an iterative procedure to estimate muscle lines of action in vivo a method is described to estimate the line of action of muscles in the three dimensional space from serial images of parallel muscle sections obtained in vivo by means of ct or mri scanning the external shape of a muscle reconstructed from the series of parallel sections is mathematically divided into a series of imaginary slices directed arbitrarily in the three dimensional space the line of action is estimated initially as a regression line through the centroids of these mathematical slices a new series of mathematical slices is constructed perpendicular to the regression line and a new estimate of the line of action is obtained from their centroids this procedure is repeated until the estimated line of action is perpendicular to the mathematical slices it can then be considered as a reliable estimate of the line of action the accuracy of the method has been tested for various reconstruction parameters and muscle shapes the results of these tests show that the accuracy is relatively independent of the direction in which the sectional images have been made and that except for relatively short and thick muscles the estimated lines of action deviated less than about 2 from the theoretical one the presented method is a relatively simple mathematical technique which can be used easily for muscles reconstructed in vivo from routinely obtained sectional mri or ct images

parameter estimation of delay dynamical system from a scalar time series under external noise we introduce an adaptive learning rules for estimating all unknown parameters of delay dynamical system using a scalar time series sufficient condition for synchronization is derived using krasovskii lyapunov theory this scheme is highly applicable in secure communication since multiple messages can be transmitted through multiple parameter modulations one of the advantage of this method is that parameter estimation is also possible even when only one time series of the transmitter is available we present numerical examples for mackey glass system with periodic delay time which are used to illustrate the validity of this scheme the corresponding numerical results and the effect of external noise are also studied

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

diffusions for global optimizations we seek a global minimum of u 0 1 n to r the solution to d dt x t nabla u x t will find local minima the solution to dx t nabla u x t dt sqrt 2t dw t where w is standard n dimensional brownian motion and the boundaries are reflecting will concentrate near the global minima of u at least when temperature t is small the equilibrium distribution for x t is gibbs with density pi t x alpha exp u x t this suggests setting t t t downarrow 0 to find the global minima of u we give conditions on u x and t t such that the solution to dx t nabla u x t dt sqrt 2t dw t converges weakly to a distribution concentrated on the global minima of u

estimating parameters of a nonlinear dynamical system a method based on a modified newton raphson scheme is presented to estimate parameters of a nonlinear dynamical system from the time series data of the variables the method removes some of the problems associated with the standard synchronization based methods an important achievement of this method is that it is possible to determine the exact form of dynamical equations for systems with quadratic nonlinearity

a three dimensional mathematical model of the human masticatory system predicting maximum possible bite forces abstract a three dimensional mathematical model of the human masticatory system containing 16 muscle forces and two joint reaction forces is described the model allows simulation of static bite forces and concomitant joint reaction forces for various bite point locations and mandibular positions the system parameters for the model were obtained from a cadaver head maximum possible bite forces were computed using optimization techniques the optimization criterion we used was the minimizing of the relative activity of the most active muscle the model predicts that at each specific bite point bite forces can be generated in a wide range of directions and that the magnitude of the maximum bite force depends on its direction the relationship between bite force direction and its maximum magnitude depends on bite point location and mandibular position in general the direction of the largest possible bite force does not coincide with the direction perpendicular to the occlusal plane

adaptive schemes for the numerical solution of sdes a comparison the efficient numerical solution of stochastic differential equations is important for applications in many fields adaptive schemes well developed in the deterministic setting may be one possible way to reduce computational cost we review the two main step size control algorithms that have been proposed in recent years for stochastic differential systems and compare their efficiency in a simulation study

identification of delays and discontinuity points of unknown systems by using synchronization of chaos in this paper we present an approach in which synchronization of chaos is used to address identification problems in particular we are able to identify i the discontinuity points of systems described by piecewise dynamical equations and ii the delays of systems described by delay differential equations delays and discontinuities are widespread features of the dynamics of both natural and manmade systems the foremost goal of the paper is to present a general and flexible methodology that can be used in a broad variety of identification problems

application and validation of a three dimensional mathematical model of the human masticatory system in vivo abstract a previously described three dimensional mathematical model of the human masticatory system predicting maximum possible bite forces in all directions and the recruitment patterns of the masticatory muscles necessary to generate these forces was validated in in vivo experiments the morphological input parameters to the model for individual subjects were collected using mri scanning of the jaw system experimental measurements included recording of maximum voluntary bite force magnitude and direction and surface emg from the temporalis and masseter muscles for bite forces with an angle of 0 10 and 20 relative to the normal to the occlusal plane the predicted maximum possible bite forces were between 0 9 and 1 2 times the measured ones and the average ratio of measured to predicted maximum bite force was close to unity the average measured and predicted muscle recruitment patterns showed no striking differences nevertheless some systematic differences dependent on the bite force direction were found between the predicted and the measured maximum possible bite forces in a second series of simulations the influence of the direction of the joint reaction forces on these errors was studied the results suggest that they were caused primarily by an improper determination of the joint force directions

domain formation in transitions with noise and a time dependent bifurcation parameter the characteristic size for spatial structure that emerges when the bifurcation parameter in model partial differential equations is slowly increased through its critical value depends logarithmically on the size of added noise numerics and analysis are presented for the real ginzburg landau and swift hohenberg equa tions

flots et series de taylor stochastiques we study the expansion of the solution of a stochastic differential equation as an infinite sum of iterated stochastic stratonovitch integrals this enables us to give a universal and explicit formula for any invariant diffusion on a lie group in terms of lie brackets as well as a universal and explicit formula for the brownian motion on a riemannian manifold in terms of derivatives of the curvature tensor the first of these formulae contains and extends to the non nilpotent case the results of doss 6 sussmann 17 yamato 18 fliess and normand cyrot 7 krener and lobry 19 and kunita 11 on the representation of solutions of stochastic differential equations

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

adaptive modified function projective synchronization of unknown chaotic systems with different order abstract this paper investigates the modified function projective synchronization mfps between two different dimensional chaotic systems with fully unknown or partially unknown parameters via increased order based on the lyapunov stability theorem and adaptive control method a unified adaptive controller and parameters update law can be designed for achieving the mfps of the two different chaotic systems with different orders numerical simulations are presented to show the effectiveness of the proposed synchronization scheme

convergence and stability of the backward euler method for jump diffusion sdes with super linearly growing diffusion and jump coefficients abstract this paper firstly investigates convergence of the backward euler method for stochastic differential equations sdes driven by brownian motion and compound poisson process different from existing works we do not impose a globally lipschitz condition on the jump coefficient but formulate a more relaxed condition to allow for its super linear growth it is shown that the mean square convergence order of this method can be arbitrarily close to 1 2 under mild assumptions imposed on sdes allowing for possibly super linearly growing drift diffusion and jump coefficients an exact order 1 2 is recovered when further differentiability assumption is put on the coefficients furthermore the considered method is able to inherit the mean square stability of a wider class of levy noise driven sdes for all stepsizes these results are finally supported by some numerical experiments

step size control in the numerical solution of stochastic differential equations abstract we introduce a variable step size algorithm for the pathwise numerical approximation of solutions to stochastic ordinary differential equations the algorithm is based on a new pair of embedded explicit runge kutta methods of strong order 1 5 1 0 where the method of strong order 1 5 advances the numerical computation and the difference between approximations defined by the two methods is used for control of the local error we show that convergence of our method is preserved though the discretization times are not stopping times any more and further we present numerical results which demonstrate the effectiveness of the variable step size implementation compared to a fixed step size implementation

convergence rate of weak local linearization schemes for stochastic differential equations with additive noise there exists a diversity of weak local linearization ll schemes for the integration of stochastic differential equations with additive noise which differ in the algorithms employed for the numerical implementation of the weak local linear discretizations despite convergence results for these discretizations have been already developed the convergence of the weak ll schemes has not been considered up to date in this work a general result concerning the convergence rate of the weak ll schemes is presented as well as specificities for a number of particular schemes as an application the convergence of weak ll schemes for equations driven by poisson processes is presented in addition

mean square and asymptotic stability of the stochastic theta method stability analysis of numerical methods for ordinary differential equations odes is motivated by the question for what choices of stepsize does the numerical method reproduce the characteristics of the test equation we study a linear test equation with a multiplicative noise term and consider mean square and asymptotic stability of a stochastic version of the theta method we extend some mean square stability results in saito and mitsui siam j numer anal 33 1996 pp 2254 2267 in particular we show that an extension of the deterministic a stability property holds we also plot mean square stability regions for the case where the test equation has real parameters for asymptotic stability we show that the issue reduces to finding the expected value of a parametrized random variable we combine analytical and numerical techniques to get insights into the stability properties for a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region this allows us to prove a number of results the technique introduced is widely applicable and we use it to show that a fully implicit method suggested by kloeden and platen numerical solution of stochastic differential equations springer verlag 1992 has an asymptotic stability extension of the deterministic a stability property we also use the approach to explain some numerical results reported in milstein platen and schurz siam j numer anal 35 1998 pp 1010 1019

variable step size control in the numerical solution of stochastic differential equations we introduce a variable step size method for the numerical approximation of pathwise solutions to stochastic differential equations sdes the method which is dependent on a representation of bro

mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations in this paper we first study the mean square stability of numerical methods for stochastic delay differential equations under a coupled condition on the drift and diffusion coefficients this condition admits that the diffusion coefficient can be highly nonlinear i e it does not necessarily satisfy a linear growth or global lipschitz condition it is proved that for all positive stepsizes the classical stochastic theta method with q 0 5 is asymptotically mean square stable and the split step theta method with q 0 5 is exponentially mean square stable conditional stability results for the methods with q 0 5 and prove that the method possesses a bounded absorbing set in mean square independent of initial data

asymptotically efficient runge kutta methods for a class of itoˆ and stratonovich equations in certain applications of stochastic differential equations approximate solutions must be found that depend only on samples of the driving process it is known that the order of convergence of such approximations is limited and that some are asymptotically efficient in the sense that they minimize the leading coefficient in the expansion of mean square errors as power series in the sample step size this article develops asymptotically efficient runge kt tta methods that involve evaluations either of the coefficients of an ito equation or of the coefficients of the corresponding stratonovich equation simpler approximations which converge with the maximum possible order but which are not asymptotically efficient are also defined the runge kutta methods for ito equations differ from those designed for ordinary differential equations in that they involve terms in the square root of the sample step size the approximations are tested along with the classical euler method on five examples the simulatio

structure identification of uncertain general complex dynamical networks with time delay it is well known that many real world complex networks have various uncertain information such as unknown or uncertain topological structure and node dynamics the structure identification problem has theoretical and practical importance for uncertain complex dynamical networks at the same time time delay often appears in the state variables or coupling coefficients of various practical complex networks this paper initiates a novel approach for simultaneously identifying the topological structure and unknown parameters of uncertain general complex networks with time delay in particular this method is also effective for uncertain delayed complex dynamical networks with different node dynamics moreover the proposed method can be easily extended to monitor the on line evolution of network topological structure finally three representative examples are then given to verify the effectiveness of the proposed approach

synchronization based estimation of all parameters of chaotic systems from time series by a simple combination of adaptive scheme and linear feedback with the updated feedback strength for a large class of chaotic systems it is proved rigorously by using the invariance principle of differential equations that all unknown model parameters can be estimated dynamically this approach supplies a systematic and analytical procedure for estimating parameters from time series and it is simple to implement in practice in addition this method is quite robust against the effect of noise and able to respond rapidly to changes in operating parameters of the experimental system lorenz and r ossler hyperchaos systems are used to illustrate the validity of this technique

numerical methods for stochastic systems preserving symplectic structure stochastic hamiltonian systems with multiplicative noise phase flows of which preserve symplectic structure are considered to construct symplectic methods for such systems sufficiently general fully implicit schemes i e schemes with implicitness both in deterministic and stochastic terms are needed a new class of fully implicit methods for stochastic systems is proposed increments of wiener processes in these fully implicit schemes are substituted by some truncated random variables a number of symplectic integrators is constructed special attention is paid to systems with separable hamiltonians some results of numerical experiments are presented they demonstrate superiority of the proposed symplectic methods over very long times in comparison with nonsymplectic ones

identifying the topology of networks with discrete time dynamics we suggest a method for identifying the topology of networks with discrete time dynamics based on the dynamical evolution supported by the networks the frobenius matrix norm and lasalle s invariance principle are used in this work the network concerned can be directed or undirected weighted or unweighted and the local dynamics of each node can be nonidentical the connections among the nodes can be all unknown or partially known finally several examples are illustrated to verify the theoretical results

simple adaptive feedback controller for identical chaos synchronization based on the invariance principle of differential equations a simple adaptive feedback scheme is proposed to strictly synchronize almost all chaotic systems unlike the usual linear feedback the variable feedback strength is automatically adapted to completely synchronize two almost arbitrary identical chaotic systems so this scheme is analytical and simple to implement in practice moreover it is quite robust against the effect of noise the famous lorenz and rossler hyperchaos systems are used as illustrative examples

synchronization of neural networks based on parameter identification and via output or state coupling for neural networks with all the parameters unknown we focus on the global robust synchronization between two coupled neural networks with time varying delay that are linearly and unidirectionally coupled first we use lyapunov functionals to establish general theoretical conditions for designing the coupling matrix neither symmetry nor negative positive definiteness of the coupling matrix are required under less restrictive conditions the two coupled chaotic neural networks can achieve global robust synchronization regardless of their initial states second by employing the invariance principle of functional differential equations a simple analytical and rigorous adaptive feedback scheme is proposed for the robust synchronization of almost all kinds of coupled neural networks with time varying delay based on the parameter identification of uncertain delayed neural networks finally numerical simulations validate the effectiveness and feasibility of the proposed technique

seeker optimization algorithm for parameter estimation of time delay chaotic systems time delay chaotic systems have some very interesting properties and their parameter estimation has received increasing interest in the recent years it is well known that parameter estimation of a chaotic system is a nonlinear multivariable and multimodal optimization problem for which global optimization techniques are required in order to avoid local minima in this work a seeker optimization algorithm soa based method is proposed to address this issue in the soa search direction is based on the empirical gradients by evaluating the response to the position changes and step length is based on uncertainty reasoning by using a simple fuzzy rule the performance of the algorithm is evaluated on two typical test systems moreover two state of the art algorithms i e particle swarm optimization and differential evolution are also considered for comparison the simulation results show that the proposed algorithm is better than or at least as good as the other two algorithms and can effectively solve the parameter estimation problem of time delay chaotic systems

order conditions of stochastic runge kutta methods by b series in this paper general order conditions and a global convergence proof are given for stochastic runge kutta methods applied to stochastic ordinary differential equations sodes of stratonovich type this work generalizes the ideas of b series as applied to deterministic ordinary differential equations odes to the stochastic case and allows a completely general formalism for constructing high order stochastic methods either explicit or implicit some numerical results will be given to illustrate this theory

balanced implicit methods for stiff stochastic systems this paper introduces some implicitness in stochastic terms of numerical methods for solving stiff stochastic differential equations and especially a class of fully implicit methods the balanced methods their order of strong convergence is proved numerical experiments compare the stability properties of these schemes with explicit ones

conditions of parameter identification from time series we study the problem of synchronization based parameters identification of dynamical systems from time series through theoretical analysis and numerical examples we show that some recent research reports on this issue are not perfect or even incorrect long time full rank and finite time full rank conditions of gram matrix are pointed out which are sufficient for parameters identification of dynamical systems the influence of additive noise on the proposed parameter identifier is also investigated the mean filter is used to suppress the estimation fluctuation caused by the noise

parameter and state estimation of experimental chaotic systems using synchronization we examine the use of synchronization as a mechanism for extracting parameter and state information from experimental systems we focus on important aspects of this problem that have received little attention previously and we explore them using experiments and simulations with the chaotic colpitts oscillator as an example system we explore the impact of model imperfection on the ability to extract valid information from an experimental system we compare two optimization methods an initial value method and a constrained method each of these involves coupling the model equations to the experimental data in order to regularize the chaotic motions on the synchronization manifold we explore both time dependent and time independent coupling and discuss the use of periodic impulse coupling we also examine both optimized and fixed or manually adjusted coupling for the case of an optimized time dependent coupling function u t we find a robust structure which includes sharp peaks and intervals where it is zero this structure shows a strong correlation with the location in phase space and appears to depend on noise imperfections of the model and the lyapunov direction vectors for time independent coupling we find the counterintuitive result that often the optimal rms error in fitting the model to the data initially increases with coupling strength comparison of this result with that obtained using simulated data may provide one measure of model imperfection the constrained method with time dependent coupling appears to have benefits in synchronizing long data sets with minimal impact while the initial value method with time independent coupling tends to be substantially faster more flexible and easier to use we also describe a method of coupling which is useful for sparse experimental data sets our use of the colpitts oscillator allows us to explore in detail the case of a system with one positive lyapunov exponent the methods we explored are easily extended to driven systems such as neurons with time dependent injected current they are expected to be of value in nonchaotic systems as well software is available on request

realization of parameters identification in only locally lipschitzian dynamical systems with multiple types of time delays most of the existing results for identifying parameters in dynamical systems through adaptive coupling techniques always require a uniform lipschitz condition on system vector fields however it is not this requirement but the only locally lipschitz condition that is more suitable for many nonlinear systems in real applications also time delays sometimes are essential for modeling real systems so that in addition to the discrete form time delays in the distributed form could be considered this paper thus considers the problem of parameters identification in only locally lipschitzian dynamical systems with multiple types of time delays through adaptive coupling techniques in particular the paper proposes some linear independence conditions for realization of parameters identification in both nonautonomous and autonomous systems and draws a comparison between these conditions and the persistent of excitation condition which is used frequently in the literature moreover the paper proposes an index t

high strong order methods for non commutative stochastic ordinary differential equation systems and the magnus formula in recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations sodes as sodes are often more appropriate than their deterministic counterparts in many modelling situations however unlike the deterministic case numerical methods for sodes are considerably less sophisticated due to the difficulty in representing the possibly large number of random variable approximations to the stochastic integrals although burrage and burrage high strong order explicit runge kutta methods for stochastic ordinary differential equations applied numerical mathematics 22 1996 81 101 were able to construct strong local order 1 5 stochastic runge kutta methods for certain cases it is known that all extant stochastic runge kutta methods suffer an order reduction down to strong order 0 5 if there is non commutativity between the functions associated with the multiple wiener processes this order reduction down to that of the euler maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques an additional difficulty in solving sodes arises even in the linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple wiener processes thus in this present paper first the work of magnus on the exponential solution of differential equations for a linear operator communications on pure and applied mathematics 7 1954 649 673 applied to deterministic non commutative linear problems will be applied to non commutative linear sodes and methods of strong order 1 5 for arbitrary linear non commutative sode systems will be constructed hence giving an accurate approximation to the general linear problem secondly for general nonlinear non commutative systems with an arbitrary number d of wiener processes it is shown that strong local order i runge kutta methods with d 1 stages can be constructed by evaluated a set of lie brackets as well as the standard function evaluations a method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of wiener processes finally some numerical results are presented which illustrate the efficacy of these approaches c 1999 elsevier science b v all rights reserved

stability of weak numerical schemes for stochastic differential equations abstract this paper considers numerical stability and convergence of weak schemes solving stochastic differential equations a relatively strong notion of stability for a special type of test equations is proposed these are stochastic differential equations with multiplicative noise for different explicit and implicit schemes the regions of stability are also examined

an integration factor method for stochastic and stiff reaction diffusion systems stochastic effects are often present in the biochemical systems involving reactions and diffusions when the reactions are stiff existing numerical methods for stochastic reaction diffusion equations require either very small time steps for any explicit schemes or solving large nonlinear systems at each time step for the implicit schemes here we present a class of semi implicit integration factor methods that treat the diffusion term exactly and reaction implicitly for a system of stochastic reaction diffusion equations our linear stability analysis shows the advantage of such methods for both small and large amplitudes of noise direct use of the method to solving several linear and nonlinear stochastic reaction diffusion equations demonstrates good accuracy efficiency and stability properties this new class of methods which are easy to implement will have broader applications in solving stochastic reaction diffusion equations arising from models in biology and physical sciences

parameter identification of dynamical systems from time series in this paper synchronization based parameter identification of dynamical systems from time series is carefully revisited it is shown based on rigorous theoretical analysis and concrete counterexamples that some recent research reports on this issue are incomplete or even incorrect a linear independence condition is pointed out which is sufficient for such parameter identification of general dynamical systems

some issues in discrete approximate solution for stochastic differential equations abstract an evaluation method for numerical schemes of stochastic differential equations is treated discussing the source of errors in the discrete numerical solution we highlight the effect of pseudo random numbers upon the numerical solution and point out the significance of the independencies of the series of them required in the numerical schemes to discriminate the stochastic and deterministic parts in the errors more clearly we propose a new two dimensional linear test equation of multiplicative type whose analytical solution can be obtained readily our results are illustrated through some numerical examples

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

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

predictor corrector methods of runge kutta type for stochastic differential equations in this paper we construct predictor corrector pc methods based on the trivial predictor and stochastic implicit runge kutta rk correctors for solving stochastic differential equations using the colored rooted tree theory and stochastic b series the order condition theorem is derived for constructing stochastic rk methods based on pc implementations we also present detailed order conditions of the pc methods using stochastic implicit rk correctors with strong global order 1 0 and 1 5 a two stage implicit rk method with strong global order 1 0 and a four stage implicit rk method with strong global order 1 5 used as the correctors are constructed in this paper the mean square stability properties and numerical results of the pc methods based on these two implicit rk correctors are reported

identifying the topology of a coupled fitzhugh nagumo neurobiological network via a pinning mechanism topology identification of a network has received great interest for the reason that the study on many key properties of a network assumes a special known topology different from recent similar works in which the evolution of all the nodes in a complex network need to be received this brief presents a novel criterion to identify the topology of a coupled fitzhugh nagumo fhn neurobiological network by receiving the membrane potentials of only a fraction of the neurons meanwhile although incomplete information is received the evolution of all the neurons including membrane potentials and recovery variables are traced based on schur complement and lyapunov stability theory the exact weight configuration matrix can be estimated by a simple adaptive feedback control the effectiveness of the proposed approach is successfully verified by neural networks with fixed and switching topologies

numerical solution of stochastic differential equations with constant diffusion coefficients we present runge kutta methods of high accuracy for stochastic differential equations with constant diffusion coefficients we analyze l sub 2 convergence of these methods and present convergence proofs for scalar equations a second order method is derived and for systems a method of order one and one half is derived we further consider a variance reduction technique based on hermite expansions for evaluating expectations of functions of sample solutions numerical examples in two dimensions are presented

numerical methods for strong solutions of stochastic differential equations an overview this paper gives a review of recent progress in the design of numerical methods for computing the trajectories sample paths of solutions to stochastic differential equations we give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important with a particular focus on computational biology applications and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes we present the stochastic taylor series expansion as the fundamental mechanism for constructing effective numerical methods give general results that relate local and global order of convergence and mention the magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem we also present various classes of explicit and implicit methods for strong solutions based on the underlying structure of the problem finally we discuss implementation issues relating to maintaining the brownian path efficient simulation of stochastic integrals and variable step size implementations based on various types of control

gradient particle solutions of fokker planck equations for noisy delay bifurcations a gradient particle method is used to compute probability densities for processes with delay bifurcations that are sensitive to very small noise the method is particularly useful for computing the probability density in the transition region where asymptotic approximations may not be valid for a one dimensional steady bifurcation problem and a two dimensional fitzhugh nagumo model we solve the fokker planck equation fpe and compare the results with direct simulations and asymptotic approximations

a variable stepsize implementation for stochastic differential equations stochastic differential equations sdes arise from physical systems where the parameters describing the system can only be estimated or are subject to noise much work has been done recently on developing higher order runge kutta methods for solving sdes numerically fixed stepsize implementations of numerical methods have limitations when for example the sde being solved is stiff as this forces the stepsize to be very small this paper presents a completely general variable stepsize implementation of an embedded runge kutta pair for solving sdes numerically in this implementation there is no restriction on the value used for the stepsize and it is demonstrated that the integration remains on the correct brownian path

synchronization control of stochastic delayed neural networks abstract in this paper synchronization control of stochastic neural networks with time varying delays has been considered a novel control method is given using the lyapunov functional method and linear matrix inequality lmi approach several sufficient conditions have been derived to ensure the global asymptotical stability in mean square for the error system and thus the drive system synchronize with the response system also the estimation gains can be obtained with these new and effective methods synchronization can be achieved simulation results are given to verify the theoretical analysis in this paper

adaptive feedback control algorithm this paper is motivated by giving the detailed proofs and some interesting remarks on the results the author obtained in a series of papers phys rev lett 93 214101 2004 phys rev e 71 037203 2005 69 067201 2004 where an adaptive feedback algorithm was proposed to effectively stabilize and synchronize chaotic systems this note proves in detail the strictness of this algorithm from the viewpoint of mathematics and gives some interesting remarks for its potential applications to chaos control a synchronization in addition a significant comment on synchronization based parameter estimation is given which shows some techniques proposed in literature less strict and ineffective in some cases

a class of weak second order split drift stochastic runge kutta schemes for stiff sde systems abstract in this paper we present a family of stochastic runge kutta srk schemes for stiff systems of ito stochastic differential equations sdes derivation of these schemes called sdsrk to denote split drift srk is based on expressing the drift as a sum of two terms then a class of sdsrk schemes with stochastic weak second order is introduced for a subclass of these schemes that is appropriate for stiff sdes an analysis of mean square stability ms stability is given then by using the obtained stability function and the order conditions we determine the parameters of two schemes called sdsrk4a and sdsrk4b having deterministic and stochastic weak orders 2 from which the first one is fully drift implicit with better stability properties and the second one is semi implicit with reasonable stability properties but with much less computational cost numerical examples are presented to demonstrate the efficiency and accuracy of the new schemes