refined iterative algorithms based on arnoldi s process for large unsymmetric eigenproblems abstract arnoldi s method has been popular for computing the small number of selected eigenvalues and the associated eigenvectors of large unsymmetric matrices however the approximate eigenvectors or ritz vectors obtained by arnoldi s method cannot be guaranteed to converge in theory even if the approximate eigenvalues or ritz values do in order to circumvent this potential danger a new strategy is proposed that computes refined approximate eigenvectors by small sized singular value decompositions it is shown that refined approximate eigenvectors converge to eigenvectors if ritz values do moreover the resulting refined algorithms converge more rapidly we report some numerical experiments and compare the refined algorithms with their counterparts the iterative arnoldi and arnoldi chebyshev algorithms the results show that the refined algorithms are considerably more efficient than their counterparts

adiabatic small noise fluctuations around anticipated synchronization a perspective from scalar master slave dynamics abstract in this paper we extract and highlight some essential ingredients and properties that characterize the phenomenon of anticipated synchronization when external additive noise sources perturb the master and slave dynamics our results rely on a minimal scalar setup able to exhibit the more fundamental aspects of this phenomenon where the fluctuations around the average dynamics are worked out in a small noise and adiabatic approximations allowing to describe their dynamics through linear delay langevin equations in this context we find necessary conditions that guarantee anticipated synchronization in the mean value fluctuations around this condition are studied through the stationary correlation of the delayed difference between the master and slave dynamics it is shown that external noise properties can be inferred by measuring this object conditions for minimizing the dynamical fluctuations around the anticipated synchronization in mean value are found a detailed analysis of the dependence on the characteristic parameters is presented

delay fokker planck equations perturbation theory and data analysis for nonlinear stochastic systems with time delays we study nonlinear stochastic systems with time delayed feedback using the concept of delay fokker planck equations introduced by guillouzic l heureux and longtin we derive an analytical expression for stationary distributions using first order perturbation theory we demonstrate how to determine drift functions and noise amplitudes of this kind of systems from experimental data in addition we show that the fokker planck perspective for stochastic systems with time delays is consistent with the so called extended phase space approach to time delayed systems

interaction equations for short and long dispersive waves abstract we show the time local well posedness for a system of nonlinear dispersive equations for the water wave interaction formula it is shown that for any initial data u0 v0 hs r hs 1 2 r s 0 the solution for the above equation uniquely exists in a subset ofc t t hs c t t hs 1 2 and depends continuously on the data by virtue of a special structure of the nonlinear coupling the solution is stable under a singular limiting process

small delay approximation of stochastic delay differential equations delay differential equations evolve in an infinite dimensional phase space in this paper we consider the effect of external fluctuations noise on delay differential equations involving one variable thus leading to univariate stochastic delay differential equations sdde s for small delays a univariate nondelayed stochastic differential equation approximating such a sdde is presented another approximation complementary to the first is also obtained using an average of the sdde s drift term over the delayed dynamical variable which defines a conditional average drift this second approximation is characterized by the fact that the diffusion term is identical to that of the original sdde for small delays our approach yields a steady state probability density and a conditional average drift which are in close agreement with numerical simulations of the original sdde we illustrate this scheme with the delayed linear langevin equation and a stochastic version of the delayed logistic equation the technique can be used with any type of noise and is easily generalized to multiple delays s1063 651x 99 08304 x

polynomial characterizations of the approximate eigenvectors by the refined arnoldi method and an implicitly restarted refined arnoldi algorithm abstract it is well known that a ritz vector obtained by the lanczos method and the arnoldi method can be characterized by an elegant explicit polynomial whose roots are just the other m 1 ritz values of the matrix a in question from the m dimensional krylov subspace involved analogous results hold for the recently developed harmonic lanczos and arnoldi methods and the roots of the corresponding polynomial are now the other m 1 harmonic ritz values of a from the same subspace in this paper we investigate a polynomial characterization of the refined arnoldi method proposed by the author in recent years we derive a polynomial characterization of the refined arnoldi method and apply the implicitly restarting scheme proposed by sorensen to the refined arnoldi method the roots of this polynomial are used as shifts called refined shifts within an implicitly restarted refined arnoldi algorithm the shifts are interpreted to have the same nature as the exact shifts used within the implicitly restarted arnoldi algorithm proposed by sorensen numerical experiments compare the use of refined shifts with exact shifts within an implicitly restarted arnoldi algorithm the results also show that implicitly restarting the refined arnoldi method is far superior to the saad s explicitly restarted scheme

a small delay and correlation time limit of stochastic differential delay equations with state dependent colored noise we consider a general stochastic differential delay equation sdde with state dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero the work is motivated by an experiment involving an electrical circuit with noisy delayed feedback an ornstein uhlenbeck process is used to model the colored noise the main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by kurtz and protter and a maximal inequality for sums of a stationary sequence of random variables by peligrad and utev

fokker planck equations for time delayed systems via markovian embedding for stochastic systems with discrete time delay the fokker planck equation fpe of the one time probability density function pdf does not provide a complete self contained probabilistic description it explicitly involves the two time pdf and represents in fact only the first member of an infinite hierarchy we here introduce a new approach to obtain a fokker planck description by using a markovian embedding technique and a subsequent limiting procedure on this way we find a closed complete fpe in an infinite dimensional space from which one can derive a hierarchy of fpes while the first member is the well known fpe for the one time pdf we obtain as second member a new representation of the equation for the two time pdf from a conceptual point of view our approach is simpler than earlier derivations and it yields interesting insight into both the physical meaning and the mathematical structure of delayed processes we further propose an approximation for the two time pdf which is a central quantity in the description of these non markovian systems as it directly gives the correlation between the present and the delayed state application to a prototypical bistable system reveals that this approximation captures the non trivial effects induced by the delay remarkably well despite its surprisingly simple form moreover it outperforms earlier approaches for the one time pdf in the regime of large delays

on low regularity of the ostrovsky stepanyams and tsimring equation abstract in this paper we prove by using the fourier restriction norm method that the initial value problem of the ostrovsky stepanyams and tsimring equation u t u x x x h u x h u x x x u u x 0 x r t 0 where 0 and h denotes the usual hilbert transformation is locally well posed in the sobolev space h s r for any s 5 4 which improve our former result in zhao and cui 2009 5

stochastic multiresonance for a fractional linear oscillator with time delayed kernel and quadratic noise abstract the stochastic resonance for a fractional oscillator with time delayed kernel and quadratic trichotomous noise is investigated applying linear system theory and laplace transform the system output amplitude spa for the fractional oscillator is obtained it is found that the spa is a periodical function of the kernel delayed time stochastic multiplicative phenomenon appears on the spa versus the driving frequency versus the noise amplitude and versus the fractional exponent the non monotonous dependence of the spa on the system parameters is also discussed

influence of strong noise on the decline and propagation of population in the delayed malthus verhulst model abstract the effects of strong noise on the decline and propagation processes of a population in the malthus verhulst model with time delay are investigated by a stochastic simulation time delays in two different processes are concurrent in ecosystems the simulation results indicate that the stability of the population is enhanced by the decreasing multiplicative noise intensity and the increasing delay time the replacement of old individuals with young ones is accelerated by an increasing multiplicative noise intensity an increasing additive noise intensity and a decreasing delay time an increasing multiplicative noise intensity will drive the population of species to fluctuate more largely

local well posedness of the ostrovsky stepanyams and tsimring equation in sobolev spaces of negative indices abstract in this paper we prove by using the method of bilinear estimate in the bourgain type spaces that the initial value problem of the ost equation u t u x x x h u x h u x x x u u x 0 x r t 0 where 0 and h denotes the usual hilbert transformation is locally well posed in the sobolev space h s r when s 1

control of noise induced oscillations by delayed feedback abstract we propose a method to control noise induced motion based on using delayed feedback in the form of the difference between the delayed and the current states of the system the method is applied to two different types of systems namely a self oscillator near andronov hopf bifurcation and a threshold system in both cases we demonstrate that by variation of time delay one can effectively control coherence and timescales of stochastic oscillations the entrainment of the basic period of oscillations by time delay is discovered we give explanations of the phenomena observed and provide a theory for the system near bifurcation

an analysis of the rayleigh ritz method for approximating eigenspaces this paper concerns the rayleigh ritz method for computing an approximation to an eigenspace x of a general matrix a from a subspace w that contains an approximation to x the method produces a pair n x that purports to approximate a pair l x where x is a basis for x and ax xl in this paper we consider the convergence of n x as the sine e of the angle between x and w approaches zero it is shown that under a natural hypothesis called the uniform separation condition the ritz pairs n x converge to the eigenpair l x when one is concerned with eigenvalues and eigenvectors one can compute certain refined ritz vectors whose convergence is guaranteed even when the uniform separation condition is not satisfied an attractive feature of the analysis is that it does not assume that a has distinct eigenvalues or is diagonalizable

effects of time delay on symmetric two species competition subject to noise noise and time delay act simultaneously on real ecological systems the lotka volterra model of symmetric two species competition with noise and time delay was investigated in this paper by means of stochastic simulation we find that i the time delay induces the densities of the two species to periodically oscillate synchronously ii the stationary probability distribution function of the two species densities exhibits a transition from multiple to single stability as the delay time increases iii the characteristic correlation time for the sum of the two species densities squared exhibits a nonmonotonic behavior as a function of delay time our results have the implication that the combination of noise and time delay could provide an efficient tool for understanding real ecological systems

a new stochastic gompertz diffusion process with threshold parameter computational aspects and applications abstract in this paper we propose a new homogeneous stochastic gompertz diffusion model with a threshold parameter this can be considered an extension of the homogeneous three parameter gompertz process with the addition of a fourth parameter from the corresponding kolmogorov equations and ito s stochastic differential equations we obtain the transition probability density function and the moments of this process specifically the trend functions the parameters are estimated by considering discrete sampling of the sample path of the model and by using maximum likelihood methodology estimation of the threshold parameter requires us to solve a non linear equation which is achieved by the newton raphson method simulated model data are considered and the methodology in question is applied to estimate the parameters the values obtained are compared with those used in the simulation finally the model is applied to model the evolution of the trend of the dynamic variable average monthly salary cost for all sectors and broken down construction industry services in spain for the period 1985 2005

analytical results for fundamental time delayed feedback systems subjected to multiplicative noise we study the stochastic behavior of fundamental time delayed feedback systems subjected to multiplicative noise we derive exact results for the first and second moments and the autocorrelation function for a particular class of systems we show how the variance depends on the amplitude of the multiplicative noise furthermore we identify parameter regions of stationary solutions with finite and infinite variances finally we suggest that delay induced levy flights may occur in time delayed feedback systems involving multiplicative noise

exact statistics for linear time delayed oscillators subjected to gaussian excitation abstract large classes of stochastic systems of interest to acoustics and vibrations have time delays the presence of these time delays hinders efforts to provide physical insight into the system behavior because of the difficulty of finding exact solutions this paper addresses this issue by presenting exact steady state solutions for the probability density functions pdf s for linear time delayed oscillators subjected to gaussian excitation of arbitrary correlation a key parameter influencing these pdf s is shown to be the ratio of mean generalized kinetic energy to the mean generalized potential energy in a single damped harmonic oscillator this ratio is unity which implies an equi partition of generalized energy time delays cause deviations in the equi partition of generalized energy which creates regions of preferential total phase and causes the amplitude pdf to shift from a rayleigh distribution to a hoyt distribution

stochastic resonance in time delayed bistable systems driven by weak periodic signal abstract we study theoretically a bistable system with time delayed feedback driven by a weak periodic force the effective potential function and the steady state probability density are derived the delay time and the strength of its feedback can change the shapes of the potential wells in the adiabatic approximation the signal to noise ratio snr of the system with a weak periodic force is obtained the time delayed feedback modulates the magnitude of snr by changing the shape of the potential and the effective strength of the signal the maximum of snr decreases with increasing the feedback intensity when is negative or positive the time delay can suppress or promote the stochastic resonance phenomenon

on a perturbation of the benjamin ono equation abstract we prove using the fourier restriction norm method that the initial value problem associated to a perturbation of the benjamin ono equation u t u u x h u x x h u x u x x 0 where x r t 0 0 and h denotes the usual hilbert transform is locally well posed in the sobolev space h s r for any s 1 2 and globally well posed in h s r when s 0 for s 1 we also prove some ill posedness issues

frequency adaptation in controlled stochastic resonance utilizing delayed feedback method two pole approximation for response function stochastic resonance sr enhanced by time delayed feedback control is studied the system in the absence of control is described by a langevin equation for a bistable system and possesses a usual sr response the control with the feedback loop the delay time of which equals to one half of the period 2 of the input signal gives rise to a noise induced oscillatory switching cycle between two states in the output time series while its average frequency is just smaller than in a small noise regime as the noise intensity d approaches an appropriate level the noise constructively works to adapt the frequency of the switching cycle to and this changes the dynamics into a state wherein the phase of the output signal is entrained to that of the input signal from its phase slipped state the behavior is characterized by power loss of the external signal or response function this paper deals with the response function based on a dichotomic model a method of delay coordinate series expansion which reduces a non markovian transition probability flux to a series of memory fluxes on a discrete delay coordinate system is proposed its primitive implementation suggests that the method can be a potential tool for a systematic analysis of sr phenomenon with delayed feedback loop we show that a d dependent behavior of poles of a finite laplace transform of the response function qualitatively characterizes the structure of the power loss and we also show analytical results for the correlation function and the power spectral density

theory of oscillatory firing induced by spatially correlated noise and delayed inhibitory feedback a network of leaky integrate and fire neurons with global inhibitory feedback and under the influence of spatially correlated noise is studied we calculate the spectral statistics of the network power spectrum of the population activity cross spectrum between spike trains of different neurons as well as of a single neuron power spectrum of spike train cross spectrum between external noise and spike train within the network as shown by comparison with numerical simulations our theory works well for arbitrary network size if the feedback is weak and the amount of external noise does not exceed that of the internal noise by means of our analytical results we discuss the quality of the correlation induced oscillation in a large network as a function of the transmission delay and the internal noise intensity it is shown that the strongest oscillation is obtained in a system with zero internal noise and adiabatically long delay i e the delay period is longer than any other time scale in the system for a neuron with a strong intrinsic frequency the oscillation becomes strongly anharmonic in the case of a long delay time we also discuss briefly the kind of synchrony introduced by the feedback induced oscillation

fokker planck description for a linear delayed langevin equation with additive gaussian noise we construct an equivalent probability description of linear multi delay langevin equations subject to additive gaussian white noise by exploiting the time convolutionless transform and a time variable transformation we are able to write a fokker planck equation fpe for the 1 time and for the 2 time probability distributions valid irrespective of the regime of stability of the langevin equations we solve exactly the derived fpes and analyze the aging dynamics by studying analytically the conditional probability distribution we discuss explicitly why the initially conditioned distribution is not sufficient to describe fully out a non markov process as both preparation and observation times have bearing on its dynamics as our analytic procedure can also be applied to linear langevin equations with memory kernels we compare the non markov dynamics of a one delay system with that of a generalized langevin equation with an exponential as well as a power law memory application to a generalization of the green kubo formula is also presented

a refined shift and invert arnoldi algorithm for large unsymmetric generalized eigenproblems abstract the shift and invert arnoldi method has been popularly used for computing a number of eigenvalues close to a given shift and or the associated eigenvectors of a large unsymmetric matrix pair but there is no guarantee for the approximate eigenvectors ritz vectors obtained by this method to converge even though the subspace is good enough in order to correct this problem a refined shift and invert arnoldi method is proposed that uses certain refined ritz vectors to approximate the desired eigenvectors the refined ritz vectors can be computed cheaply and reliably by small sized singular value decompositions it is shown that the refined method converges a refined shift and invert arnoldi algorithm is developed and several numerical examples are reported comparisons are drawn on the refined algorithm and the shift and invert arnoldi algorithm indicating that the former is considerably more efficient than the latter

ito versus stratonovich white noise limits for systems with inertia and colored multiplicative noise we consider the dynamics of systems in the presence of inertia and colored multiplicative noise we study the limit where the particle relaxation time and the correlation time of the noise both tend to zero we show that the limiting equation for the particle position depends on the magnitude of the particle relaxation time relative to the noise correlation time in particular the limiting equation should be interpreted either in the ito or stratonovich sense with a crossover occurring when the two fast time scales are of comparable magnitude at the crossover the limiting stochastic differential equation is neither of ito nor of stratonovich type this means that after adiabatic elimination the governing equations have different drift fields leading to different physical behavior depending on the relative magnitude of the two fast time scales our findings are supported by numerical simulations

non gaussian noise weakened stability in a foraging colony system with time delay abstract in this paper the dynamical properties in a foraging colony system with time delay and non gaussian noise were investigated using delay fokker planck approach the stationary probability distribution spd the associated relaxation time art and normalization correlation function ncf are obtained respectively the results show that i the time delay and non gaussian noise can induce transition from a single peak to double peaks in the spd i e a type of bistability occurring in a foraging colony system where time delay and non gaussian noise not only cause transitions between stable states but also construct the states themselves numerical simulations are presented and are in good agreement with the approximate theoretical results ii there exists a maximum in the art as a function of the noise intensity this maximum for art is identified as the characteristic of the non gaussian noise weakened stability of the foraging colonies in the steady state iii the art as a function of the noise correlation time exhibits a maximum and a minimum where the minimum for art is identified as the signature of the non gaussian noise enhanced stability of the foraging colonies and iv the time delay can enhance the stability of the foraging colonies in the steady state while the departure from gaussian noise can weaken it namely the time delay and departure from gaussian noise play opposite roles in art or ncf

stochastic order parameter equation of isometric force production revealed by drift diffusion estimates we address two questions that are central to understanding human motor control variability what kind of dynamical components contribute to motor control variability i e deterministic and or random ones and how are those components structured to this end we derive a stochastic order parameter equation for isometric force production from experimental data using drift diffusion estimates we show that the force variability increases with the required force output because of a decrease of deterministic stability and an accompanying increase of noise intensity a structural analysis reveals that the deterministic component consists of a linear control loop while the random component involves a noise source that scales with force output in addition we present evidence for the existence of a subject independent overall noise level of human isometric force production

modeling delayed processes in biological systems delayed processes are ubiquitous in biological systems and are often characterized by delay differential equations ddes and their extension to include stochastic effects ddes do not explicitly incorporate intermediate states associated with a delayed process but instead use an estimated average delay time in an effort to examine the validity of this approach we study systems with significant delays by explicitly incorporating intermediate steps we show that such explicit models often yield significantly different equilibrium distributions and transition times as compared to ddes with deterministic delay values additionally different explicit models with qualitatively different dynamics can give rise to the same ddes revealing important ambiguities we also show that dde based predictions of oscillatory behavior may fail for the corresponding explicit model

residuals of refined projection methods for large matrix eigenproblems abstract in terms of the deviation of the eigenvector from a projection subspace e a priori error bounds are established for the residual norm of an approximate eigenpair obtained by a refined projection method it is shown that the residual converges to zero as the deviation tends to zero finally how to efficiently compute refined ritz vectors is discussed in the refined symmetric lanczos method

augmented moment method for stochastic ensembles with delayed couplings ii fitzhugh nagumo model dynamics of fitzhugh nagumo fn neuron ensembles with time delayed couplings subject to white noises has been studied by using both direct simulations and a semianalytical augmented moment method amm which has been proposed in a preceding paper h hasegawa phys rev e 70 021911 2004 for n unit fn neuron ensembles amm transforms original 2n dimensional stochastic delay differential equations sddes to infinite dimensional deterministic des for means and correlation functions of local and global variables infinite order recursive des are terminated at the finite level m in the level m amm mathrm amm m yielding 8 m 1 dimensional deterministic des when a single spike is applied the oscillation may be induced if parameters of coupling strength delay noise intensity and or ensemble size are appropriate effects of these parameters on the emergence of the oscillation and on the synchronization in fn neuron ensembles have been studied the synchronization shows the fluctuation induced enhancement at the transition between nonoscillating and oscillating states results calculated by amm5 are in fairly good agreement with those obtained by direct simulations

stochastic resonance in discrete kinetics with delay abstract multiplicative stochastic resonance in linear systems with delay is studied for the discrete case the overdamped linear system with symmetric multiplicative markovian dichotomous noise dn is considered the stability range of the solutions for different values of delay is analyzed the maximum of the signal to noise ratio is observed as a function of the noise intensity

influence of delayed time and colored noise in a bistable system subject to asymmetric dichotomous noise and colored noise the influence of delayed time and colored noise in a bistable system subject to asymmetric dichotomous noise and colored noise is studied applying small delay time approximation under the adiabatic limit condition the expression of the signal to noise ratio snr of the system is obtained it is found that the snr varies non monotonously with the delayed time and the correlation of the colored noise moreover the snr exhibits sr behavior when it is plotted as a function of the intensity and asymmetry of the dichotomous noise and as a function of the strength of the colored noise

multivariate markov processes for stochastic systems with delays application to the stochastic gompertz model with delay using the method of steps we describe stochastic processes with delays in terms of markov diffusion processes thus multivariate langevin equations and fokker planck equations are derived for stochastic delay differential equations natural periodic and reflective boundary conditions are discussed both ito and stratonovich calculus are used in particular our fokker planck approach recovers the generalized delay fokker planck equation proposed by guillouzic et al the results obtained are applied to a model for population growth the gompertz model with delay and multiplicative white noise

on reducible nonlinear time delayed stochastic systems fluctuation dissipation relations transitions to bistability and secondary transitions to non stationarity we show the conditions under which nonlinear time delayed dynamical systems with multiplicative noise sources can be transformed into linear time delayed systems with additive noise sources we show that for such reducible systems analytical expressions for stationary distributions can be obtained we demonstrate that fluctuation dissipation relations of reducible systems become trivial and we show that reducible systems may exhibit delay and noise induced transitions to bistability and secondary transitions to non stationarity our general findings are exemplified for three models a gompertz model a hongler model and a model involving a 1 x 2 noise amplitude

successive linear newton interpolation methods for solving the large scale nonlinear eigenvalue problems abstract we present the successive linear newton interpolation method for solving the large scale nonlinear eigenvalue problems establish locally linear convergence and give the corresponding convergence factor of the method in terms of the left and right eigenvectors in this paper to speed up the convergence rate we develop the modified successive linear newton interpolation method which updates the pole simultaneously in addition we propose the inexact versions of the modified successive linear newton interpolation method to reduce the computational cost and analyze the convergence properties numerical results demonstrate the effectiveness of our proposed methods

exact dynamics of stochastic linear delayed systems application to spatiotemporal coordination of comoving agents delayed dynamics result from finite transmission speeds of a signal in the form of energy mass or information in stochastic systems the resulting lagged dynamics challenge our understanding due to the rich behavioral repertoire encompassing monotonic oscillatory and unstable evolution despite the vast literature quantifying this rich behavior is limited by a lack of explicit analytic studies of high dimensional stochastic delay systems here we fill this gap for systems governed by a linear langevin equation of any number of delays and spatial dimensions with additive gaussian noise by exploiting laplace transforms we are able to derive an exact time dependent analytic solution of the langevin equation by using characteristic functionals we are able to construct the full time dependence of the multivariate probability distribution of the stochastic process as a function of the delayed and nondelayed random variables as an application we consider interactions in animal collective movement that go beyond the traditional assumption of instantaneous alignment we propose models for coordinated maneuvers of comoving agents applicable to recent empirical findings in pigeons and bats whereby individuals copy the heading of their neighbors with some delay we highlight possible strategies that individual pairs may adopt to reduce the variance in their velocity difference and or in their spatial separation we also show that a minimum in the variance of the spatial separation at long times can be achieved with certain ratios of measurement to reaction delay

pitchfork and hopf bifurcation thresholds in stochastic equations with delayed feedback the bifurcation diagram of a model stochastic differential equation with delayed feedback is presented we are motivated by recent research on stochastic effects in models of transcriptional gene regulation we start from the normal form for a pitchfork bifurcation and add multiplicative or parametric noise and linear delayed feedback the latter is sufficient to originate a hopf bifurcation in that region of parameters in which there is a sufficiently strong negative feedback we find a sharp bifurcation in parameter space and define the threshold as the point in which the stationary distribution function p x changes from a delta function at the trivial state x 0 to p x approximately x alpha at small x with alpha 1 exactly at threshold we find that the bifurcation threshold is shifted by fluctuations relative to the deterministic limit by an amount that scales linearly with the noise intensity analytic calculations of the bifurcation threshold are also presented in the limit of small delay tau 0 that compare quite favorably with the numerical solutions even for moderate values of tau

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

some remarks on the smoluchowski kramers approximation according to the smoluchowski kramers approximation solution qt of the equation mu ddot q t mu b q t mu dot q t mu sigma q t mu dot w t q 0 q dot q p where dot w t is the white noise converges to the solution of equation dot q t b q t sigma q t dot w t q 0 q as 0 many asymptotic problems for the last equation were studied in recent years we consider relations between asymptotics for the first order equation and the original second order equation homogenization large deviations and stochastic resonance approximation of brownian motion wt by a smooth stochastic process stationary distributions are considered

theoretical analysis of destabilization resonances in time delayed stochastic second order dynamical systems and some implications for human motor control a linear stochastic delay differential equation of second order is studied that can be regarded as a kramers model with time delay an analytical expression for the stationary probability density is derived in terms of a gaussian distribution in particular the variance as a function of the time delay is computed analytically for several parameter regimes strikingly in the parameter regime close to the parameter regime in which the deterministic system exhibits hopf bifurcations we find that the variance as a function of the time delay exhibits a sequence of pronounced peaks these peaks are interpreted as delay induced destabilization resonances arising from oscillatory ghost instabilities on the basis of the obtained theoretical findings reinterpretations of previous human motor control studies and predictions for future human motor control studies are provided 2006 the american physical society

effects of small time delay on a bistable system subject to levy stable noise l vy stable noise is often used to describe impulsive noise bursting in communication systems this paper investigates the effects of small time delay on a bistable system driven by an aperiodic bipolar pulse signal and l vy stable noise we obtain the dynamical probability density of the system response by solving the approximated time delayed fractional fokker planck equation ffpe via an implicit finite difference method a new approach to evaluate the system response time is presented the bit error rate ber is employed to measure the performance of the bistable system in detecting binary signals the theoretical ber is validated by the monte carlo simulation we find that the existence of time delay can change both the drift term and the diffusion coefficient in time delayed ffpe for small noise intensity the time delay extends the system response time and thus reduces the detection performance however effects of this kind will fade away with the increase of noise intensity

fokker planck perspective on stochastic delay systems exact solutions and data analysis of biological systems stochastic delay systems with additive noise are examined from the perspective of fokker planck equations for a linear system the exact stationary probability density is derived by means of a delay fokker planck equation we show how to determine the delay equation of the linear system from experimental data and corroborate a fundamental result previously obtained by kuchler and mensch we also propose a method to derive delay equations of nonlinear systems from experimental data to this end the theory of multivariate fokker planck equations is used the applicability of this method is demonstrated for stochastic models describing tracking and pointing movements of humans

stochastic resonance in a tumor immune system subject to bounded noises and time delay abstract immunotherapy is one of the most recent approaches in cancer therapy a mathematical model of tumor immune interaction subject to a periodic immunotherapy treatment imitated by a periodic signal correlative and bounded stochastic fluctuations and time delays is investigated by numerical simulations for its signal power amplification spa within the tailored parameter regime the synchronous response of tumor growth to the immunotherapy stochastic resonance sr versus both the noises and delays is obtained the details are as follows i the peak values of spa versus the noise intensity a in the proliferation term of tumor cells decrease as the frequency of periodic signal increases i e an increase of the frequency restrains the sr ii an increase of the amplitude of periodic signal restrains the sr versus a but boosts up the sr versus the noise intensity b in the immune term iii there is an optimum cross correlated degree between the two bounded noises at which the system exhibits the strongest sr versus the delay time the reaction time of tumor cell population to their surrounding environment constraints iv upon increasing the delay time double sr versus the delay time the time taken by both the tumor antigen identification and tumor stimulated proliferation of effectors emerges these results may be helpful for an immunotherapy treatment for the sufferer

fdfd method based on refined shift and invert arnoldi technique for periodic leaky wave antennas the finite difference frequency domain fdfd method can be used to analyze the properties of arbitrary complex close open periodic guided wave structures for the open problem it probably results in a large scale nonsymmetrical generalized eigenvalue problem in the analysis of the leaky wave antennas because the leakage constant attenuation constant increases a lot in the working frequency range the extraction process of the eigenvalues complex propagation constants is difficult to converge when the shift and invert arnoldi technique is used to resolve this large scale nonsymmetrical generalized eigenvalue problem in this paper a refined shift and invert arnoldi algorithm is adopted to speed the convergence rate by using refined ritz vectors to approximate the desired eigenvectors since the refined method always converges a simple fdfd method with six field components can be used the difference equations of this method are much simpler than those of the previous method because the longitudinal field components don t have to be eliminated in order to approximately transform the generalized eigenvalue problem to a standard eigenvalue problem consequently the accuracy of the method can be improved and the boundary conditions can be implemented easily

stochastic time delayed systems driven by correlated noises steady state analysis abstract in the present paper a class of stochastic time delayed systems driven by correlated gaussian white noises are considered novikov s theorem is used to derive delay fokker planck equations in the case of small time delays approximate stationary solutions are obtained as a special case the delay fokker planck equation and the approximate stationary probability density function is obtained for a bistable system numerical simulations show that the small delay approximation is a good approximation in the case of small delay furthermore the effects of the correlated noises and the feedback are investigated the critical curve separating the unimodal and bimodal regions of the stationary probability distribution is shown to be affected by the degree of the correlation of the noises and e the strength of the feedback both and e can change the curve of the stationary probability density function from a bimodal to a unimodal structure

on decay properties and asymptotic behavior of solutions to a non local perturbed kdv equation abstract we consider the kdv equation with an additional non local perturbation term defined through the hilbert transform also known as the ost equation we prove that the solutions u t x have a pointwise decay in spatial variable u t x 1 1 x 2 provided that the initial data has the same decaying and moreover we find the asymptotic profile of u t x when x next we show that decay rate given above is optimal when the initial data is not a zero mean function and in this case we derive an estimate from below 1 x 2 u t x for x large enough in the case when the initial datum is a zero mean function we prove that the decay rate above is improved to 1 1 x 2 e for 0 e 1 finally we study the local well posedness of the ost equation in the framework of lebesgue spaces

coherence of noisy oscillators with delayed feedback a theory ofe ect ofdelayed f on the coherence ofthe noisy self sustained oscillations is developed in the gaussian approximation a closed system ofequations is derived f the phase di usion constant and the mean frequency a comparison with numerics shows that the theory works well for weak feedback and strong noise c

functional characterization of linear delay langevin equations we present an exact functional characterization of linear delay langevin equations driven by any noise structure defined through its characteristic functional this method relies on the possibility of finding an explicitly analytical expression for each realization of the delayed stochastic process in terms of those of the driving noise general properties of the transient dissipative dynamics are analyzed the corresponding interplay with a color gaussian noise is presented as a full application of our functional method we study a model for population growth with non gaussian fluctuations the gompertz model driven by multiplicative white shot noise

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation

transitions induced by time delays and cross correlated sine wiener noises in a tumor immune system interplay abstract a model of tumor immune system interplay with time delays and cross correlated sine wiener noises is investigated by numerical simulations for the stationary probability distribution spd and stationary mean value x s t of tumor cell population results show that i the structure of the spd transfers from bimodal to unimodal as the reaction time of the cell population to environmental constraints increases ii conversely as the time of immune system to develop specific immune competence and the correlation time of noises increase the structure of the spd transfers from unimodal to bimodal iii as the cross correlated intensity between noises increases the transitions induced by are suppressed

a new maximal inequality and invariance principle for stationary sequences we derive a new maximal inequality for stationary sequences under a martingale type condition introduced by maxwell and woodroofe ann probab 28 2000 713 724 then we apply it to establish the donsker invariance principle for this class of stationary sequences a markov chain example is given in order to show the optimality of the conditions imposed

rational krylov sequence methods for eigenvalue computation abstract algorithms to solve large sparse eigenvalue problems are considered a new class of algorithms which is based on rational functions of the matrix is described the lanczos method the arnoldi method the spectral transformation lanczos method and rayleigh quotient iteration all are special cases but there are also new algorithms which correspond to rational functions with several poles in the simplest case a basis of a rational krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of hessenberg matrices

effects of time delay on transient behavior of a time delayed metastable system subjected to cross correlated noises abstract the effects of time delay on the transient properties of a time delayed metastable system subjected to cross correlated noises are studied by means of a stochastic simulation method it is found that i both additive noise and multiplicative noise can produce the noise enhanced stability nes effect ii the time delay induces critical behavior on the nes i e there is a critical value of the delay time c 1 2 2 above which the time delay increases the stability of the system enhanced by the additive noise and below which the nes effect induced by the additive noise disappears iii there exists another critical value of the delay time c 2 3 0 above which the time delay increases the stability of the system enhanced by the multiplicative noise and below which the time delay decreases it

delay fokker planck equations novikov s theorem and boltzmann distributions as small delay approximations we study time delayed stochastic systems that can be described by means of so called delay fokker planck equations using novikov s theorem we first show that the theory of delay fokker planck equations is on an equal footing with the theory of ordinary fokker planck equations subsequently we derive stationary distributions in the case of small time delays in the case of additive noise systems these distributions can be cast into the form of boltzmann distributions involving effective potential functions

on the well posedness for the generalized ostrovsky stepanyams and tsimring equation abstract in this work we prove that the initial value problem associated with equation u t u xxx h u x h u xxx u k u x 0 x r t 0 where 0 and h denotes the usual hilbert transform is locally well posed in sobolev spaces h s r for s 0 and k 1 2 3 and globally well posed in l 2 r

stationary distributions of stochastic processes described by a linear neutral delay differential equation stationary distributions of processes are derived that involve a time delay and are defined by a linear stochastic neutral delay differential equation the distributions are gaussian distributions the variances of the gaussian distributions are either monotonically increasing or decreasing functions of the time delays the variances become infinite when fixed points of corresponding deterministic processes become unstable

the smoluchowski kramers limit of stochastic differential equations with arbitrary state dependent friction we study a class of systems of stochastic differential equations describing diffusive phenomena the smoluchowski kramers approximation is used to describe their dynamics in the small mass limit our systems have arbitrary state dependent friction and noise coefficients we identify the limiting equation and in particular the additional drift term that appears in the limit is expressed in terms of the solution to a lyapunov matrix equation the proof uses a theory of convergence of stochastic integrals developed by kurtz and protter the result is sufficiently general to include systems driven by both white and ornstein uhlenbeck colored noises we discuss applications of the main theorem to several physical phenomena including the experimental study of brownian motion in a diffusion gradient

stationary solutions of linear stochastic delay differential equations applications to biological systems recently k uchler and mensch stochastics stochastics rep 40 23 1992 derived exact stationary probability densities for linear stochastic delay differential equations this paper presents an alternative derivation of these solutions by means of the fokker planck approach introduced by guillouzic phys rev e 59 3970 1999 61 4906 2000 applications of this approach which is argued to have greater generality are discussed in the context of stochastic models for population growth and tracking movements

system reliability of randomly vibrating structures computational modeling and laboratory testing the problem of determination of system reliability of randomly vibrating structures arises in many application areas of engineering we discuss in this paper approaches based on monte carlo simulations and laboratory testing to tackle problems of time variant system reliability estimation the strategy we adopt is based on the application of girsanov s transformation to the governing stochastic differential equations which enables estimation of probability of failure with significantly reduced number of samples than what is needed in a direct simulation study notably we show that the ideas from girsanov s transformation based monte carlo simulations can be extended to conduct laboratory testing to assess system reliability of engineering structures with reduced number of samples and hence with reduced testing times illustrative examples include computational studies on a 10 degree of freedom nonlinear system model and laboratory computational investigations on road load response of an automotive system tested on a four post lest rig c 2015 elsevier ltd all rights reserved

stochastic delay differential equations for genetic regulatory networks time delay is an important aspect in the modelling of genetic regulation due to slow biochemical reactions such as gene transcription and translation and protein diffusion between the cytosol and nucleus in this paper we introduce a general mathematical formalism via stochastic delay differential equations for describing time delays in genetic regulatory networks based on recent developments with the delay stochastic simulation algorithm the delay chemical master equation and the delay reaction rate equation are developed for describing biological reactions with time delay which leads to stochastic delay differential equations derived from the langevin approach two simple genetic regulatory networks are used to study the impact of intrinsic noise on the system dynamics where there are delays

a new algorithm for computing the inertia of eigenproblems ax λx and ax λbx abstract the inertia of a n n complex matrix a is defined to be an integer triple in a a a a where a is the number of eigenvalues of a with positive real parts a is the number of eigenvalues with negative real parts and a is the number of eigenvalues with zero real parts in this paper we show that the inertia can be computed by gereshgorin theorem and block shift and invert algorithm for equation ax bx or ax x and this algorithm is compared by function eig m and sptarn m in matlab

stochastic resonance in a bistable system subject to multi time delayed feedback and aperiodic signal we discuss in detail the effects of the multi time delayed feedback driven by an aperiodic signal on the output of a stochastic resonance sr system the effective potential function and dynamical probability density function pdf are derived to measure the performance of the sr system in the presence of a binary random signal the bit error rate ber defined by the dynamical pdf is employed as is commonly used in digital communications we find that the delay time strength of the feedback and number of time delayed terms can change the effective potential function and the effective amplitude of the signal and then affect the ber of the sr system the numerical simulations strongly support the theoretical results the goal of this investigation is to explore the effects of the multi time delayed feedback on sr and give a guidance to nonlinear systems in the application of information processing

a refined iterative algorithm based on the block arnoldi process for large unsymmetric eigenproblems when the matrix in question is unsymmetric the approximate eigenvectors or ritz vectors obtained by orthogonal projection methods including arnoldi s method and the block arnoldi method cannot be guaranteed to converge in theory even if the corresponding approximate eigenvalues or ritz values do in order to circumvent this danger a new strategy has been proposed by the author for arnoldi s method the strategy used is generalized to the block arnoldi method in this paper it discards ritz vectors and instead computes refined approximate eigenvectors by small sized singular value decompositions it is proved that the new strategy can guarantee the convergence of refined approximate eigenvectors if the corresponding ritz values do the resulting refined iterative algorithm is realized by the block arnoldi process numerical experiments show that the refined algorithm is much more efficient than the iterative block arnoldi algorithm