can easily be obtained from the evolution equation we are thus able to determine the parameter dependence of the steady state distribution and in particular the influence of a correlation time of the fluctuations which does not vanish on the steady state distribution

on mean square stability of two step maruyama methods for nonlinear neutral stochastic delay differential equations the asymptotic mean square stability of two step maruyama methods is considered for nonlinear neutral stochastic differential equations with constant time delay nsddes under the one sided lipschitz condition and the linear growth condition it is proved that a family of implicit two step maruyama methods can preserve the stability of the analytic solution in mean square sense numerical results for both a nonlinear nsdde and a system show that the family of two step maruyama methods have better stability than previous two step maruyama methods

hydrodynamic limit of the stationary boltzmann equation in a slab we study the stationary solution of the boltzmann equation in a slab with a constant external force parallel to the boundary and complete accommodation condition on the walls at a specified temperature we prove that when the force is sufficiently small there exists a solution which converges in the hydrodynamic limit to a local maxwellian with parameters given by the stationary solution of the corresponding compressible navier stokes equations with no slip boundary conditions corrections to this maxwellian are obtained in powers of the knudsen number with a controlled remainder

chaos noise and synchronization reconsidered in a recent paper maritan and banavar phys rev lett 72 1451 1994 reported the synchronization of identical chaotic systems by additive noise we relate such a synchronization to the maximum lyapunov exponent of a single system and discuss the underlying mechanisms of the effect in the case of the lorenz equations the nonvanishing mean of the noise mimics a parameter change leading to synchronization for the logistic map a state dependence of the fluctuations is induced by the boundary conditions

stochastic bifurcations and coherencelike resonance in a self sustained bistable noisy oscillator we investigate the influence of additive gaussian white noise on two different bistable self sustained oscillators duffing van der pol oscillator with hard excitation and a model of a synthetic genetic oscillator in the deterministic case both oscillators are characterized with a coexistence of a stable limit cycle and a stable equilibrium state we find that under the influence of noise their dynamics can be well characterized through the concept of stochastic bifurcation consisting in a qualitative change of the stationary amplitude distribution for the duffing van der pol oscillator analytical results obtained for a quasiharmonic approach are compared with the result of direct computer simulations in particular we show that the dynamics is different for isochronous and anisochronous systems moreover we find that the increase of noise intensity in the isochronous regime leads to a narrowing of the spectral line this effect is similar to coherence resonance however in the case of anisochronous systems this effect breaks down and a new phenomenon anisochronous based stochastic bifurcation occurs

the van kampen expansion for the fokker planck equation of a duffing oscillator in rodriguez and van kampen s 1976 paper a method of extracting information from the fokker planck equation without having to solve the equation is outlined the fokker planck equation for a duffing oscillator excited by white noise is expanded about the intensity of the forcing function this expansion is carried to order mathcal o alpha 1 2 however no studies are made of the effects of the order of the expansion or variation of the parameters nor are comparisons made to experimental results in the present paper the expansion is carried to a higher order mathcal o alpha 3 2 results are presented and compared to monte carlo experiments using both white and colored noise and parametric studies are performed on the intensity of the forcing function and the damping coefficient it is found that the expansion method works well for the case of white noise and for colored noise where the correlation time is less than 0 1 sec but fails to give certain details it is also found that the system behaves as expected when the parameters are varied

perturbation analysis of a stationary nonequilibrium flow generated by an external force the stationary flow of a gas in a slab under the action of a constant external force parallel to the walls is analyzed in the context of the bhatnagar gross krook model kinetic equation the force produces spatial gradients along the coordinate normal to the walls by performing a perturbation expansion in powers of the force we obtain the hydrodynamic fields up to fifth order in the force then the velocity distribution function and all its moments are evaluated to third order the expansion coefficients are polynomials in the space variable of a degree increasing linearly with the expansion order although the series expansion is only asymptotic it shows how the state of the system is modified by a variation of the external force beyond the linear regime

numerical treatment of stochastic differential equations we define general runge kutta approximations for the solution of stochastic differential equations sde these approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift the explicit form of the correction term is given concerning the order of convergence we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than o h 3 this order is attained e g by the stochastic analogue of heun s method in the n dimensional case the highest order of convergence is in general only o h 2 attained by euler s method the order o h 3 can only be reached if nabla x sigma r sigma s nabla x sigma s sigma r for the diffusion matrix

positive lyapunov exponents in the kramers oscillator the maximum lyapunov exponent is computed numerically for the double well oscillator in a heat bath positive exponents are found in a wide range of friction coefficients in the low damping regime

phase diagram of the random frequency oscillator the case of ornstein uhlenbeck noise abstract we study the stability of a stochastic oscillator whose frequency is a random process with finite time memory represented by an ornstein uhlenbeck noise this system undergoes a noise induced bifurcation when the amplitude of the noise grows the critical curve that separates the absorbing phase from an extended non equilibrium steady state corresponds to the vanishing of the lyapunov exponent that measures the asymptotic logarithmic growth rate of the energy we derive various expressions for this lyapunov exponent by using different approximation schemes this allows us to study quantitatively the phase diagram of the random parametric oscillator

generalized lyapunov exponents of the random harmonic oscillator cumulant expansion approach the cumulant expansion is used to estimate generalized lyapunov exponents of the random frequency harmonic oscillator three stochastic processes are considered gaussian white noise ornstein uhlenbeck and poisson shot noise in some cases nontrivial numerical difficulties arise these are mostly solved by implementing an appropriate importance sampling monte carlo scheme we analyze the relation between random frequency oscillators and many particle systems with pairwise interactions like the lennard jones gas

temperature profile of a dilute gas undergoing a plane poiseuille flow abstract the temperature and the pressure profile of a dilute or moderately rarefied gas undergoing a plane poiseuille flow are calculated and compared with simulation data for a gas of hard spheres as first presented by malek mansour baras and garcia physica a 240 1997 225 the boltzmann equation is solved by the moment method the desired solution including terms up to second order in the applied force driving the flow requires the use of a 19 moments approximation it comprises results based on somewhat simpler 13 moments approximation which are qualitatively reasonable for the present problem

anderson localization as a parametric instability of the linear kicked oscillator we rigorously analyze the correspondence between the one dimensional standard anderson model and a related classical system the kicked oscillator with noisy frequency we show that the anderson localization corresponds to a parametric instability of the oscillator the localization length being related to the rate of exponential growth of the energy of the oscillator analytical expression for a weak disorder is obtained which is valid both inside the energy band and at the band edge

stationary energy probability density of oscillators driven by a random external force we derive rigorous analytical results for the stationary energy probability density function of linear and nonlinear oscillators driven by additive gaussian noise our study focuses on two cases i a harmonic oscillator subjected to gaussian colored noise with an arbitrary correlation function and ii nonlinear oscillators with a general potential driven by gaussian white noise we also derive analytical expressions for the stationary moments of the energy and investigate the partition of the mean energy between kinetic and potential energy to illustrate our general results we consider specifically the case of exponentially correlated noise for i and power law and bistable potentials for ii our theoretical results are substantiated by langevin simulations between the kinetic energy and the potential energy of the oscillator our studies focus on two cases first we consider a harmonically bound particle driven by general additive gaussian colored noise second we consider a particle in a general potential driven by gaussian white noise in both cases we develop a general theory and study specific illustrative examples the paper is organized as follows in sec ii we formulate the langevin equations for a particle with inertia in a general potential driven by an additive random force we also derive the general relation between the stationary energy pdf and the stationary pdf for the particle s position and velocity section iii is devoted to the motion of a particle in a harmonic potential driven by colored gaussian noise with an arbitrary correlation function the stationary energy pdf as well as the first and second moments of the energy is derived analytically we apply these results to the case of exponentially correlated gaussian noise and to gaussian white noise for these specific cases we also determine the partition of the energy between kinetic and potential energy we analyze the effect of the potential shape on the stationary energy pdf for oscillators driven by an additive gaussian white noise in sec iv w e apply our analytical results to two specific cases a general power law potential and a bistable potential the first type of oscillator without friction and with additive noise 36 and multiplicative noise 37 has recently been studied in the context of anomalous diffusion the second type the duffing oscillator has been investigated as an alternative to the linear oscillator for harvesting energy from external random sources 28 29 35 38 39

noise enhanced stability in fluctuating metastable states we derive general equations for the nonlinear relaxation time of brownian diffusion in randomly switching potential with a sink for piece wise linear dichotomously fluctuating potential with metastable state we obtain the exact average lifetime as a function of the potential parameters and the noise intensity our result is valid for arbitrary white noise intensity and for arbitrary fluctuation rate of the potential we find noise enhanced stability phenomenon in the system investigated the average lifetime of the metastable state is greater than the time obtained in the absence of additive white noise we obtain the parameter region of the fluctuating potential where the effect can be observed the system investigated also exhibits a maximum of the lifetime as a function of the fluctuation rate of the potential