fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

singular perturbation analysis of boundary value problems for differential difference equations iii turning point problems this paper continues a study of a class of boundary value problems for linear second order differential difference equations in which the second order derivative is multiplied by a small parameter siam j appl math 42 1982 pp 502 531 45 1985 pp 687 707 the previous papers focused on problems involving boundary and interior layer phenomena rapid oscillations and resonance behavior the problems studied here have solutions which exhibit turning point behavior i e transition regions between rapid oscillations and exponential behavior the presence of the shift terms can induce large amplitudes and multiphase behavior over parts of the interval a combination of exact solutions singular perturbation methods and numerical computations are used in these studies

fitted reproducing kernel method for singularly perturbed delay initial value problems in this paper a fitted reproducing kernel is proposed for solving singularly perturbed initial value problems based on it an effective numerical method is proposed for a class of singularly perturbed delay initial value problem numerical results show that the present reproducing kernel is superior to the usual reproducing kernel for solving singularly perturbed initial value problems

exponential methods for singularly perturbed ordinary differential difference equations abstract a variety of exponential methods based on piecewise analytical solutions of advection reaction diffusion operators is proposed for the numerical solution of linear ordinary differential difference equations with small delay these methods are shown to be exact in exact arithmetic for linear ordinary differential equations with constant coefficients and right hand side are non standard and include three point non local approximations for the coefficients right hand side and dependent variable also a variety of exponential methods based on the piecewise analytical solution of advection diffusion operators with non local approximations is presented it is shown that exponential methods based on the advection diffusion reaction operator provide more accurate results than those based on the analytical solution of the advection diffusion operator it is also shown that at least for the problems considered here non local approximations for either the coefficients or the right hand side of the differential equations do not result in an increase of accuracy over those based on local approximations

annular chaotic areas on etudie l existence et les proprietes d acier chaotiques annulaires pour l application t de r 2 sur lui meme t x y f x y g x y

singular perturbation analysis of boundary value problems for differential difference equations vi small shifts with rapid oscillations this paper continues the authors study of boundary value problems for singularly perturbed linear second order differential difference equations with small shifts this study was initiated in the companion paper siam j appl math 54 1994 pp 249 272 in this paper the study is extended to problems that have solutions that exhibit rapid oscillations restrictions on the sizes of the shifts in terms of the small parameter are found such that generally the shifted terms cannot be replaced with truncated taylor series in particular it is shown that even when the shifts are small relative to the width of an oscillation they can affect the solution to leading order the conclusion is that oscillatory solutions are more sensitive to small delays than are layer solutions it is shown that a suitably modified version of the standard wkb method can be used to obtain leading order oscillatory solutions of these differential difference equations these preliminary studies of differential difference eq

singular perturbation analysis of boundary value problems for differential difference quations ii rapid oscillations and resonances this paper continues a study of a class of boundary value problems for linear second order differential difference equations in which the second order derivative is multiplied by a small parameter siam j appl math 42 1982 pp 502 531 the previous paper focused on problems involving boundary and interior layer phenomena here the problems studied have solutions exhibiting rapid oscillations the presence of the shift terms can induce large amplitudes multiphase behavior and resonance phenomena n particular we study two types of resonance phenomena namely global and local resonance a combination of exact solutions singular perturbation methods and numerical computations are used in these studies in a companion paper siam j appl math 45 1985 pp 708 734 we study problems with solutions which exhibit turning point behavior

local c 1 stability versus global c 1 unstability for iterative roots abstract stability of iterative roots is important in the numerical computation of iterative roots known results show that under some conditions iterative roots of strictly monotonic self mappings are c 0 stable in both the local sense and the global sense in this paper we discuss the c 1 stability for iterative roots of strictly increasing self mappings on a compact interval between two fixed points we prove that those iterative roots are locally c 1 stable but globally c 1 unstable

hyperbolic partial differential difference equation in the mathematical modeling of neuronal firing and its numerical solution abstract first we assume a simple neuronal model based on stein s model richard b stein a theoretical analysis of neuronal variability biophys j 5 1965 173 194 in which after a refractory period excitatory and inhibitory exponentially decaying inputs of constant size occur at random intervals and sum until a threshold is reached we briefly discuss the distribution of time intervals between successive neuronal firings the firing rate as a function of input frequency the strength duration curve and the role of inhibition then a first order partial differential difference equation for the distribution of neuronal firing intervals is derived and a numerical scheme based on finite difference is constructed for solving such type of initial and boundary value problems the proposed method is analyzed for stability and convergence finally some test examples are given to validate convergence and the computational efficiency of the present scheme

power spectra and dynamical invariants for delay differential and difference equations abstract the mackey glass and ikeda delay differential equations ddes are models of feedback systems in which attractor dimension is proportional to the delay power spectra from these models are investigated analytically and numerically as the delay increases much beyond the system response time an exponentially decaying spectrum with a near periodic superimposed modulation appears it is shown that each peak in this modulation is associated with a mode of oscillation predicted by linear stability analysis around the fixed point the number of such modes within a characteristic autocorrelation time of the solution agrees with the lyapunov dimension of the attractor the disappearance of this modulation at higher delay values is also explained the decay rate of the spectrum in the mid to high frequency range is found to agree with the sum of the positive lyapunov exponents i e with the metric entropy realistic models with a distribution of delays display similar spectra as their memory kernel narrows into a delta function for very large delay the dde approaches an infinite dimensional continuous time difference equation ctde with each point on a delay interval evolving independently and according to a chaotic discrete time map ctde spectra which have many features of finite delay dde spectra can be computed analytically if the spectrum of this map is known for constant initial functions they have a 1 f 2 form at low frequencies our analysis suggests that the spectrum of the chaotic map is at the origin of the peak shapes in dde spectra at low frequency our findings highlight the relevance of the linear prrperties of ddes to the characterization of their nonlinear properties

relaxation oscillations in a class of delay differential equations we study a class of delay differential equations which have been used to model hematological stem cell regulation and dynamics under certain circumstances the model exhibits self sustained oscillations with periods which can be significantly longer than the basic cell cycle time we show that the long periods in the oscillations occur when the cell generation rate is small and we provide an asymptotic analysis of the model in this case this analysis bears a close resemblance to the analysis of relaxation oscillators such as the van der pol oscillator except that in our case the slow manifold is infinite dimensional despite this a fairly complete analysis of the problem is possible

hyperelliptic quasi periodic and soliton solutions of the nonlinear schrodinger equation on considere la construction de solutions par des methodes de geometrie algebriques et en particulier par les fonctions theta hyperelliptiques

an asymptotic theory of large deviations for markov jump processes we present new asymptotic methods for the analysis of markov jump processes the methods based on the wkb and other singular perturbation techniques are applied directly to the kolmogorov equations and not to approximate equations that come e g from diffusion approximations for time homogeneous processes we construct approximations to the stationary density function and the mean first passage time from a given domain examples involving a random walk and a problem in queueing theory are presented to illustrate our methods for a class of time inhomogeneous processes we construct long time approximations to the transition probability density function and the probability of large deviations from a stable state the law of large numbers is obtained as a special case

e uniformly convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general ddes we consider some problems arising from singularly perturbed general differential difference equations first we construct in a new way and analyze a fitted operator finite difference method fofdm which is first order e uniformly convergent with the aim of having just one function evaluation at each step attempts have been made to derive a higher order method via shishkin mesh to which we refer as the fitted mesh finite difference method fmfdm this fmfdm is a direct method and e uniformly convergent with the nodal error as o n 2ln 2n which is an improvement over the existing direct methods i e those which do not use any acceleration of convergence techniques e g richardson s extrapolation or defect correction etc for such problems on a mesh of shishkin type that lead the error as o n 1lnn where n denotes the total number of sub intervals of 0 1 comparative numerical results are presented in support of the theory

string equations for the kdv hierarchy and the grassmannian the grassmannian model for the korteweg de vries hierarchy is used to describe the translational galilean and scaling self similar solutions of this hierarchy these solutions are characterized by the string equations appearing in two dimensional quantum gravity in particular the subsets of the sato grassmannian corresponding to solutions of the string equations are found the well known adler moser rational solutions are obtained as well as a three parameter family of solutions associated with the painleve ii equation

maximum norm a posteriori error estimates for a singularly perturbed differential difference equation with small delay we consider an adaptive grid approach to solve a singularly perturbed differential difference equation with small delay the mesh is constructed adaptively by equidistributing a arc length monitor function we derive a first order maximum norm a posteriori estimate for the full discretization scheme a first order rate of convergence independent of the perturbation parameter and the small delay parameter is established a singularly perturbed differential difference equation with small delay is discretized on an adaptive grid which is formed by equidistributing arc length monitor function we first derive first order maximum norm a posteriori estimates for the full discretization scheme of these problems then a first order rate of convergence independent of the perturbation parameter and the small delay parameter is established numerical results are provided that support our theoretical estimates

numerical methods based on additive splittings for hyperbolic partial differential equations we derive and analyze several methods for systems of hyperbolic equations with wide ranges of signal speeds these techniques are also useful for problems whose coefficients have large mean values about which they oscillate with small amplitude our methods are based on additive splittings of the operators into components that can be approximated independently on the different time scales some of which are sometimes treated exactly the efficiency of the splitting methods is seen to depend on the error incurred in splitting the exact solution operator this is analyzed and a technique is discussed for reducing this error through a simple change of variables a procedure for generating the appropriate boundary data for the intermediate solutions is also presented

on the birkhoff factorization problem for the heisenberg magnet and nonlinear schrodinger equations a geometrical description of the heisenberg magnet hm equation with classical spins is given in terms of flows on the quotient space g h where g is an infinite dimensional lie group and h is a subgroup of g it is shown that the hm flows are induced by an action of r2 on g h and that the hm equation can be integrated by solving a birkhoff factorization problem for g for the hm flows that are laurent polynomials in the spectral variable we derive an algebraic transformation between solutions of the nonlinear schrodinger nls and heisenberg magnet equation the birkhoff factorization problem for g is treated in terms of the geometry of the segal wilson grassmannian gr h the solution of the problem is given in terms of a pair of baker functions for special subspaces in gr h the baker functions are constructed explicitly for subspaces that yield multisoliton solutions of nls and hm equations

asymptotic analysis of a state dependent m g 1 queueing system we present new asymptotic methods for the analysis of queueing systems these methods are applied to a state dependent m g 1 queue we formulate problems for and compute approximations to i the stationary density of the unfinished work ii the mean length of time until the end of a busy period iii the mean length of a busy period and iv the mean time until the unfinished work reaches or exceeds a specified capacity the methods are applied to the full kolmogorov equations scaled so that the arrival rate is rapid and the mean service is small thus we do not truncate equations as in diffusion approximations for state independent m g 1 queues our results are shown to agree with the known exact solutions we include comparisons both analytic and numerical between our results and those obtained from diffusion approximations

upwind and midpoint upwind difference methods for time dependent differential difference equations with layer behavior abstract in this paper we have considered an important class of time dependent singularly perturbed convection diffusion problems with retarded terms which often arise in computational neuroscience we used taylors series to approximate the retarded terms and the resulting time dependent singularly perturbed differential equation is approximated using parameters uniform numerical methods based on euler implicit upwind and midpoint upwind finite difference schemes we discretize the continuous problem using implicit euler scheme in the time direction with a constant step size and the resulting system of equations is approximated using upwind and midpoint upwind difference schemes on a piecewise uniform mesh we will prove the uniform convergence of these two schemes numerical experiments support the convergence results

a review on singularly perturbed differential equations with turning points and interior layers singular perturbation problems with turning points arise as mathematical models for various physical phenomena the problem with interior turning point represent one dimensional version of stationary convection diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin boundary turning point problems on the other hand arise in geophysics and in modeling thermal boundary layers in laminar flow in this paper we review some existing literature on asymptotic and numerical analysis of singularly perturbed turning point and interior layer problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye towards the goal of developing general methods to solve such problems since major work in this area started after 1970 so this paper limits its coverage to the work done by numerous researchers between 1970 and 2011

numerical analysis of singularly perturbed delay differential equations with layer behavior in this paper we present a numerical method to solve boundary value problems for singularly perturbed differential difference equations with negative shift in recent papers the term negative shift has been used for delay siam j appl math 54 1994 249 siam j appl math 42 1982 502 siam j appl math 45 1985 687 siam j appl math 45 1985 708 siam j appl math 54 1994 273 similar bvps are associated with expected first exit time problem of the membrane potential in models for neuron and in variational problem in control theory the stability and convergence analysis of the method is discussed also the effect of small shift on the boundary layer solution in both the cases i e boundary layer on the left side as well as the right side is discussed by considering numerical experiments several numerical examples are solved using the presented method compared the computed result with exact solution and plotted the graphs of the solution of the problems

asymptotic and numerical analysis of singular perturbation problems a survey this paper is intended to be a brief survey of the asymptotic and numerical analysis of singular perturbation problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye toward the goal of developing general methods to solve such problems a summary of the results of some recent methods is presented and this leads to conclusions and recommendations about what methods to use on singular perturbation problems finally some areas of current research are indicated a bibliography of about 130 items is provided

differential difference akns equations and homogeneous heisenberg algebras the well known evolution equations associated to the homogeneous heisenberg algebras of kac moody algebras g 1 akns systems are extended by differential difference equations that can be written in zero curvature form

particular solutions of forced generalized airy equations differential equations with delayed forcing terms arise naturally in the explicit solution of differential delay equations 11 in our present studies of singular perturbation analyses of boundary value problems for secondorder differential difference equations with a turning point see 2 for earlier results we have had to deal with forcing terms consisting of shifted airy and lommel functions and their derivatives one example of the type of equation which arises is the inhomogeneous airy equation see the appendix for a derivation y xy c ai x a c bi x a c l x a d ai x a d bi x a d3 l x a 1 where a is a constant 0 a co ci di i 1 2 3 are constants ai bi are airy functions and l is a lommel function see 31 satisfying l xl 1 2

lmi based stability for singularly perturbed nonlinear impulsive differential systems with delays of small parameter abstract in this paper a class of singularly perturbed nonlinear impulsive delay differential systems is considered where the time delays include a small parameter based on lyapunov krasovskii functional and free weighting matrix method some novel delay dependent global asymptotic stability results are derived in terms of linear matrix inequalities lmis which can be easily solved using efficient convex optimization algorithms the results show that the system stability is very sensitive to the singular perturbation and the small parameter in time delays a numerical example is given to show the effectiveness of the proposed results

the n m th korteweg de vries hierarchy and the associated w algebra a differential integrable hierarchy which is called the n m th korteweg de vries kdv hierarchy whose lax operator is obtained by properly adding m pseudo differential terms to the lax operator of the nth kdv hierarchy is discussed herein this new hierarchy contains both the higher kdv hierarchy and multifield representation of the kadomtsev petviashvili kp hierarchy as subsystems and naturally appears in multimatrix models the n 2m 1 coordinates or fields of this hierarchy satisfy two algebras of compatible poisson brackets which are local and polynomial each poisson structure generates an extended w1 and w algebras respectively w n m is called the generating algebra of the extended w algebra this algebra which corresponds with the second poisson structure shares many features of the usual wn algebra it is shown that there exist m distinct reductions of the n m th kdv hierarchy which are obtained by imposing suitable second class constraints the most drastic reduction corresponds

a numerical method for singularly perturbed turning point problems with an interior layer the objective of this paper is to present a numerical method for solving singularly perturbed turning point problems exhibiting an interior layer the method is based on the asymptotic expansion technique and the reproducing kernel method rkm the original problem is reduced to interior layer and regular domain problems the regular domain problems are solved by using the asymptotic expansion method the interior layer problem is treated by the method of stretching variable and the rkm four numerical examples are provided to illustrate the effectiveness of the present method the results of numerical examples show that the present method can provide very accurate approximate solutions

higher order muscl scheme for transport equation originating in a neuronal model we consider a transport equation with mixed boundary conditions dirichlet and neumann originating in a neuronal model this equation contains point wise delay and advanced argument the objective of this paper is to construct and analyze higher order muscl scheme to find a numerical solution the developed scheme is stable and convergent the importance of this scheme is that it is valid for large as well as for small values of point wise delay and advance some test examples are included to examine the behavior of the solution and to verify the order of convergence the effect of point wise delay and advance arguments on the solution is shown graphically

akns hierarchy self similarity string equations and the grassmannian in this paper the galilean scaling and translational self similarity conditions for the akns hierarchy are analysed geometrically in terms of the infinite dimensional grassmannian the string equations of the one matrix model correspond to the galilean self similarity condition for this hierarchy we describe in terms of the initial data for the zero curvature 1 form of the akns hierarchy the moduli space of these self similar solutions in the sato grassmannian as a by product we characterize the points in the segal wilson grassmannian corresponding to the sachs rational solutions of the akns equation and to the nakamura hirota rational solutions of the nls equation

singular perturbation analysis of boundary value problems for differential difference equations v small shifts with layer behavior in this paper an investigation is initiated of boundary value problems for singularly perturbed linear second order differential difference equations with small shifts i e where the second order derivative is multiplied by a small parameter and the shift depends on the small parameter similar boundary value problems are associated with expected first exit times of the membrane potential in models for neurons in particular this paper focuses on problems with solutions that exhibit layer behavior at one or both of the boundaries the analyses of the layer equations using laplace transforms lead to novel results it is shown that the layer behavior can change its character and even be destroyed as the shifts increase but remain small in the companion paper siam j appi math 54 1994 pp 273283 similar boundary value problems with solutions that exhibit rapid oscillations are studied

numerical analysis of singularly perturbed delay differential turning point problem abstract in this paper we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts similar boundary value problems are encountered while simulating several real life processes for instance first exit time problem in the modelling of neuronal variability a rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem in the development of numerical methods for constructing an approximation to the solution of the problem a special type of mesh is generated to tackle the delay term along with the turning point then to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term an exponential fitting parameter is used several numerical examples are presented to support the theory developed in the paper

on stabilization of nonstandard singularly perturbed systems with small delays in state and control a nonstandard singularly perturbed linear time invariant system with a general type of small delay in state and control variables is considered two much simpler parameter free systems the reduced order and the boundary layer ones associated with the original system are introduced since the original system is nonstandard the reduced order one is descriptor differential algebraic a composite feedback control exponentially stabilizing the original system is constructed based on stabilizing controls of the reduced order and boundary layer systems an illustrative example is presented

numerical treatment of a mathematical model arising from a model of neuronal variability in this paper we describe a numerical approach based on finite difference method to solve a mathematical model arising from a model of neuronal variability the mathematical modelling of the determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in dendrites includes a general boundary value problem for singularly perturbed differential difference equation with small shifts in the numerical treatment for such type of boundary value problems first we use taylor approximation to tackle the terms containing small shifts which converts it to a boundary value problem for singularly perturbed differential equation a rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives up to third order then a parameter uniform difference scheme is constructed to solve the boundary value problem so obtained a parameter uniform error estimate for the numerical scheme so constructed is established though the convergence of the difference scheme is almost linear but its beauty is that it converges independently of the singular perturbation parameter i e the numerical scheme converges for each value of the singular perturbation parameter however small it may be but remains positive several test examples are solved to demonstrate the efficiency of the numerical scheme presented in the paper and to show the effect of the small shift on the solution behavior

an iterative numerical method for singularly perturbed reaction diffusion equations with negative shift in this paper a numerical method based on an iterative scheme is proposed for a singularly perturbed delay differential equation of reaction diffusion type in this method the solution of the delay problem is obtained as the limit of the solutions to a sequence of the non delay problems the numerical solutions of the non delay problems are obtained by applying existing finite difference schemes and finite element method for the non delay singularly perturbed equations error estimates are derived in the supremum norm numerical results illustrating the theory are also included

hybrid method for numerical solution of singularly perturbed delay differential equations this paper is concerned with the numerical technique for a singularly perturbed second order differential difference equation of the convection diffusion type with small delay parameter d whose solution has a single boundary layer we analyse three difference operators l k n k 1 2 3 a simple upwind scheme midpoint upwind scheme and a hybrid scheme respectively on a shishkin mesh to approximate the solution of the problem the hybrid algorithm uses central difference in the boundary layer region and midpoint upwind scheme outside the boundary layer we have concluded that the hybrid scheme gives better accuracy the paper concludes with a few numerical results exhibiting the performance of these three schemes

a study of splitting scheme for hyperbolic conservation laws with source terms abstract this study deals with the convergence of a numerical scheme for conservation laws including source terms a splitting method for source term integration is presented more precisely the convergence of the numerical solution towards the entropy solution is proved in the scalar case because of the effect of source term the constructed scheme is total variation bounded numerical experiments for one dimensional shallow water equation are presented to demonstrate the performance of the scheme

an analysis of the fractional step method the fractional step method for solving the incompressible navier stokes equations in primitive variables is analyzed as a block lu decomposition in this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved in addition it is shown that poor temporal accuracy first order is not due to boundary conditions but due to the method itself a generalized block lu decomposition that overcomes this difficulty is presented allowing arbitrarily high temporal order of accuracy the generalized decomposition is shown to be useful for a wide range of problems including steady problems technical issues such as stability and the appropriate pressure update scheme are also addressed numerical simulations of the steady incompressible navier stokes equations in a square domain confirm the theoretical results 18 refs 4 figs

e uniformly convergent non standard finite difference methods for singularly perturbed differential difference equations with small delay abstract non standard finite difference methods nsfdms now a days are playing an important role in solving the real life problems governed by odes and or by pdes many differential models of sciences and engineerings for which the existing methodologies do not give reliable results these nsfdms are solving them competitively to this end in this paper we consider second order linear singularly perturbed differential difference equations using the second of the five non standard modeling rules of mickens r e mickens nonstandard finite difference models of differential equations world scientific singapore 1994 the new finite difference methods are obtained for the particular cases of these problems this rule suggests us to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures most of the significant properties of the governing differential equation s both theoretically and numerically we show that these nsfdms are e uniformly convergent

uniformly convergent non standard finite difference methods for singularly perturbed differential difference equations with delay and advance a new class of fitted operator finite difference methods are constructed via non standard finite difference methods nsfdm s for the numerical solution of singularly perturbed differential difference equations having both delay and advance arguments the main idea behind the construction of our method s is to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures significant properties of the governing differential equation and thus provides the reliable numerical results unlike other fofdms constructed in standard ways the methods that we present in this paper are fairly simple to construct and thus enrich the class of fitted operator methods by adding these new methods these methods are shown to be e uniformly convergent with order two which is the highest possible order of convergence obtained via any fitted operator method for the problems under consideration this paper further clarifies several doubts e g why a particular scheme is not suitable for the whole range of values of the associated parameters and what could be the possible remedies finally we provide some numerical examples which illustrate the theoretical findings copyright 2005 john wiley sons ltd

singular perturbation analysis of boundary value problems for differential difference equations a study is made of a class of boundary value problems for linear second order differential difference equations in which the highest order derivative is multiplied by a small parameter consideration of the exact solution of simple model equations provides insight into the appropriate use of singular perturbation techniques for more general problems the resulting analysis leads to several novel features which are not present in problems without differences

quintic b spline method for solving second order linear and nonlinear singularly perturbed two point boundary value problems in this paper we have studied a numerical scheme to solve second order singularly perturbed two point linear and nonlinear boundary value problems the boundary layer of this type of problems exhibits at one end left or right point of the domain due to the presence of perturbation parameter the quintic b spline method and suitable piecewise uniform shishkin mesh have been used linear and nonlinear second order singularly perturbed boundary value problems have been solved by the present method the convergence analysis is also provided and the method is shown to have uniform convergence of fourth order numerical results have demonstrated the efficiency of the present method

non standard difference schemes for singular perturbation problems revisited non standard finite difference techniques for solving various linear two point singularly perturbed boundary value problems are considered they consist of replacing the original differential equation by a related differential equation with a small deviating argument this related problem is then discretized using finite differences this discretized problem is solved using standard matrix methods the accuracy and robustness of these numerical schemes are then carefully investigated and compared to standard centered finite differences in particular earlier claims for the superiority of such schemes over standard finite differences are shown to be invalid

some global bifurcations of two dimensional endomorphisms by use of critical lines many studies have been devoted to discrete dynamical systems represented by homeomorphisms and diffeomorphisms since the works by smale l and gavrilov and shilnikov 2 31 but comparatively few works have been devoted to endomorphisms that is to maps with a nonunique inverse however the mathematical models derived from the description of evolutive phenomena in several applicative fields are often represented by discrete dynamical systems or maps with a nonunique inverse this is particularly true in the socio economic field 4 61 but examples can also be found in ecology 7 81 biology 9 l l and control systems 121 this work concerns the study of the global dynamics of recurrences represented by twodimensional endomorphisms of applicative interest in the economic context useful notions for the local and global analysis of nonlinear endomorphisms have been introduced by gumowski and mira ll 13 141 for two dimensional endomorphisms in particular this is the notion of critical lines or critical curves representing the twodimensional extension of the critical points in one dimensional maps and the notions of absorbing area and chaotic area bounded by elements of critical lines which determine phase plane domains where chaotic solutions are located these notions which will be recalled in the next section together with some bifurcations involving the chaotic areas are the basic concepts and properties that we shall use in the analysis of the subsequent sections the aim of this paper is to illustrate by an example how powerful the critical lines may be in studying the global dynamics and bifurcations of plane nonlinear endomorphisms particularly when they are used in combination with more traditional techniques it is known that in diffeomorphisms global bifurcations which qualitatively change the structure of a chaotic attractor called interior crises or catastrophes by some authors 15 181 are characterized by the stable manifold of a saddle approaching and touching the chaotic attractor we shall find that this phenomenology also occurs in the endomorphisms of our model with a particular role being played by the critical lines in section 3 we shall describe our mathematical model a two dimensional piecewise linear map which we call map f together with some of its elementary properties and the definition of three classes of endomorphisms into which the analysis can be conveniently subdivided endomorphisms of the first class have been the object of a previous work 191 here we shall show in section 4 how several bifurcations involving chaotic areas can be explained by use of critical lines the main result is the analytical determination of the bifurcation which discriminates between the global attractivity of an absorbing area and the nonglobal attractivity

periodic solutions and chaos in a delay differential equation modelling haematopoesis simple mathematical models have been devised to describe the process of haematopoesis the formation and differentiation of blood cells within the human body l 21 haematopoesis is an example of a biological control system with an inherent delay in it this is due to the time required for primitive blood cells to mature such a delayed response is known to cause oscillatory behaviour in control systems and it has been found that similar oscillatory instabilities in blood cell counts are associated with diseases such as aplastic anaemia 2 this suggest that the study of such equations may be of biological importance mathematically they are of interest because they can exhibit complex aperiodic behaviour a model put forward for consideration by mackey glass 3 is

on the τ function of matrix hierarchies of integrable equations expressions of the baker and of the functions for algebraic geometrical solutions of the akns d hierarchy of integrable equations are obtained in terms of functions along with a formula connecting them

the height of a binary search tree the limiting distribution perspective we study the height of the binary search tree the most fundamental data structure used for searching we assume that the binary search tree is built from a random permutation of n elements under this assumption we study the limiting distribution of the height as n we show that the distribution has six asymptotic regions scales these correspond to different ranges of k and n where pr hn k is the height distribution in the critical region the so called central region where most of the probability mass is concentrated the limiting distribution satisfies a non linear integral equation while we cannot solve this equation exactly we show that both tails of the distribution are roughly of a double exponential form from our analysis we conclude that the average height e hn a log n 3 2 a a 1 log log n where a 4 311 is the unique solution of x logx x xlog2 1 0 x 1 while the variance var hn o 1 the second term in the expansion of e kn and the rate of growth of the variance were also recently obtained by b reed who used probabilistic arguments while m drmota established the growth of the variance by analytic methods our analysis makes certain assumptions about the forms of some asymptotic expansions as well as their asymptotic matching

on the node fractional iterates let m be a bounded convex set of r n let f m r n be a continuous non bijective map we study solutions g of the functional equation g m x g g m 1 x f x m 2 this functional equation represents the problem of the fractional iteration for f

τ functions and zero curvature equations of toda akns type the connection between functions and zero curvature equations for the homogeneous construction of the basic module l 0 over the simplest affine kac moody algebra a 1 1 is studied

a connection between fractional iteration and graph theory we consider the problem of the fractional iteration for n dimensional maps we have two objectives in this paper the first objective is to establish a connection between fractional iteration and graph theory the second objective is to make a study concerning the problem of fractional iteration for n dimensional maps

a numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations abstract a boundary value problem for second order singularly perturbed delay differential equation is considered when the delay argument is sufficiently small to tackle the delay term the researchers m k kadalbajoo k k sharma numerical analysis of singularly perturbed delay differential equations with layer behavior appl math comput 157 2004 11 28 r e o malley jr singular perturbation methods for ordinary differential equations springer verlag new york 1991 used taylor s series expansion and presented an asymptotic as well as numerical approach to solve such type boundary value problem but the existing methods in the literature fail in the case when the delay argument is bigger one because in this case the use of taylor s series expansion for the term containing delay may lead to a bad approximation in this paper to short out this problem we present a numerical scheme for solving such type of boundary value problems which works nicely in both the cases i e when delay argument is bigger one as well as smaller one to handle the delay argument we construct a special type of mesh so that the term containing delay lies on nodal points after discretization the proposed method is analyzed for stability and convergence to demonstrate the efficiency of the method and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution several test examples are considered

numerical study of singularly perturbed differential difference equation arising in the modeling of neuronal variability the objective of this paper is to present a numerical study of a class of boundary value problems of singularly perturbed differential difference equations spdde which arise in computational neuroscience in particular in the modeling of neuronal variability the mathematical modeling of the determination of the expected time for the generation of action potential in the nerve cells by random synaptic inputs in dendrites includes a general boundary value problem for singularly perturbed differential difference equation with shifts the problem considered in this paper exhibit turning point behavior which add to the complexity in the construction of numerical approximation to the solution of the problem as well as in obtaining theoretical estimates on the solution exponentially fitted finite difference scheme based on il in allen southwell fitting is used on a specially designed mesh some numerical examples are given to validate convergence and computational efficiency of the proposed numerical scheme effect of the shifts on the layer structure is illuminated for the considered examples

numerical method for a singularly perturbed convection diffusion problem with delay this paper deals with the singularly perturbed boundary value problem for a linear second order delay differential equation for the numerical solution of this problem we use an exponentially fitted difference scheme on a uniform mesh which is accomplished by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form it is shown that one gets first order convergence in the discrete maximum norm independently of the perturbation parameter numerical results are presented which illustrate the theoretical results

kac moody lie algebras and soliton equations ii lax equations associated with a1 1 the soliton equations associated with sl 2 eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment they are shown to be generated by a single family of commuting hamiltonians on a subalgebra of the loop algebra of sl 2 the conserved densities and fluxes of the usual anks hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems the hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment

a uniformly almost second order convergent numerical method for singularly perturbed delay differential equations abstract the purpose of this paper is to present a uniform finite difference method for the numerical solution of a second order singularly perturbed delay differential equation the problem is solved by using a hybrid difference scheme on a shishkin type mesh the method is shown to be uniformly convergent with respect to the perturbation parameter numerical experiments illustrate in practice the result of convergence proved theoretically

sufficient conditions for the uniform convergence of a difference scheme for a singularly perturbed turning point problem a number of results exist in the literature for singularly perturbed differential equations without turning points in particular a number of difference schemes have been proposed that satisfy a stronger than normal convergence criteria known as uniform convergence this guarantees that the schemes model the boundary layers well we wish to examine whether these schemes will also be uniformly convergent if the equation has turning points to this end we derive sufficient conditions for uniform convergence which are satisfied not only by these schemes but by a more general class of schemes we show that the rate of convergence is determined by a characteristic parameter of the problem which may be less than one we confirm these theoretical results by numerical calculations

reformulation of boundary value problems into standard form boundary value problems in ode s arising in various applications are frequently not in the standard form required by the currently existing software however many problems can be converted to such a form thus enabling the practitioner to take advantage of the availability and reliability of this general purpose software here various conversion devices are surveyed

computational method based on reproducing kernel for solving singularly perturbed differential difference equations with a delay abstract in this paper reproducing kernel hilbert space method rkhsm based on collocation scheme is used to solve singularly perturbed second order differential difference equations we implement rkhsm without gram schmidt orthogonalization process for singularly perturbed differential difference equation with boundary layer behavior and also oscillatory behavior with small delay rkhsm in this study is based on the division of the problem domain into two subintervals one with boundary layer and the other one without such a boundary layer several numerical examples are studied to demonstrate the accuracy of the present method results of the present scheme indicate that new algorithm has the following advantages small computational work fast convergence and high precision

numerical methods on shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy abstract a numerical study is made for a class of boundary value problems of second order differential equations in which the highest order derivative is multiplied by a small parameter and both the differentiated convection and undifferentiated reaction terms are with negative shift we analyze three difference operators l k h k 1 2 3 a simple upwind scheme midpoint upwind scheme and a hybrid scheme respectively on a shishkin mesh to approximate the solution of the problem theoretical error bounds are established the accuracy by one of the grid adaptation strategies namely grid redistribution is examined by solving the considered problem using hybrid method for some values of and n as a result the accuracy gets improved by this grid adaptation strategy with almost the same computational cost a few numerical results exhibiting the performance of these three schemes are presented

uniform numerical method for singularly perturbed delay differential equations this paper deals with the singularly perturbed initial value problem for a linear first order delay differential equation a numerical method is constructed for this problem which involves an appropriate piecewise uniform mesh on each time subinterval the difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter numerical results are presented

a priori estimates and analysis of a numerical method for a turning point problem bounds are obtained for the derivatives of the solution of a turning point problem these results suggest a modification of the el mistikawy werle finite difference scheme at the turning point a uniform error estimate is obtained for the resulting method and illustrative numerical results are given i introduction we will examine the following two point boundary value problem with dirichlet data at the endpoints l la ly eyx x p x yx x q x y x f x for i x 1