almost sure exponential stability of numerical solutions to stochastic delay hopfield neural networks stability of numerical solutions to stochastic delay differential equations have received an increasing attention but there has been so far little work on the stability analysis of numerical solutions to stochastic delay hopfield neural networks the aim of this paper is to study the almost sure exponential stability of numerical solutions to stochastic delay hopfield neural networks by using two approaches the euler method and the backward euler method under some reasonable conditions both the euler scheme and the backward euler scheme are proved to be almost sure exponential stability in particular the euler method and the backward euler method are mainly based on the semimartingale convergence theorem

convergence and stability of numerical solutions to sddes with markovian switching a class of linear stochastic delay differential equations with markovian switching is considered the main aim of this paper is to investigate the convergence and stability of the euler method of the equations it is proved that the euler method is convergent with strong order p 1 2 the ms stable properties of the euler scheme are also studied

stability analysis of split step θ milstein method for a class of n dimensional stochastic differential equations abstract in this paper we introduce a split step theta milstein sstm method for n dimensional stochastic delay differential equations sddes the exponential mean square stability of the numerical solutions is analyzed and in accordance with previous findings we prove that the method is exponentially mean square stable if the employed time step is smaller than a given and easily computable upper bound in particular according to our investigation larger time steps can be used in the case 1 2 1 than in the case 0 1 2 numerical results are presented which reveal that the sstm method is conditionally mean square stable and that in the case 1 2 1 the interval of time steps for which the sstm method is theoretically shown to be mean square stable is significantly larger than in the case 0 1 2 it is worth mentioning that the sstm method has never been employed or analyzed for the numerical approximation of sddes at least to the very best of our knowledge

numerical solution of stochastic differential problems in the biosciences stochastic differential equations sdes models play a prominent role in many application areas including biology epidemiology and population dynamics mostly because they can offer a more sophisticated insight through physical phenomena than their deterministic counterparts do so suitable numerical methods must be introduced to simulate the solutions of the resulting stochastic differential systems in this work we take into account both euler taylor expansion and runge kutta type methods for stochastic ordinary differential equations sodes and the euler maruyama method for stochastic delay differential equations sddes focusing on the most relevant implementation issues the corresponding matlab codes for both sodes and sddes problems are tested on mathematical models arising in the biosciences math comput simulation numerical analysis of a campbell like stochastic delay model preprint

the αth moment stability for the stochastic pantograph equation in this paper we investigate the ath moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the razumikhin technique especially the linear stochastic pantograph equations and the semi implicit euler method applying them are considered the convergence result of the semi implicit euler method is obtained the stability conditions of the analytic solution of those equations and the numerical method are given finally some experiments are given

mean square stability of analytic solution and euler maruyama method for impulsive stochastic differential equations from the view of algebra the mean square stability of analytic solutions and numerical solutions for impulsive stochastic differential equations are considered by the logarithmic norm the conditions under which the analytic and numerical solutions for a linear impulsive stochastic differential equation are mean square stable ms stable respectively are obtained the conditions are simple and easy to use some numerical experiments are given to illustrate the results

friction induced vibration in periodic linear elastic media for a one dimensional finite elastic continuum with distributed contacts and periodic boundary conditions the presence of unstable waves is investigated the stability of the waves is evaluated and explanations for instabilities under a constant coefficient of friction are provided a negative slope in the coefficient of friction as a function of sliding speed is not a necessary condition for the occurrence of dynamic instability dynamic instability occurs in the form of self excited unstable travelling waves the stabilizing effects of external and internal damping were studied low and high frequency terms of the travelling waves are stabilized by adding external and internal damping respectively responses corresponding to unstable eigenvalues can dominate the system response it is presumed that this can lead to squeaking or squealing noise in applications

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

oscillation analysis of numerical solutions for delay differential equations with real coefficients abstract the paper is mainly concerned with the oscillations of numerical solutions by the linear methods for a kind of linear delay differential equations with positive and negative coefficients some conditions under which the numerical solutions are oscillatory are obtained and it is proved that oscillations of the analytical solutions are preserved by the numerical solutions under mild conditions numerical experiments are given to demonstrate our results

vector valued stochastic delay equations a weak solution and its markovian representation abstract a class of stochastic delay equations in a type 2 umd banach space e driven by the cylindrical wiener process is studied we investigate two concepts of solutions weak and generalised strong and give conditions under which they are equivalent we present an evolution equation approach in a banach space e p e l p 1 0 e proving that the solutions can be reformulated as e p valued markov processes based on the markovian representation we prove the existence and continuity of the solutions the results are applied to stochastic reaction diffusion equations with delays

using waveform relaxation methods to approximate neutral stochastic functional differential equation systems this paper extends the waveform relaxation wr method to neutral stochastic functional differential equations the linear convergence theory of the continuous time wr method is established in the mean square sense when the coefficients of stochastic differential equation systems satisfy the lipschitz condition and the contractive mapping the discrete time wr method based on the euler scheme which is used in an actual implementation was also studied it turns out that the sequence produced by this method converges linearly to the euler approximate solution which is convergent in addition we prove that the preceding methods are convergent superlinearly if the neutral term does not been split finally the theory is applied to a one dimensional model problem and checked against results obtained by numerical experiments

convergence of numerical solutions to neutral stochastic delay differential equations with markovian switching recently numerical solutions of stochastic differential equations have received a great deal of attention it is surprising that there are not any numerical methods established for neutral stochastic delay differential equations yet in the paper the euler maruyama method for neutral stochastic delay differential equations is developed the key aim is to show that the numerical solutions will converge to the true solutions under the local lipschitz condition

a geometrically non linear model of rotating shafts with internal resonance and self excited vibration a geometrically non linear model of the rotating shaft is introduced which includes karman non linearity non linear curvature effects large displacements and rotations as well as gyroscopic effects through applying timoshenko type assumptions the shear effects are also included in the model convenient matrix descriptions are used in order to facilitate the analysis based on galerkin and continuation methods the model is used to analyze the phenomenon of internal resonance the influence of some of the system s parameters on the amplitude and frequency of self excited vibration is investigated

new approximations and policy implications from a delayed dynamic model of a fast pandemic abstract we study an seiqr susceptible exposed infectious quarantined recovered model due to young et al 2019 for an infectious disease with time delays for latency and an asymptomatic phase for fast pandemics where nobody has prior immunity and everyone has immunity after recovery the seiqr model decouples into two nonlinear delay differential equations ddes with five parameters one parameter is set to unity by scaling time the simple subcase of perfect quarantining and zero self recovery before quarantine with two free parameters is examined first the method of multiple scales yields a hyperbolic tangent solution and a long wave short delay approximation yields a first order ordinary differential equation ode with imperfect quarantining and nonzero self recovery the long wave approximation is a second order ode these three approximations each capture the full outbreak from infinitesimal initiation to final saturation low dimensional dynamics in the ddes is demonstrated using a six state non delayed reduced order model obtained by galerkin projection numerical solutions from the reduced order model match the dde over a range of parameter choices and initial conditions finally stability analysis and numerics show how a well executed temporary phase of social distancing can reduce the total number of people affected the reduction can be by as much as half for a weak pandemic and is smaller but still substantial for stronger pandemics an explicit formula for the greatest possible reduction is given

on piterbarg s max discretisation theorem for homogeneous gaussian random fields abstract motivated by the papers of piterbarg 17 and husler 7 in this paper the asymptotic relation between the maximum of a continuous dependent homogeneous gaussian random field and the maximum of this field sampled at discrete time points is studied it is shown that for the weakly dependent case these two maxima are asymptotically independent dependent or coincide when the grid of the discrete time points is a sparse grid pickands grid or dense grid respectively while for the strongly dependent case these two maxima are asymptotically totally dependent if the grid of the discrete time points is sufficiently dense and asymptotically dependent if the grid points are sparse or pickands grids

extreme values and high boundary crossings of locally stationary gaussian processes we consider the large values of a locally stationary gaussian process which satisfies berman s condition on the long range dependence we present some limit results on the exceedances of the process above a certain general smooth high boundary this allows deriving the limiting distribution of the maximum up to time t for example in the case of a standardized process with a constant boundary or in the case of a nonstandardized process with a smooth trend

retarded differential equations retarded differential equations rdes are differential equations having retarded arguments they arise in many realistic models of problems in science engineering and medicine where there is a time lag or after effect numerical techniques for such problems may be regarded as extensions of dense output methods for ordinary differential equations odes but scalar rdes are inherently infinite dimensional with a richer structure than their ode counterparts we give background material develop a theoretical foundation for the basic numerics and give some results not previously published

delay dependent stability of stochastic split step θ methods for stochastic delay differential equations abstract in this paper the delay dependent asymptotic mean square stability of the stochastic split step method for a scalar linear stochastic delay differential equation with real coefficients is studied the full stability region of this method is given by using root locus technique the necessary and sufficient condition with respect to the equation coefficients time stepsize and method parameter is derived it is proved that the stochastic split step backward euler can preserve the asymptotic mean square stability of the underlying system completely in addition the numerical stability regions of the stochastic split step method and the stochastic method are compared with each other at last we validate our conclusions by numerical experiments

exponential stability of equidistant euler maruyama approximations of stochastic differential delay equations our aim is to study under what conditions the exact and numerical solution based on equidistant nonrandom partitions of integration time intervals to a stochastic differential delay equation sdde share the property of mean square exponential stability our approach is trying to avoid the use of lyapunov functions or functionals we show that under a global lipschitz assumption an sdde is exponentially stable in mean square if and only if for some sufficiently small stepsize d the euler maruyama em method is exponentially stable in mean square we then replace the global lipschitz condition with a finite time convergence condition and establish the same if and only if result the important feature of this result is that it transfers the asymptotic problem into a finite time convergence problem replacing the exact and em numerical solution with stochastic processes we also discuss whether a family of stochastic processes share the stability property this new approach allows us to discuss i whether a family of sddes share the stability property and ii whether an sdde with variable time lag shares stability property with the modified em method as another application of this general approach we consider a linear sdde with variable time lag and establish an if and only if result it should also be mentioned that the questions addressed results proved as well as style of analysis borrow heavily from 14 but the computations involved to cope with time delay are nontrivial

discretisation of stochastic control problems for continuous time dynamics with delay as a main step in the numerical solution of control problems in continuous time the controlled process is approximated by sequences of controlled markov chains thus discretising time and space a new feature in this context is to allow for delay in the dynamics the existence of an optimal strategy with respect to the cost functional can be guaranteed in the class of relaxed controls weak convergence of the approximating extended markov chains to the original process together with convergence of the associated optimal strategies is established

stability analysis of general linear methods for nonlinear neutral delay differential equations abstract the present paper is concerned with the numerical solution to initial value problems of neutral delay differential equations nddes the sufficient conditions for the stability and asymptotic stability of the general linear methods for a class r of nonlinear systems of nddes are derived numerical test is given that confirm the theoretical results in the end

instabilities and bifurcations in a rotating shaft abstract flexural vibrations of a slender balanced rotating shaft are considered the shaft is of circular cross section it rotates at a constant rate about its longitudinal axis of symmetry and is made of a viscoelastic material the effects of large transverse displacements and external sources of dissipation are also included the resulting non linear model is analyzed for stability and bifurcations of the trivial shaft configuration the undeformed straight position the instabilities are predicted by using the linearized shaft model and the post critical behaviors are determined by applying center manifold theory to the full non linear equations of motion the bifurcations encountered are simple hopf double hopf and double zero eigenvalues it is determined that synchronous whirl non synchronous whirl and competing mode types of behavior are possible for various ranges of the parameter values the results obtained prove that the modal truncations typically used when analyzing continuous shaft instabilities are mathematically justifiable

convergence analysis of fitted numerical method for a singularly perturbed nonlinear volterra integro differential equation with delay abstract in this paper the initial value problem for a quasilinear singularly perturbed delay volterra integro differential equation was considered by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder term in integral form a fitted difference scheme is constructed and analysed it is shown that the method displays first order uniform convergence in perturbation parameter some numerical results are given to confirm the theoretical analysis

asymptotic mean square stability of explicit runge kutta maruyama methods for stochastic delay differential equations as relatively little is known about runge kutta type method applied to stochastic delay differential equations we present an explicit runge kutta maruyama rkm method for solving them the mean square stability properties of the numerical solutions generated by the rkm method are investigated and a sufficient condition for stability is obtained and applied to the s rock type methods for stochastic delay differential equations numerical examples are provided to confirm theoretical results

an analysis of stability of milstein method for stochastic differential equations with delay this paper deals with the adapted milstein method for solving linear stochastic delay differential equations it is proved that the numerical method is mean square ms stable under suitable conditions the obtained result shows that the method preserves the stability property of a class of linear constant coefficient problems this is also verified by several numerical examples

numerical modelling in biosciences using delay differential equations our principal purposes here are i to consider from the perspective of applied mathematics models of phenomena in the biosciences that are based on delay differential equations and for which numerical approaches are a major tool in understanding their dynamics ii to review the application of numerical techniques to investigate these models we show that there are prima facie reasons for using such models i they have a richer mathematical framework compared with ordinary differential equations for the analysis of biosystem dynamics ii they display better consistency with the nature of certain biological processes and predictive results we analyze both the qualitative and quantitative role that delays play in basic time lag models proposed in population dynamics epidemiology physiology immunology neural networks and cell kinetics we then indicate suitable computational techniques for the numerical treatment of mathematical problems emerging in the biosciences comparing them with those implemented by the bio modellers

convergence and stability of the semi implicit euler method for a linear stochastic differential delay equation the paper deals with convergence and stability of the semi implicit euler method for a linear stochastic differential delay equation it is proved that the semi implicit euler method is convergent with strong order p the conditions under which the method is ms stable and gms stable are determined and the numerical experiments are given

mean square convergence of one step methods for neutral stochastic differential delay equations this paper deals with strong approximations of the solutions of neutral stochastic differential delay equations nsddes in ito sense a general framework for the strong convergence of a class of drift implicit one step schemes to the solutions of nsddes is established two examples to illustrate the applicability of our results are provided

a derivative free explicit method with order 1 0 for solving stochastic delay differential equations a new explicit stochastic scheme of order 1 is proposed for solving stochastic delay differential equations sddes with sufficiently smooth drift and diffusion coefficients and a scalar wiener process the method is derivative free and is shown to be stable in mean square a stability theorem for the continuous strong approximation of the solution of a linear test equation by the milstein method is also proved which shows the stepsize restriction for stability is larger than those given previously in the literature the case of linear sddes is further investigated in order to compare the stepsize restrictions for stability of these two methods numerical experiments are given to illustrate the obtained stability properties

convergence analysis of semi implicit euler methods for solving stochastic equations with variable delays and random jump magnitudes in this paper we study the following stochastic equations with variable delays and random jump magnitudes dx t f x t x t t t dt g x t x t t t dw t h x t x t t t c n t 1 dn t 0 t t x t j t r t 0 we establish the semi implicit euler approximate solutions for the above systems and prove that the approximate solutions converge to the analytical solutions in the mean square sense as well as in the probability sense some known results are generalized and improved

numerical bifurcation analysis of immunological models with time delays in recent years a large number of mathematical models that are described by delay differential equations ddes have appeared in the life sciences to analyze the models dynamics numerical methods are necessary since analytical studies can only give limited results in turn the availability of efficient numerical methods and software packages encourages the use of time delays in mathematical modelling which may lead to more realistic models we outline recently developed numerical methods for bifurcation analysis of ddes and illustrate the use of these methods in the analysis of a mathematical model of human hepatitis b virus infection

extremes on different grids and continuous time of stationary processes abstract this paper investigates asymptotical distributions of extremes on different grids and continuous time of stationary processes the findings show that these joint extremes over certain threshold dependent grids are asymptotically completely dependent dependent and independent respectively for dense pickands and sparse grids furthermore an integral representation of the asymptotic function involved in the approximation is obtained

parametric stabilization of a gyroscopic system this paper studies the stabilization of a gyroscopic system using parametric stabilization near a combination resonance the gyroscopic system is near its primary instability i e the bifurcation parameter is such that the system possesses a double zero eigenvalue the stability of the system is studied for the linear hamiltonian system the damped linear system the forced linear hamiltonian system and finally the damped and forced linear system the addition of the periodic excitation near the critical combination resonance provides the system with an extended stability region when the excitation frequency is slightly above the combination resonance a non linear numerical example shows that these results may persist for the non linear problem the results of this work are then discussed in relation to an example gyroscopic problem a rotating shaft with periodically perturbed rotation rate

the existence and asymptotic estimations of solutions to stochastic pantograph equations with diffusion and levy jumps in this paper we consider a class of stochastic pantograph differential equations with levy jumps spdewljs by using the burkholder davis gundy inequality and the kunita s inequality we prove the existence and uniqueness of solutions to spdewljs whose coefficients satisfying the lipschitz conditions and the local lipschitz conditions meantime we establish the p th exponential estimations and almost surely asymptotic estimations of solutions to spdewljs

mean square exponential stability of stochastic genetic regulatory networks with time varying delays in this paper we study the mean square exponential stability of stochastic genetic regulatory networks with time varying delays two kinds of time varying delays are considered one is differentiable with bounded delay derivative the other is continuous without constraints on the delay derivative in order to investigate the mean square exponential stability in stochastic genetic regulatory networks some novel rate dependent independent mean square exponential stability criteria are derived by constructing lyapunov krasovskii functional the sufficient conditions are given in terms of linear matrix inequalities moreover illustrative examples are used to substantiate the effectiveness and less conservativeness of our results

control design for output tracking of delayed boolean control networks abstract this paper investigates the output tracking control of delayed boolean control networks dbcns by using the algebraic state space representation method firstly the output tracking control problem of dbcns is formulated and the dynamics of dbcns is converted into an equivalent algebraic form secondly based on the algebraic form and controllability of dbcns a necessary and sufficient condition is presented for the output tracking control of dbcns without state constraints thirdly a necessary and sufficient condition is presented for the output tracking control of dbcns with state constraints by constructing a constrained reachability matrix the study of an illustrative example shows the effectiveness of the obtained new results

a finite difference method for a singularly perturbed delay integro differential equation we consider the singularly perturbed initial value problem for a linear first order volterra integro differential equation with delay our purpose is to construct and analyse a numerical method with uniform convergence in the perturbation parameter the numerical solution of this problem is discretized using implicit difference rules for differential part and the composite numerical quadrature rules for integral part on a layer adapted mesh error estimations for the approximate solution are established numerical examples supporting the theory are presented

ms stability of the euler maruyama method for stochastic differential delay equations the aim of this paper is to investigate the stability of the euler maruyama method for the stochastic differential equations with time delay the definition of ms stability of numerical methods is established the condition of the mean square stability of analytical solution is obtained for a linear scalar system with multiplicative noise and the ms stability of the numerical scheme is proved

rival approaches to mathematical modelling in immunology in order to formulate quantitatively correct mathematical models of the immune system one requires an understanding of immune processes and familiarity with a range of mathematical techniques selection of an appropriate model requires a number of decisions to be made including a choice of the modelling objectives strategies and techniques and the types of model considered as candidate models the authors adopt a multidisciplinary perspective

introduction to the numerical analysis of stochastic delay differential equations abstract we consider the problem of the numerical solution of stochastic delay differential equations of ito form d x t f x t x t d t g x t x t d w t t 0 t and x t t for t 0 with given f g wiener noise w and given 0 with a prescribed initial function we indicate the nature of the equations of interest and give a convergence proof for explicit single step methods some illustrative numerical examples using a strong euler maruyama scheme are provided

numerical solutions of neutral stochastic functional differential equations this paper examines the numerical solutions of neutral stochastic functional differential equations nsfdes d x t u x t f x t dt g x t dw t t geq 0 the key contribution is to establish the strong mean square convergence theory of the euler maruyama approximate solution under the local lipschitz condition the linear growth condition and contractive mapping these conditions are generally imposed to guarantee the existence and uniqueness of the true solution so the numerical results given here are obtained under quite general conditions although the way of analysis borrows from x mao lms j comput math 6 2003 pp 141 161 to cope with u x t several new techniques have been developed

reconstruction of time delayed feedback systems from time series for various classes of time delay systems we propose the methods of their model delay differential equation reconstruction from time series the methods are based on the characteristic location of extrema in the time series of time delay systems and the projection of infinite dimensional phase space of these systems to suitably chosen low dimensional subspaces we verify our methods by using them for the recovery of time delay differential equations from their chaotic solutions and for modelling experimental systems with delay induced dynamics from their chaotic time series

existence and uniqueness of the solutions and convergence of semi implicit euler methods for stochastic pantograph equations abstract in this paper the sufficient conditions of existence and uniqueness of the solutions for stochastic pantograph equation are given i e the local lipschitz condition and the linear growth condition under the lipschitz condition and the linear growth condition it is proved that the semi implicit euler method is convergence with strong order 1 2

algebraic formulation and topological structure of boolean networks with state dependent delay abstract this paper investigates the algebraic formulation topological structure and set stability of boolean networks with state dependent delay sdd firstly using the algebraic state space representation assr method the dynamics of boolean networks with sdd is converted into an equivalent augmented system secondly based on the equivalent augmented system some necessary and sufficient conditions are presented to calculate fixed points and cycles of boolean networks with sdd thirdly it is proved that the set stability of boolean networks with sdd is equivalent to the set stability of augmented system and a necessary and sufficient condition is presented by constructing a kind of set stability matrix finally the obtained results are applied to the strategy consensus of networked evolutionary games with memories

recovery of systems with a linear filter and nonlinear delay feedback in periodic regimes we propose a set of methods for the estimation of the parameters of time delay systems with a linear filter and nonlinear delay feedback performing periodic oscillations the methods are based on an analysis of the system response to regular external perturbations and are valid only for systems whose dynamics can be perturbed the efficiency of the methods is illustrated using both numerical and experimental data

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

entropy production and fluctuation theorems for langevin processes under continuous non markovian feedback control continuous feedback control of langevin processes may be non markovian due to a time lag between the measurement and the control action we show that this requires one to modify the basic relation between dissipation and time reversal and to include a contribution arising from the noncausal character of the reverse process we then propose a new definition of the quantity measuring the irreversibility of a path in a nonequilibrium stationary state which can also be regarded as the trajectory dependent total entropy production this leads to an extension of the second law which takes a simple form in the long time limit as an illustration we apply the general approach to linear systems that are both analytically tractable and experimentally relevant

variational calculation of the limit cycle and its frequency in a two neuron model with delay we consider a model system of two coupled hopfield neurons which is described by delay differential equations taking into account the finite signal propagation and processing times when the delay exceeds a critical value a limit cycle emerges via a supercritical hopf bifurcation first we calculate its frequency and trajectory perturbatively by applying the poincar e lindstedt method then the perturbation series are resummed by means of the shohat expansion in good agreement with numerical values however with increasing delay the accuracy of the results from the shohat expansion worsens we thus apply variational perturbation theory vpt to the perturbation expansions to obtain more accurate results which moreover hold even in the limit of large delays

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

mean square exponential stability for stochastic control systems with discrete time state feedbacks and their numerical schemes in simulation aiming at the qualitative numerical simulation this article concerns the mean square exponential stability mses of the stochastic systems with discrete time state feedbacks and their numerical schemes a time delay dependent stability criterion is obtained for the underlying system the euler maruyama and backward euler maruyama schemes are established and their mses are proved under the same criterion which means that the numerical schemes preserve the stability of the continuous model the illustrative example at the end of this letter shows that the theoretical upper bound for the sampling period limited in the stability criterion is much greater than that obtained by the stability criteria from the literature

a delay decomposition approach to fuzzy markovian jumping genetic regulatory networks with time varying delays in this paper we investigated the asymptotic stability of fuzzy markovian jumping genetic regulatory networks with time varying delays by delay decomposition approach the genetic regulatory networks are composed of s modes and the network switches from one mode to another according to a markovian chain with known transition probabilities purpose of this work is to establish some easy to verify conditions under which the dynamics of the true concentrations of the messenger ribonucleic acid mrna and protein is asymptotically stable irrespective of the norm bounded modeling errors a new lyapunov krasovskii functional lkf is constructed by nonuniformly dividing the whole delay intervals into multiple segments choosing proper functionals with different weighting matrices corresponding to different segments in the lkfs employing these new lkfs for the case of constant time delays and time varying delays some new delay dependent stability criteria are established with markovian jumping parameters by t s fuzzy model maximum admissible upper bounds maubs for the time varying delays are determined all the conditions presented here are in the form of linear matrix inequalities lmi which are efficiently solved by the lmi toolbox in matlab numerical examples are given to show the effectiveness of the proposed method

stochastic stability of markovian jumping genetic regulatory networks with mixed time delays abstract in this paper the stability analysis problem is investigated for a class of markovian jumping genetic regulatory networks grns with mixed time delays discrete time delays and distributed time delays and stochastic perturbations the main purpose of the addressed stability analysis problem is to establish some easy to verify conditions under which the dynamics of the true concentrations of the messenger ribonucleic acid and protein is asymptotically stable by utilizing a more general lyapunov krasovskii functional based on the idea of delay decomposing and the lmi linear matrix inequality technique we derive sufficient delay dependent conditions ensuring the asymptotically stability of the grns with mixed time delays and noise perturbations in terms of lmi finally simulation examples are exploited to illustrate the effectiveness of the developed theoretical results

exponential stability of numerical solutions to sddes with markovian switching abstract the main aim of this paper is to investigate the exponential stability of the euler method for a class of autonomous stochastic delay differential equations with markovian switching the definition of exponential mean square stability of numerical method is introduced it is proved that the euler scheme is exponentially stable in mean square sense several examples are given for illustration

on piterbarg s max discretisation theorem for multivariate stationary gaussian processes abstract let x t t 0 be a stationary gaussian process with zero mean and unit variance a deep result derived in piterbarg 2004 23 which we refer to as piterbarg s max discretisation theorem gives the joint asymptotic behaviour t of the continuous time maximum m t max t 0 t x t and the maximum m t max t r x t with r 0 t a uniform grid of points of distance t under some asymptotic restrictions on the correlation function piterbarg s max discretisation theorem shows that for the limit result it is important to know the speed t approaches 0 as t the present contribution derives the aforementioned theorem for multivariate stationary gaussian processes

numerical solutions of stochastic differential delay equations under local lipschitz condition in this paper a variant of the euler maruyama method is used to define the numerical solutions for stochastic differential delay equations sddes with variable delay the key aim is to show that the numerical solutions will converge to the true solutions of the sddes under the local lipschitz condition

s rock methods for stochastic delay differential equations with one fixed delay abstract we propose stabilized explicit stochastic runge kutta methods of strong order one half for ito stochastic delay differential equations with one fixed delay the family of the methods is constructed by embedding runge kutta chebyshev methods of order one for ordinary differential equations the values of a damping parameter of the methods are determined appropriately in order to obtain excellent mean square stability properties numerical experiments are carried out to confirm their order of convergence and stability properties

convergence of numerical solutions to stochastic delay differential equations with jumps this paper studies a class of stochastic delay differential equations with jumps sddejs explicit solutions can hardly be obtained for the sddejs appropriate numerical approximation schemes such as the euler scheme are needed to apply sddejs in practice or to study their properties in this paper it is proved that the euler approximation solutions converge to the analytic solution for sddejs under weaker conditions than the linear growth condition and global lipschitz condition an example is given for illustration

stability analysis of volterra delay integro differential equations and their backward differentiation time discretization we study the stability of the analytical solutions of initial value problems of a general class of systems of volterra delay integro differential equations numerical methods based on backward differentiation formulae and repeated quadrature formulae are suggested nonlinear and linear stability conditions for the presented methods are derived

robust stability for uncertain genetic regulatory networks with interval time varying delays this paper addresses the problem of robust stability of uncertain genetic regulatory networks grns with interval time varying delays we derive some new delay range dependent and delay derivative dependent independent stability criteria by employing some free weighting matrices and linear matrix inequalities in our analysis we carefully consider the relationship between the time varying state delays and their bounds when calculating the upper bound of the derivative of lyapunov functional we hence show that the rigorous requirement of other literatures that the time derivatives of time varying delays must be smaller than one is abandoned in the proposed scheme the new criteria are applicable to both fast and slow time varying delays finally four numerical examples are presented to illustrate the effectiveness and the less conservativeness of the developed results

convergence and stability of numerical solutions to a class of index 1 stochastic differential algebraic equations with time delay in this paper we study the convergence and stability of the stochastic theta method stm for a class of index 1 stochastic delay differential algebraic equations first in the case of constrained mesh i e the stepsize is a submultiple of the delay it is proved that the method is strongly consistent and convergent with order 1 2 in the mean square sense then the result is further extended to the case of non constrained mesh where we employ linear interpolation to approximate the delay argument later under a sufficient condition for mean square stability of the analytical solution it is proved that when the stepsizes are sufficiently small the stm approximations reproduce the stability of the analytical solution finally some numerical experiments are presented to illustrate the theoretical findings

convergence and stability of balanced methods for stochastic delay integro differential equations abstract this paper deals with a family of balanced implicit methods for the stochastic delay integro differential equations it is shown that the balanced methods which own the implicit iterative scheme in the diffusion term give strong convergence rate of at least 1 2 it proves that the mean square stability for the stochastic delay integro differential equations is inherited by the strong balanced methods and the weak balanced methods with sufficiently small stepsizes several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean square stabilities

convergence in nonlinear filtering for stochastic delay systems we study an approximation scheme for a nonlinear filtering problem when the state process x is the solution of a stochastic delay diffusion equation and the observation process is a noisy function of x s for s in t tau t where tau is a constant the approximating state is the piecewise linear euler maruyama scheme and the observation process is a noisy function of the approximating state the rate of convergence of this scheme is computed

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

computational approaches to parameter estimation and model selection in immunology one of the significant challenges in biomathematics and other areas of science is to formulate meaningful mathematical models our problem is to decide on a parametrized model which is in some sense most likely to represent the information in a set of observed data in this paper we illustrate the computational implementation of an information theoretic approach associated with a maximum likelihood treatment to modelling in immunology the approach is illustrated by modelling lcmv infection using a family of models based on systems of ordinary differential and delay differential equations the models which use parameters that have a scientific interpretation are chosen to fit data arising from experimental studies of virus cytotoxic t lymphocyte kinetics the parametrized models that result are arranged in a hierarchy by the computation of akaike indices the practical illustration is used to convey more general insight because the mathematical equations that comprise the models are solved numerically the accuracy in the computation has a bearing on the outcome and we address this and other practical details in our discussion

numerical analysis of semilinear stochastic evolution equations in banach spaces the solution of stochastic evolution equations generally relies on numerical computation here usually the main idea is to discretize the spde spatially obtaining a system of sdes that can be solved by e g the euler scheme in this paper we investigate the discretization error of semilinear stochastic evolution equations in lp spaces resp banach spaces the space discretization may be done by galerkin approximation for the time discretization we consider the implicit euler the explicit euler scheme and the crank nicholson scheme in the last section we give some examples i e we consider an spdes driven by nuclear wiener noise approximated by wavelets and delay equation approximated by finite differences

existence uniqueness and almost surely asymptotic estimations of the solutions to neutral stochastic functional differential equations driven by pure jumps in this paper we are concerned with neutral stochastic functional differential equations driven by pure jumps nsfdewpjs we prove the existence and uniqueness of the solution to nsfdewpjs whose coefficients satisfying the local lipschitz condition in addition we establish the p th exponential estimations and almost surely asymptotic estimations of the solution for nsfdewjs

the semi implicit euler method for stochastic differential delay equation with jumps abstract the class of jump diffusion sddes that admits explicit solutions is rather limited consequently there is a need for the systematic use of discrete time approximations in corresponding simulations in this paper we shall deal with convergence of the semi implicit euler method for nonlinear stochastic differential delay equation driven by wiener processes and poisson processes it is proved that the semi implicit euler method is convergent with strong order p 1 2

on a stable one leg methods for solving nonlinear volterra functional differential equations abstract this paper is concerned with the stability of the one leg methods for nonlinear volterra functional differential equations vfdes the contractivity and asymptotic stability properties are first analyzed for quasi equivalent and a stable one leg methods by use of two lemmas proven in this paper to extend the analysis to the case of strongly a stable one leg methods nevanlinna and liniger s technique of introducing new norm is used to obtain the conditions for contractivity and asymptotic stability of these methods as a consequence it is shown that one leg methods 1 2 1 with linear interpolation are unconditionally contractive and asymptotically stable and 2 step adams type method and 2 step bdf method are conditionally contractive and asymptotically stable the bounded stability of the midpoint rule is also proved with the help of the concept of semi equivalent one leg methods

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

preserving stability implicit euler method for nonlinear volterra and neutral functional differential equations in banach space this paper is concerned with the contractivity and asymptotic stability properties of the implicit euler method iem for nonlinear functional differential equations fdes these properties are first analyzed for volterra fdes and then the analysis is extended to the case of neutral fdes nfdes such an extension is particularly important since nfdes are more general and have received little attention in the literature the main result we establish is that the iem with linear interpolation can completely preserve these stability properties of the analytical solution to such fdes

the split step backward euler method for linear stochastic delay differential equations in this paper the numerical approximation of solutions of linear stochastic delay differential equations sddes in the ito sense is considered we construct split step backward euler ssbe method for solving linear sddes and develop the fundamental numerical analysis concerning its strong convergence and mean square stability it is proved that the ssbe method is convergent with strong order c 12 in the mean square sense the conditions under which the ssbe method is mean square stable ms stable and general mean square stable gms stable are obtained some illustrative numerical examples are presented to demonstrate the order of strong convergence and the mean square stability of the ssbe method

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

convergence of numerical solutions for variable delay differential equations driven by poisson random jump measure we study the following stochastic differential delay equations driven by poisson random jump measuredx t f x t x email protected t dt g x t x email protected t dw t email protected r nh x t x email protected t u n dt du 0

periodic solutions and chaos in a non linear model for the delayed cellular immune response abstract we model the cellular immune response using a set of non linear delayed differential equations we observe that the stationary solution becomes unstable above a critical immune response time the exponents characterizing the approach to this bifurcation point as well as the critical slow dynamics are obtained in the periodic regime the minimum virus load is substantially reduced with respect to the stationary solution further increasing the delay time the dynamics display a series of bifurcations evolving to a chaotic regime characterized by a set of 2d portraits