simple treatment of correlated multiplicative and additive noises two simple rules are derived for obtaining the solution of a problem with correlated multiplicative and additive noise in terms of the results for systems with non correlated noise it is shown that the mean free passage time decreases as the correlation time increases and its influence becomes important for not too small correlation times

multivariate pade approximation abstract this paper is a survey on the multivariate pade approximation two types of approximants are considered those which can approximate general meromorphic functions f h g where both h and g are holomorphic and those which are specialized to the approximation of functions of the same form where g is a polynomial algorithms are described together with the different techniques used for proving convergence

eigenvalues and eigenfunctions of the fokker planck equation for the extremely underdamped brownian motion in a double well potential the eigenvalues and eigenfunctions of the fokker planck equation describing the extremely underdamped brownian motion in a symmetric double well potential are investigated by transforming the fokker planck equation to energy and position coordinates and by performing a suitable averaging over the position coordinate a differential equation depending only on energy is derived for finite temperatures this equation is solved by numerical integration whereas in the weak noise limit an analytic result for the lowest nonzero eigenvalue is obtained furthermore by using a boundary layer theory near the critical trajectory the correction term to the zero friction limit result is found

least squares orthogonal polynomials abstract in this paper the new notion of least squares orthogonal polynomials is introduced their existence and uniqueness is studied and some methods for their recursive computation are given the location of their zeros in the case of a positive definite functional is studied the paper ends with applications to pade type approximation and gaussian quadrature methods

average trajectories and fluctuations from noisy nonlinear maps the average trajectories and fluctuations around them resulting from an ensemble of noisy nonlinear maps are analyzed the bifurcation diagram for the average value obtained from the computer simulation of noisy maps ensemble is discussed first then a deterministic average equation of motion describing in an approximate way the time evolution of the average value and of the variance is analyzed numerically this equation predicts the existence of the bifurcation gap and of the exceptional attractors for special initial points the scaling properties of the average value and of the variance are obtained with the help of this equation

numerical integration of the fluctuating hydrodynamic equations an approach to numerically integrate the landau lifshitz fluctuating hydrodynamic equations is outlined the method is applied to one dimensional systems obeying the nonlinear fourier equation and the full hydrodynamic equations for a dilute gas static spatial correlation functions are obtained from computer generated sample trajectories time series they are found to show the emergence of long range behavior whenever a temperature gradient is applied the results are in very good agreement with those obtained from solving the correlation equations directly

szego kernel asymptotics and morse inequalities on cr manifolds let x be an abstract compact orientable cr manifold of dimension 2n 1 n 2 and let l k be the k th tensor power of a cr complex line bundle l over x we assume that condition y q holds at each point of x in this paper we obtain a scaling upper bound for the szego kernel on 0 q forms with values in l k for large k after integration this gives weak morse inequalities analogues of the holomorphic morse inequalities of demailly by a refined spectral analysis we obtain also strong morse inequalities we apply the strong morse inequalities to the embedding of some convex concave manifolds

recursive interpolation extrapolation and projection abstract the recursive projection algorithm derived in a previous paper is related to several well known methods of numerical analysis such as the conjugate gradient method rosen s method and henrici s it is connected with the general interpolation problem with extrapolation methods with orthogonal projection on a subspace and with fourier expansions several other connections and applications are presented

on the relaxation of fluctuations in the steady state of the stratonovich model a detailed investigation has been carried out of the relaxation of fluctuations in the steady state of the stratonovich model also known as the random growing rate model the autocorrelation time of this system driven by parametric white noise in the linear term has been measured for an electronic circuit model computed by digital simulation and calculated by use of a continued fraction expansion method the results are all consistent with each other and with those obtained from the matrix continued fraction method of jung and risken 1985 provided that explicit account is taken of the ways in which any macroscopic real physical system is bound to differ from the idealisation represented by the original form of the stratonovich model in particular it is necessary to recognise the existence of weak additive noise and a small additive constant the physical origins of some earlier seemingly discrepant calculations and experimental data are discussed

a path integral method for coarse graining noise in stochastic differential equations with multiple time scales abstract we present a new path integral method to analyze stochastically perturbed ordinary differential equations with multiple time scales the objective of this method is to derive from the original system a new stochastic differential equation describing the system s evolution on slow time scales for this purpose we start from the corresponding path integral representation of the stochastic system and apply a multi scale expansion to the associated path integral kernel of the corresponding lagrangian as a concrete example we apply this expansion to a system that arises in the study of random dispersion fluctuations in dispersion managed fiber optic communications moreover we show that for this particular example the new path integration method yields the same result at leading order as an asymptotic expansion of the associated fokker planck equation

comment on asymptotic properties of coupled langevin equations the asymptotic behavior of coupled langevin equations in the limit of weak noise is studied by general normal form techniques in the vicinity of a pitchfork bifurcation the non gaussian behavior of the critical variable is established the conditional probability of the noncritical variable around the center manifold is determined it is shown that in certain cases the distribution of this later variable may be non gaussian

additive noise induced turing transitions in spatial systems with application to neural fields and the swift hohenberg equation this work studies the spatio temporal dynamics of a generic integral differential equation subject to additive random fluctuations it introduces a combination of the stochastic center manifold approach for stochastic differential equations and the adiabatic elimination for fokker planck equations and studies analytically the systems stability near turing bifurcations in addition two types of fluctuation are studied namely fluctuations uncorrelated in space and time and global fluctuations which are constant in space but uncorrelated in time we show that the global fluctuations shift the turing bifurcation threshold this shift is proportional to the fluctuation variance applications to a neural field equation and the swift hohenberg equation reveal the shift of the bifurcation to larger control parameters which represents a stabilization of the system all analytical results are confirmed by numerical simulations of the occurring mode equations and the full stochastic integral differential equation to gain some insight into experimental manifestations the sum of uncorrelated and global additive fluctuations is studied numerically and the analytical results on global fluctuations are confirmed qualitatively

a new method of analysis of the effect of weak colored noise in nonlinear dynamical systems a systematic method for obtaining the asymptotic behavior of a dynamical system forced by colored noise in the limit of small intensity is developed it is based on the search of wkb solutions to the fokker planck equation for the joint probability density of the system and noise in which the perturbation expansion is continued to the first correction beyond the hamilton jacobi limit the method can be applied to noise with correlation time of order unity it is illustrated on the normal form of a pitchfork bifurcation where it is pointed out that additive noise can induce a shift of the most probable value this prediction is confirmed by numerical simulation of the stochastic differential equations

diffusion over barriers eigenanalysis of the smoluchowski equation for bistable potentials abstract a variational method is used to get the eigenvalues and eigenfunctions of the one dimensional smoluchowski equation for two types of symmetric bistable potential fields the effects of barrier height and width on the rate of escape of particles over the barrier first non zero eigenvalue are evaluated numerically and the results are compared with kramers theory in order to assess its range of application

asymptotic properties of coupled nonlinear langevin equations in the limit of weak noise i cusp bifurcation we show how a singular perturbation technique based on the introduction of properly scaled variables enables us to derive the asymptotic properties of coupled langevin equations in the limit of weak noise this technique can be applied when the macroscopic steady state is asymptotically or marginally stable in the close vicinity of a cusp bifurcation point a simple prescription for the adiabatic elimination of the fast variable is established the critical variable exhibits amplified non gaussian fluctuations on a slow time scale the properties of the fast variable depend on the nonlinearity of the system under consideration because of its coupling to the critical variable it may exhibit amplified fluctuations of non gaussian nature

noise and bifurcations the influence of while noise on bifurcating dynamical systems is investigated using both fokker planck and functional integral methods noise leads to fuzzy bifurcations where physically relevant quantities become smooth functions of the bifurcation parameters we study dynamical and probabilistic quantities such as invariant measures liapunov exponents correlation functions and exit times the behavior of these quantities near the deterministic bifurcation point changes for distinct values of the control parameter therefore the very concept of bifurcation point becomes meaningless and must be replaced by the notion of bifurcation region

non linear dynamics of systems coupled with external noise some exact results abstract we solve exactly a class of non linear stochastic differential equations describing a variety of physical and chemical relevant systems coupled to external sources of noise the static and dynamic solutions are valid in the whole space of parameters including the critical points of non equilibrium instabilities critical slowing down is found and an unexpected long time tail evolution is predicted near and at the critical point this last effect is completely due to the coupling with the external noise

bifurcations under weak noise dynamical systems driven by gaussian white noise share some general asymptotic properties in the limit of weak noise which are reminiscent of equilibrium thermodynamics these properties and applications to bifurcating systems are discussed

on the theory of biorthogonal polynomials let x u be a distribution in x e r for every u in a real parameter set q subject to additional technical conditions we study mth degree monic polynomials pm that satisfy the biorthogonality conditions roo pm z d9 x az o i 1 2 m m 1 j oo for a distinct sequence u1 u2 e q necessary and sufficient conditions for existence and uniqueness are established as well as explicit determinantal and integral representations we also consider loci of zeros existence of rodrigues type formulae and reducibility to standard orthogonality the paper is accompanied by several examples of biorthogonal systems

resolution of subgrid microscale interactions enhances the discretisation of nonautonomous partial differential equations we consider a general method for the macroscale modelling of nonlinear nonautomomous partial differential equations or difference equations our method is able to resolve the complex fluctuations which arise in nonliner nonautonomous systems these fluctuations are not accounted for by popular multiscale averaging procedures and homogenization our method is illustrated with the example of a forced burgers equation coarse grained macroscale spatial discretisations of nonlinear nonautonomous partial differential difference equations are given novel support by centre manifold theory dividing the physical domain into overlapping macroscale elements empowers the approach to resolve significant subgrid microscale structures and interactions between neighbouring elements the crucial aspect of this approach is that centre manifold theory organises the resolution of the detailed subgrid microscale structure interacting via the nonlinear dynamics within and between neighbouring elements the techniques and theory developed here may be applied to soundly discretise on a macroscale many dissipative nonautonomous partial differential difference equations such as the forced burgers equation adopted here as an illustrative example

bifurcations in fluctuating systems the center manifold approach bifurcations in fluctuating dynamical systems are studied using the ideas of center manifold reduction the method provides not only a systematic procedure for the reduction of the system to a small number of variables but also a classification scheme for the different kinds of dynamical behavior possible near bifurcation points the joint probability density factorizes into a stationary gaussian densityp v u in the fast variablesv and a time dependent densityp u f in the slow variablesu describing the dynamics on the center manifoldv v0 u p u t obeys a reduced fokker planck equation that can be written in a normal form by means of local nonlinear transformations both additive and multiplicative white noise are considered as is colored noise the results extend and formalize haken s concept of adiabatic elimination of fast variables

adiabatic elimination for systems of brownian particles with nonconstant damping coefficients we discuss the problem of eliminating the momentum variable in the phase space langevin equations for a system of brownian particles in two related situations i position dependent damping and ii existence of hydrodynamic interactions we discuss the problems associated with the conventional elimination and we develop an alternative elimination procedure in the lagevin framework which leads to the correct smoluchowski equation we give a heuristic argument on the basis of stochastic differential equations for the smoluchowski limit and establish rigorously the limit for the general case of position dependent friction and diffusion coefficents

temporal localization of limit cycles in a noise driven chemical oscillator abstract we study the influence of multiplicative random external perturbation on the selkov model exhibiting limit cycles the dynamics are described by stochastic differential equations the numerically computed ensemble averages of the concentrations exhibit temporal localization of the limit cycles for small amplitudes of the noise the localization time is smaller when the amplitude of the noise is higher to understand this behavior a stochastic complex ginzburg landau equation is derived using normal form transformations on the stochastic evolution equations in the case of weak noise the equations for the radial and the phase components of the amplitude separate and can be solved exactly the solution is used to compute the damping factors which were found to be gaussian we also show that asymptotically the radial part fluctuates near the average value and the phase is found to be a gaussian random variable with time dependent mean and variance and may be responsible for the temporal localization of the limit cycle

the effect of multiplicative noise on the relaxation time of a real non linear physical system a comparison of experiment and theory for the random growing rate model rgrm it is found that the theoretical behaviour of the relaxation time t for the random growing rate model rgrm under the influence of multiplicative noise of intensity q a monotonic decrease of t 1 towards a limiting value as q to infinity differs markedly from that actually measured in electronic simulators of that system a new electronic analogue experiment is described in which a distinct minimum in t 1 q has been observed for the first time with clear evidence for an increase of t 1 q with q at large q in good qualitative agreement with an earlier analogue experiment the discrepancy between experiment and the theoretical solutions of the idealised equation is attributed to the profound influence exerted by the very weak additive noise which must also in some measure always be present in a real physical system

a new approximate stationary probability distribution for processes driven by ornstein uhlenbeck noise a novel approximate probability distribution for a class of nonlinear processes driven by multiplicative linear ornstein uhlenbeck noise is presented a comparison is made with other approximate distributions reported in the literature

stochastic analysis of symmetry breaking bifurcations master equation approach the multivariate master equation for a general reaction diffusion system is solved perturbatively in the vicinity of a bifurcation point leading to symmetry breaking transitions the possibility to express the result through a brazovskii type of potential is examined and a comparison with the langevin analysis of walgraefet al adv chem phys 49 311 1982 is performed

consistent spectral predictors for dynamic causal models of steady state responses dynamic causal modelling dcm for steady state responses ssr is a framework for inferring the mechanisms that underlie observed electrophysiological spectra using biologically plausible generative models of neuronal dynamics in this paper we examine the dynamic repertoires of nonlinear conductance based neural population models and propose a generative model of their power spectra our model comprises an ensemble of interconnected excitatory and inhibitory cells where synaptic currents are mediated by fast glutamatergic and gabaergic receptors and slower voltage gated nmda receptors we explore two formulations of how hidden neuronal states depolarisation and conductances interact through their mean and variance mean field model or through their mean alone neural mass model both rest on a nonlinear fokker planck description of population dynamics which can exhibit bifurcations phase transitions we first characterise these phase transitions numerically by varying critical model parameters we elicit both fixed points and quasiperiodic dynamics that reproduce the spectral characteristics 2 100 hz of real electrophysiological data we then introduce a predictor of spectral activity using centre manifold theory and linear stability analysis this predictor is based on sampling the system s jacobian over the orbits of hidden neuronal states this predictor behaves consistently and smoothly in the region of phase transitions which permits the use of gradient descent methods for model inversion we demonstrate this by inverting generative models dcms of ssrs using simulated data that entails phase transitions

noise induced stabilization of bumps in systems with long range spatial coupling the position of a localized region of active neurons a bump has been proposed to encode information for working memory the head direction system and feature selectivity in the visual system stationary bumps are ordinarily stable but including spike frequency adaptation in the neural dynamics causes a stationary bump to become unstable to a moving bump through a supercritical pitchfork bifurcation in bump speed adding spatiotemporal noise to the network supporting the bump can cause the average speed of the bump to decrease to almost zero reversing the effect of the adaptation and restabilizing the bump this restabilizing occurs for noise levels lower than those required to break up the bump the restabilizing can be understood by examining the effects of noise on the normal form of the pitchfork bifurcation where the variable involved in the bifurcation is bump speed this noisy normal form can be further simplified to a persistent random walk in which the probability of changing direction is related to the noise level through an arrhenius type rate the probability density function of position for the continuous time version of this random walk satisfies the telegrapher s equation and the closed form solution of this pde allows us to find expressions for the mean and variance of the average speed of the particle the bump undergoing the random walk this noise induced stabilization is a novel example in which moderate amounts of noise have a beneficial effect on a system specifically stabilizing a spatiotemporal pattern 2001 elsevier science b v all rights reserved

a breakdown free lanczos type algorithm for solving linear systems lanczos type algorithms for solving systems of linear equations have their foundations in the theory of formal orthogonal polynomials and the method of moments which leads to a determinantal formula for their iterates the various lanczos type algorithms mainly differ by the way of computing the coefficients entering into the recurrence formulae if the denominator in the formula for one of these coefficients is zero then a breakdown occurs in the algorithm and it must be stopped such a breakdown is in fact due to the non existence of some orthogonal polynomial in this paper we show how to jump over such a singularity by computing the next existing orthogonal polynomial by the block bordering method the resulting algorithm called mrz is equivalent to the nongeneric biodir algorithm which is a look ahead lanczos type algorithm but our derivation is much simpler

geometric framework for phase synchronization in coupled noisy nonlinear systems a geometric approach is introduced for understanding the phenomenon of phase synchronization in coupled nonlinear systems in the presence of additive noise we show that the emergence of cooperative behavior through a change of stability via a hopf bifurcation entails the spontaneous appearance of a gauge structure in the system arising from the evolution of the slow dynamics but induced by the fast variables the conditions for the oscillators to be synchronised in phase are obtained the role of weak noise appears to be to drive the system towards a more synchronized behavior our analysis provides a framework to explain recent experimental observations on noise induced phase synchronization in coupled nonlinear systems

dynamics of the stochastic duffing oscillator in gaussian approximation abstract the previously developed general concept of the gaussian approximation allows the description of the driven stochastic duffing oscillator by ordinary differential equations these equations are analyzed analytically as well as numerically to investigate the modification of simple bifurcations and of the feigenbaum cascade by stochastic forces furthermore the method allows a qualitative discussion of the stochastic system in the deterministic irregular region

on the stationary distribution of self sustained oscillators around bifurcation points a double expansion in powers of the damping coefficient and noise intensity is shown to be a powerful method for obtaining the stationary distribution of systems that after rescaling become weakly damped conservative ones systems undergoing hopf bifurcations belong to this class as an illustrative example the generalized van der pol oscillator is considered around its bifurcation point a calculation is carried out up to third order in both the noise intensity and the bifurcation parameter damping coefficient

semiclassical determinants of schrodinger operators on line bundles heat kernel methods are used to compute asymptotic expressions for the zeta regularized determinant and local heat invariants of a schrodinger operator acting on sections of the kth tensor power of a line bundle as k the result is an infinite series representation involving todd polynomials of the curvature

self tuning to the hopf bifurcation in fluctuating systems the problem of self tuning a system to the hopf bifurcation in the presence of noise and periodic external forcing is discussed we find that the response of the system has a non monotonic dependence on the noise strength and displays an amplified response which is more pronounced for weaker signals the observed effect is to be distinguished from stochastic resonance for the feedback we have studied the unforced self tuned hopf oscillator in the presence of fluctuations exhibits sharp peaks in its spectrum the implications of our general results are briefly discussed in the context of sound detection by the inner ear

local methods for constructing stationary distribution functions of systems of stochastic differential langevin type equations noise influence on simple bifurcation the influence is considered of two additive correlated noise effects on a two dimensional quadratic nonlinear system describing the behavior of two hydrodynamic modes using the method of gaussian approximation local characteristics of the distribution function are calculated which are used to construct the global distribution function with the aid of the method of fraction rational approximations it is shown that for a system at whose bifurcation point the asymptotic stability is lost in an expanded space of parameters bifurcation parameter in the absence of noise plus noise parameters there appears an instability zone within which the stationary distribution function does not exist the effect of noise correlation on the stationary characteristics of the system is studied

comment on bifurcations in fluctuating systems we prove using normal form techniques in a codimension one bifurcation that the conditional probability of the fast variable conditioned by the slow variables is a gaussian distribution centered in the center manifold

exact solution for the diffusion in bistable potentials we solve analytically the fokker planck equation for a one parameter family of symmetric attractive nonharmonic potentials which include double well situations the exact knowledge of the eigenfunctions and eigenvalues allows us to fully discuss the transient behavior of the probability density in particular for the bistable potentials we can give analytical expressions for the probability current over the working barrier and for the onset time which characterizes the transition from uni to bimodal probability densities

method of divergent series summation in the problem of particle diffusion in a bistable potential an infinite system for the moments of a particle moving in a potential with two minima under the action of fluctuations is considered a class of solutions is found for which a divergent series can be summed up to a simple transcendent function it is shown numerically that the true solution belongs to this class and conditions are set to define the solution uniquely

generating function for a multiplicative noise with group analysis the authors find the differential equation for the generating function or a multiplicative stochastic process and they apply to it the group analysis they give the general form of the lie generators and find the conditions for the existence of similarity solutions two classes of similarity solutions are presented and the analytic expression of the generating function is given

verification of lyapunov functions for the analysis of stochastic lienard equations versions of stochastic lienard equations perturbed by both additive and multiplicative white noise are considered we discuss existence uniqueness continuity boundedness and moment stability of solutions with the help of several lyapunov type functions the lyapunov functions are explicitly found to control uniform moment boundedness and stability a new matching condition on the interaction of resistance and restoring force plays an essential role to guarantee stability uniform boundedness of p th moments by validation of those lyapunov functions

diffusion across characteristic boundaries we consider the motion of a particle acted on by the deterministic force vector b bf b bf x t and perturbed by random forces of white noise type such a particle will leave any bounded domain omega in finite time we consider the case where b is such that partial omega consists of a trajectory or trajectories of the sytem dot x b t thus we consider the cases of an unstable limit cycle and a center we observe that these problems are such that bf b is not derivable from a potential for each problem we derive expressions for i the mean first passage time to partial omega and ii the probability distribution of exit points on partial omega our method is to employ the ito calculus to characterize the quantities i and ii as solutions of singularly perturbed elliptic boundary value problems and then to derive asymptotic representations of the solutions of those problems the results obtained are new and are of importance in a variety of applications includ

on the low dimensional modelling of stratonovich stochastic differential equations abstract we develop further ideas on how to construct low dimensional models of stochastic dynamical systems the aim is to derive a consistent and accurate model from the originally high dimensional system this is done with the support of centre manifold theory and techniques aspects of several previous approaches are combined and extended adiabatic elimination has previously been used but centre manifold techniques simplify the noise by removing memory effects and with less algebraic effort than normal forms analysis of associated fokker plank equations replace nonlinearly generated noise processes by their long term equivalent white noise the ideas are developed by examining a simple dynamical system which serves as a prototype of more interesting physical situations