discretization behaviors of equivalent control based sliding mode control systems in this note discretization behaviors of the equivalent control based sliding mode control smc systems are studied some inherent dynamical properties of the discretized second order systems are first explored upper bounds for the system steady states are established the system s steady state behaviors are discussed the analysis for the second order systems is then extended to higher order systems simulations are presented to verify the theoretical results

variable structure control of stochastic systems abstract in this paper the problem of variable structure control of stochastic svsc systems is proposed and the corresponding variable structure control strategies with complete and incomplete state information are established the concepts of stochastic sliding mode and sliding motion band are introduced which may be regarded as the basic characteristics of svsc systems robustness of svsc systems is considered and a general design procedure for svsc systems is presented the effectiveness of the proposed control method is shown by simulation results

the study of a class of rational difference equations abstract in this paper we study the global behavior of the higher order nonlinear difference equation x n 1 x n k 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 where b c and are positive k 1 2 3 and the initial conditions x 2 k 1 x 1 x 0 are positive real numbers we investigate the character of semicycles periodicity global stability and convergence to 2 k cycle solutions of mentioned equation

on the finite element method for singularly perturbed reaction diffusion problems in two and one dimensions second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered the behavior of the ordinary galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed

difference equation modelling of a variable structure system abstract variable structure systems with sliding modes have been widely discussed and used in many different fields of applications the precise behaviour at a switching surface is complicated because there the system is nonanalytic a two dimensional variable structure system is analysed in detail using a special discretisation with a difference equation model which is best in a particular sense this reveals the occurrence and structure of discrete pseudo sliding modes which give insight to the corresponding sliding modes for the continuous system necessary and sufficient conditions are obtained for the occurrence of the pseudo sliding modes and the analysis illustrated with graphs from numerical solutions

adaptive sliding mode control in discrete time systems abstract discrete time sliding mode control is considered the case of known parameters is first analyzed leading to a new definition of the so called equivalent control as the piecewise constant control that reduces to zero in finite time the distance of the system state from the sliding manifold in the presence of bounded parametric uncertainties an adaptive control scheme that guarantees the asymptotic satisfaction of the same control objective is presented the main feature of this approach is the reduction of the order of the relevant error equation and the possibility of dealing with the nonmatched uncertainties introduced by the sampling process

a monte carlo investigation of the planar lebwohl lasher lattice model abstract a monte carlo computer simulation of a planar version of the lebwohl lasher model is presented the model consists of a set of interaction centres forming a simple square lattice the pair potential is nearest neighbours attractive and varies as a second legendre polynomial of the relative orientation between the two particles five lattice sizes 5 5 10 10 20 20 60 60 and 80 80 of this two dimensional system have been simulated with monte carlo and periodic boundary conditions a study of the orientational pair correlation function indicates a power law decay in the ordered phase and an exponential decay above the pseudo transition temperature our results are consistent with the absence of a true phase transition but also indicate a low temperature phase with long short range order comparisons are made with one existing simulation and with the mean field theory results

a uniformly convergent method for a singularly perturbed semilinear reaction diffusion problem with multiple solutions this paper considers a simple central difference scheme for a singularly perturbed semilinear reaction diffusion problem which may have multiple solutions asymptotic properties of solutions to this problem are discussed and analyzed to compute accurate approximations to these solutions we consider a piecewise equidistant mesh of shishkin type which contains o n points on such a mesh we prove existence of a solution to the discretization and show that it is accurate of order n 2 in 2 n in the discrete maximum norm where the constant factor in this error estimate is independent of the perturbation parameter e and n numerical results are presented that verify this rate of convergence

on the higher order rational recursive sequence xn axn k bx n 3k the main purpose of the current paper is to prove that every positive solution of the delay difference equationx n ax n k bx n 3 k where a b 0 the initial conditions x 3 k 1 x 3 k 2 x 0 0 and k 1 2 3 converges eventually to a period k solution we also give the results of computational examples to support our theoretical discussion

variable structure systems nonlinear in the control law the most relevant results of utkin s work obtained for variable structure systems linear in the control law which slide on suitable manifolds in the state space are extended to some class of fully nonlinear plants the equivalence between the filippov solution concept is proved for this class further the approximability property convergence to the nonideal sliding motion to the ideal is shown to hold for an important subclass

on the global behavior of higher order recursive sequences our basic purpose in this paper is to establish some conditions for the global attractivity and convergence of the kth order recursive sequencex n f x n 1 x n 2 x n k n 1 2 3 we will show that if f is a real valued function with a convex basin w in r k which is differentiable with bounded differential then f has an equilibrium point our final comments are about informative examples

on the stability of discrete time sliding mode control systems the stability of discrete time sliding mode control systems is investigated and a new sliding mode condition is suggested it is shown that the control must have upper and lower bounds a numerical example is discussed as an illustration

uniform error estimates in the finite element method for a singularly perturbed reaction diffusion problem consider the problem e 2 u u f with homogeneous neumann boundary condition in a bounded smooth domain in r n the whole range 0 e 1 is treated the galerkin finite element method is used on a globally quasi uniform mesh of size h the mesh is fixed and independent of e a precise analysis of how the error at each point depends on h and e is is presented as an application first order error estimates in h which are uniform with respect to e are given

dynamics of a higher order rational difference equation abstract our aim in this paper is to investigate the dynamics of difference equation y n 1 p qy n 1 y n k n 0 1 2 where the initial conditions y k y 1 y0 are non negative k 1 2 3 and the parameters p and q are non negative we study characteristics such as the global stability the boundedness of positive solutions and the character of semicycles of the above mentioned difference equation in particular our results solve the open problem introduced by kulenovic and ladas in their monograph m r s kulenovic g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002

renormalisation group study of a lattice model of the isotropic nematic transition the isotropic nematic transition is studied using the lebwohl lasher lattice model of liquid crystal behaviour the details of the transition are investigated using the migdal kadanoff potential moving version of the real space renormalisation group the method gives good results when applied to the closely related heisenberg model of ferromagnetism a second order transition is predicted in disagreement with mean field and some computer simulation results

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

discrete time sliding mode controller design with weak pseudo sliding condition in this paper a discrete time sliding mode control law is developed to guarantee the existence of the weak pseudo sliding mode along the prescribed hyperplane meanwhile the upper bound of the sampling period is also determined in order to ensure the stability of the controlled system finally some simulation results are given to illustrate the performance of the proposed discrete time sliding mode control scheme

non linear discrete variable structure systems in quasi sliding mode abstract the problem of inducing convergent quasi sliding regimes on smooth state space surfaces of non linear single input single output discrete time controlled systems is addressed in full generality a suitable extension of the notion of relative degree is used in establishing the most salient features of quasi sliding motions in non linearly controlled systems several examples are given

pointwise approximation of corner singularities for a singularly perturbed reaction diffusion equation in an l shaped domain a singularly perturbed reaction diffusion equation is posed in a two dimensional l shaped domain subject to a continuous dirchlet boundary condition its solutions are in the holder space c 2 3 and typically exhibit boundary layers and corner singularities the problem is discretized on a tensor product shishkin mesh that is further refined in a neighboorhood of the vertex of angle 3 2 we establish almost second order convergence of our numerical method in the discrete maximum norm uniformly in the small diffusion parameter numerical results are presented that support our theoretical error estimate

absence of the kosterlitz thouless fixed points in the migdal kadanoff recursion formulas it is shown that the migdal approximate renormalization recursion formulas always bring the initial state into the high temperature region for the u 1 invariant models at the critical dimensions this means that the migdal recursion formulas fail to exhibit the kosterlitz thouless transitions this is also the case for the recursion formulas of kadanoff type the method developed here is extended to non abelian systems by some additional tricks

combined effect of explicit time stepping and quadrature for curvature driven flows the flow of a closed surface of codimension 1 in bbb r n driven by curvature is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter varepsilon 0 conforming piecewise linear finite elements with mass lumping over a quasi uniform and weakly acute mesh of size h are further used for space discretization and combined with forward differences for time discretization with uniform time step tau the resulting explicit schemes are the basis for an efficient algorithm the so called dynamic mesh algorithm and exhibit finite speed of propagation and discrete ondegeneracy no iteration is required not even to handle the obstacle constraints the zero level set of the fully discrete solution is shown to converge past singularities to the true interface provided tau h 2 approx o varepsilon 4 and no fattening occurs if the more stringent relations tau h 2 approx o varepsilon 6 are enforced then an interface rate of convergence o varepsilon is derived in the vicinity of regular points along with a companion o varepsilon 1 2 for type i singularities for smooth flows an interface rate of convergence of o varepsilonq is proven provided tau h 2 approx o varepsilon 5 and exact integration is used for the potential term the analysis is based on constructing fully discrete barriers via an explicit parabolic projection with quadrature which bears some intrinsic interest lipschitz properties of viscosity solutions of the level set approach and discrete nondegeneracy these basic ingredients are also discussed

variable structure control design for uncertain discrete time systems the main differences between the designs of discrete time variable structure systems vss and continuous time vss are the determination of the switching hyperplane and the satisfaction of the hitting sliding condition these differences yield many difficulties in the design of sliding mode control for the discrete time vss this note tries to overcome these difficulties and design a simple sliding mode control for a single input discrete time vss such that not only the discrete time uncertain system is stabilized robustly but also the chattering along the sliding mode is reduced explicitly instead of the conventional switching hyperplane in the discrete time case a switching region which includes the hyperplane s sub 0 and its neighborhood is determined and used the determination of the switching region is to locate the equilibrium point of the nominal subsystem on each hyperplane s sub p outside the switching region such that the state motions inside and outside the switching region will occur alternately until the state of the closed loop system reaches the origin moreover we prove that the robustness property is retained whenever the uncertainty satisfies a standard matching condition

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

the oscillatory character of the recursive sequence xn 1 α βxn k 1a bxn 2k 1 abstract the aim of this work is to generalize techniques of difference equations in order to apply them to a wider class we start by establishing the boundedness the oscillatory character and the global stability of the equation x n 1 x n k 1 a bx n 2 k 1 n 0 1 2 with positive parameters k 1 2 and nonnegative initial conditions

oscillation and asymptotic behavior of a class of higher order nonlinear recursive sequences abstract we investigate the global attractivity the periodic behavior and the invariant interval with the semicycles character of solutions of the equation x n 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 2 for all admissible non negative values of the parameters b c and the initial conditions x 2 k 1 x 1 x 0 we show that the solutions exhibit a periodic attitude depending on the parameters of the family of the above mentioned difference equation in comparison with each other

on the system of two difference equations xn 1 i 0kai yn ipi yn 1 i 0kbi xn iqi abstract in this paper we study the system of two difference equations of the form x n 1 i 0 k a i y n i p i y n 1 i 0 k b i x n i q i where a i b i i 0 1 k x i y i i k k 1 0 are positive numbers and p i q i i 0 k are positive constants more precisely we investigate the boundedness the persistence of the positive solutions the existence of a unique positive equilibrium and the global asymptotic stability of the positive equilibrium of the above system finally we find solutions of the system which do not oscillate about the positive equilibrium

study of a system of non linear difference equations arising in a deterministic model for hiv infection a deterministic model for hiv infection in the presence of combination therapy is introduced the model is then approximated by an implicit finite difference method a system of difference equations to solve the ivp system we initially purpose the discrete analogue of the continuous model of hiv infection then the boundedness the local and the global asymptotic stability of the equilibria of the system of difference equations the bistability phenomenon and the bifurcation phenomenon are also investigated in addition we will show that the associated approximational method is very robust since the equilibria and the stability conditions are independent on the time step finally numerical simulations of the system are presented when the system has meaningless solutions in view point of biology or our theory is incapable

dynamics of a rational difference equation using both theoretical and computational approaches this paper is concerned with the qualitative behavior of solutions to the difference equationx n 1 x n px n qx n k n 0 1 2 where the initial conditions x k x 1 x 0 are non negative email protected 1 2 3 and the parameters p q are non negative our concentration is on invariant intervals the character of semicycles the global stability and the boundedness of the above mentioned equation it is worth to mention that this difference equation was an open problem introduced by kulenovic and ladas in m r s kulenovic and g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002 our final comments are about informative examples

a parameter robust numerical method for a two dimensional reaction diffusion problem in this paper a singularly perturbed reaction diffusion partial differential equation in two space dimensions is examined by means of an appropriate decomposition we describe the asymptotic behaviour of the solution of problems of this kind a central finite difference scheme is constructed for this problem which involves an appropriate shishkin mesh we prove that the numerical approximations are almost second order uniformly convergent in the maximum norm with respect to the singular perturbation parameter some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method

a new synchronization tracking technique for uncertain discrete network with spatiotemporal chaos behaviors abstract we propose a novel scheme to achieve synchronization tracking of uncertain discrete network with spatiotemporal chaos behaviors in this work the traditional method of sliding mode control is firstly modified for researching conveniently the synchronization tracking of uncertain discrete network further the network sliding mode surface and control input are designed and their effectiveness are analyzed at the same time we also design the adaptive law to identify availably the uncertain configuration coefficient of the network sliding mode surface finally an example about the small world network is considered to illustrate the application and effectiveness of the proposed scheme

bond moving transformations for continuous spin systems we consider two and three dimensional lattices with on each lattice a d dimensional classical spin vector we restrict ourselves to nearest neighbour interactions which only depend on the angle between the spin vectors for these systems the bond moving approximation of migdal and kadanoff is used to derive renormalisation group equations which do not violate the symmetry of the lattice without further approximations these equations are then solved numerically in addition to the specific heat we also calculate critical temperatures critical exponents and phase diagrams for ferromagnets liquid crystals and other related systems for the simple cubic lattice we find a rich phase diagram which among others indicates the existence of an anti nematic phase in liquid crystals

brief sliding mode controllers design for linear discrete time systems with matching perturbations a simple methodology for designing sliding mode controllers is proposed in this paper for a class of linear multi input discrete time systems with matching perturbations if the functions of perturbations are bounded the proposed controllers cannot only satisfy the discrete sliding condition but also drive the state trajectories toward a small bounded region whose bound is dependent on the magnitude of perturbations in order to stabilize the closed loop system the determination of designed parameter of controllers is discussed neither chattering phenomenon will occur nor the knowledge of upper bound of perturbations is required numerical example is also given for demonstrating the feasibility of the proposed control scheme

geometric stability conditions for higher order difference equations is a higher order or delay autonomous difference equation if the order m is a fixed positive integer larger than 1 the real valued function f is assumed continuous on r and x x g r are the initial values in 1ym 0 addition to the fact that discretizations of scalar differential equations result in equations of type 1 direct applications of difference equations of order greater than 1 date back to more than half a century ago and nowadays these equations are increasingly encountered in both social and w x w x natural sciences see 4 and 6 for some applications and additional references a solution x of f x x s x is a fixed point or equilibrium of eq 1 like its differential analogs 1 can be and often is translated into a first order vector equation the instability or the asymptotic stability of x may then be determined by linearizing the associated vector function

robust control of discretized continuous systems using the theory of sliding modes the idea of sliding mode control smc as a robust control technique is utilized to control systems where the dynamics can be described by xdot t a a x t b b u t d t where a b and d t characterize unknown plant parameters and unexpected disturbances respectively the analysis is described in the discrete form using the euler operator the proposed controller regards the influence of unknown disturbances and parameter uncertainties as an equivalent disturbance and generates a control function estimate to cancel their influence by the mechanism of time delay the states of the uncertain plant are steered from an arbitrary initial state to a stable surface s 0 and the plant output is subsequently regulated one feature of the control is that the sliding conditions are satisfied without the discontinuous control action used in classical smc it therefore retains the positive features of smc without the disadvantage of control chattering the controller has been studied through s

sequence of discrete spin models approximating the classical heisenberg ferromagnet systematic investigation of a recent proposal of rapaport 1985 to describe the heisenberg model with discrete spin models is presented in two and three dimensions it is argued that in three dimensions a 12 state discretisation already accounts satisfactorily for the critical properties the corresponding anisotropy operators being irrelevant

a variable structure approach to feedback regulation of uncertain dynamical systems feedback control of uncertain dynamical systems is considered adopting a deterministic approach based on the theory of variable structure systems a feedback regulator is developed which guarantees global asymptotic stability or global ultimate boundedness under certain hypotheses on the bounded system uncertainties and disturbances

stabilised approximation of interior layer solutions of a singularly perturbed semilinear reaction diffusion problem a semilinear reaction diffusion two point boundary value problem whose second order derivative is multiplied by a small positive parameter varepsilon 2 is considered it can have multiple solutions the numerical computation of solutions having interior transition layers is analysed it is demonstrated that the accurate computation of such solutions is exceptionally difficult to address this difficulty we propose an artificial diffusion stabilization for both standard and stabilised finite difference methods on suitable shishkin meshes we prove existence and investigate the accuracy of computed solutions by constructing discrete sub and super solutions convergence results are deduced that depend on the relative sizes of varepsilon and n where n is the number of mesh intervals numerical experiments are given in support of these theoretical results practical issues in using newton s method to compute a discrete solution are discussed

finite element approximation of convection diffusion problems using an exponentially graded mesh we present the analysis of an h version finite element method for the approximation of the solution to convection diffusion problems the method uses piece wise polynomials of degree p 1 defined on an exponentially graded mesh optimally constructed for the approximation of exponential layers we consider a model convection diffusion problem posed on the unit square and establish robust optimal convergence rates in the energy and in the maximum norm we also present the results of some numerical computations that illustrate our theoretical findings and compare the proposed method with others found in the literature

a second order overlapping schwarz method for a 2d singularly perturbed semilinear reaction diffusion problem an overlapping schwarz domain decomposition is applied to a semilinear reaction diffusion equation posed in a smooth two dimensional domain the problem may exhibit multiple solutions its diffusion parameter e2 is arbitrarily small which induces boundary layers the schwarz method invokes a boundary layer subdomain and an interior subdomain the narrow subdomain overlap being of width o e ln h where h is the maximum side length of mesh elements and the global number of mesh nodes does not exceed o h 2 we employ finite differences on layer adapted meshes of bakhvalov and shishkin types in the boundary layer subdomain and lumped mass linear finite elements on a quasiuniform delaunay triangulation in the interior subdomain for this iterative method we present maximum norm error estimates for e 0 1 it is shown in particular that when e c ln h 1 one iteration is sufficient to get second order convergence with in the case of the shishkin mesh a logarithmic factor in the maximum norm uniformly in e numerical results are presented to support our theoretical conclusions

asymptotic error expansion and richardson extrapolation for linear fine elements the elliptic ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes this justifies the application of richardson extrapolation for increasing the accuracy

robust control of linear discrete time systems abstract given a linear discrete time system with additive disturbances a general methodology for designing a stable control algorithm is shown the approach is lyapunov based with certain liberty of tailoring the candidate function for a given problem comparison to the existing solutions based on a variable structure approach is given

bounds for solutions of a six point partial difference scheme abstract this paper is concerned with a class of partial difference equations which arise from discretising the one dimensional heat and the first order wave equations explicit bounds are found for their solutions and subexponential solutions are considered using the green s function we derive a stability criteria for arbitrary solutions of this equation

how accurate are finite elements on anisotropic triangulations in the maximum norm abstract in kopteva 2014 a counterexample of an anisotropic triangulation was given on which the exact solution has a second order error of linear interpolation while the computed solution obtained using linear finite elements is only first order pointwise accurate this example was given in the context of a singularly perturbed reaction diffusion equation in this paper we present further examples of unanticipated pointwise convergence behaviour of lagrange finite elements on anisotropic triangulations in particular we show that linear finite elements may exhibit lower than expected orders of convergence for the laplace equation as well as for certain singular equations and their accuracy depends not only on the linear interpolation error but also on the mesh topology furthermore we demonstrate that pointwise convergence rates which are worse than one might expect are also observed when higher order finite elements are employed on anisotropic meshes a theoretical justification will be given for some of the observed numerical phenomena

analysis of the migdal transformation for models with planar spins on a two dimensional lattice by numerical and analytical investigations we show that the bond moving transformation of migdal for planar spins on a two dimensional lattice possesses a very rich structure we find an infinite hierarchy of one three and higher dimensional fixed spaces in addition to phase transitions of the kosterlitz thouless type we find ising like transitions for which the critical exponents are the same as those of the q state potts model in the migdal approximation these critical exponents are constant within each fixed space the phase diagram is constructed for two specially chosen potentials

lyapunov stability and performance analysis of the implicit discrete sliding mode control discrete time sliding mode controllers with an implicit discretization of the signum function are considered with a proper choice of the equivalent part of the control the resulting controller is shown to be lyapunov stable with finite time convergence of the sliding variable to 0 the convergence of the control input as the sampling period goes to 0 to the continuous time one is shown the robustness with respect to matching perturbations is also investigated the discretization performance in terms of the error order is studied for different discretizations of the equivalent part of the input numerical and experimental results illustrate and support the analysis

global asymptotic behavior of a two dimensional difference equation modelling competition we investigate the global asymptotic behavior of solutions of the system of difference equations xn 1 xn a cyn yn 1 yn b dxn n 0 1 where the parameters a and b are in 0 1 c and d are arbitrary positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers we show that the stable manifold of this system separates the positive quadrant into basins of attraction of two types of asymptotic behavior in the case where a b we find an explicit equation for the stable manifold

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

dynamics of xn 1 xn 2k 1xn 2k 1 αxn 2l abstract we investigate the global stability of all positive solutions of the difference equation x n 1 x n 2 k 1 x n 2 k 1 x n 2 l where is a positive real number k 1 2 l 0 1 and the initial conditions are positive real numbers when x 1 1 is globally asymptotically stable also when 1 we show that this equation possesses solutions that converge to the two cycle 0 1 0 1

a parameter uniform numerical method for a singularly perturbed convection diffusion problem posed on an annulus abstract a finite difference method is constructed for a singularly perturbed convection diffusion problem posed on an annulus the method involves combining polar coordinates an upwind finite difference operator and a piecewise uniform shishkin mesh in the radial direction compatibility constraints are imposed on the data in the vicinity of certain characteristic points to ensure that interior layers do not form within the annulus a theoretical parameter uniform error bound is established and numerical results are presented to illustrate the performance of the numerical method applied to particular test problems

robustness of multivariable discrete time variable structure control a robust stability criterion for multivariable discrete time variable structure control is proposed when a variable structure controller is designed and implemented as a sampled data system achieving robustness to uncertainties can become difficult to ameliorate the situation an uncertainty estimator also formulated within the variable structure framework can be used as an embedded discrete time variable structure controller this approach requires a bounded changing rate of the uncertainties to ensure robust stability however when uncertainties vary as a function of state variables which occurs with parametric uncertainties it is not practical to assume such a bound in this paper uncertainties are assumed to consist of external disturbances and parametric uncertainties an uncertainty compensator is used to deal with the former and a robust stability criterion is developed using the small gain theorem for the latter

extended high temperature series for the classical heisenberg model in three dimensions extended series expansions are derived for the high temperature susceptibility of the classical heisenberg model on three dimensional lattices series coefficients are presented to twelfth order for the simple cubic sc and face centred cubic fcc lattices and to eleventh order for the body centred cubic bcc lattice the results are in agreement with earlier calculations apart from a small discrepancy at the tenth order on the fcc lattice extrapolation studies on the extended series are used to obtain revised estimates for the critical points kc and the susceptibility exponent gamma on the fcc lattice the authors also investigate the possibility of a confluent non analytic correction to the dominant singularity while the coefficients are consistent with the presence of such a correction term with an exponent delta 1 of 0 55 as predicted by renormalisation group rg calculations the amplitude of the correction term appears to be very small compared with that of the first analytic correction term

steady state behaviour of discretized terminal sliding mode in this paper the steady state behaviour of discretized terminal sliding mode control is studied a discrete terminal sliding mode control is designed by discretizing the continuous time system and then the steady state behaviour is analysed in terms of periodic orbits we have shown that by this discrete terminal sliding mode control the discretized second order system exhibits only period 2 motion in steady state to ensure this existence and stability conditions of all possible periodic orbits are found out analytically and shown that only period 2 conditions are satisfied by the state evolution of the discretized system in steady state next a higher order single input single output system is considered and we also establish that only period 2 motion exists in steady state

on discrete time variable structure sliding mode control the purpose of this paper is to show the limitations of discrete time variable structure sliding mode control and that the equivalent control must be used in order to have sliding in a neighborhood of the switching surface conflicting requirements for the sliding mode controller behavior in the continuous and discrete time domains are revealed and analyzed a linear control law for an uncertain discrete time linear plant with bounded uncertainties is analyzed and its superiority over nonlinear controllers is demonstrated the conclusion of the obtained results is that in the discrete time variable structure sliding mode controller design unlike in the continuous time the designer may have limited flexibility in selecting controller architectures

optimal anisotropic meshes for minimizing interpolation errors in l p norm in this paper we present a new optimal interpolation error estimate in l p norm 1 p for finite element simplicial meshes in any spatial dimension a sufficient condition for a mesh to be nearly optimal is that it is quasi uniform under a new metric defined by a modified hessian matrix of the function to be interpolated we also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the l p norm some numerical examples are also given to support the theoretical estimates

the characteristics of a higher order rational difference equation in this paper we study the global behavior of the higher order nonlinear difference equationx n email protected email protected n k email protected n 2 k 1a bx n k 1 n 0 1 where a b and a b c are positive email protected 1 2 3 and the initial conditions x 2 k 1 x 1 x 0 are positive real numbers we investigate the character of semicycles periodicity global stability and boundedness of above mentioned equation

global stability of yn 1 p qyn ryn k1 yn in this paper we investigate the global stability of difference equation y n 1 p qy n ry n k1 y n n 0 1 2 where the initial conditions y k y 1 y 0 are non negative email protected 1 2 3 and the parameters p q and r are non negative we show that when rq 1 and k is odd then this equation possesses unbounded solutions

linear finite elements may be only first order pointwise accurate on anisotropic triangulations we give a counterexample of an anisotropic triangulation on which the exact solution has a second order error of linear interpolation while the computed solution obtained using linear finite elements is only first order pointwise accurate our example is given in the context of a singularly perturbed reaction diffusion equation whose exact solution exhibits a sharp boundary layer furthermore we give a theoretical justification of the observed numerical phenomena using a finite difference representation of the considered finite element methods both standard and lumped mass cases are addressed

energy norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter to deal with anisotropic triangulations a special quasi interpolation operator is employed that may be of independent interest

molecular field theory of nematics density functional approach i bulk effects the molecular field theory of nematics is rephrased in density functional language the free energy in the nematic state is expressed as a functional expansion around the isotropic state in powers of the nematic order parameters pl and the fractional density change eta at the isotropic nematic transition the crucial quantities in the expansion are spherical harmonics of the direct correlation functions in the isotropic state sensible approximations to these direct correlation functions lead to other well known formulations of nematic field theory an exact in the authors approximation result for eta is obtained in terms of the potential parameters ul and the order parameters pl at the phase transition the authors discuss the relationship of their approach to other microscopic theories of nematics the theory can be extended to discuss inhomogeneous systems

a singularly perturbed semilinear reaction diffusion problem in a polygonal domain the semilinear reaction di usion equation i 24u b x u 0 with dirichlet bound ary conditions is considered in a convex polygonal domain the singular perturbation parameter e is arbitrarily small and the reduced equation b x u0 x 0 may have multiple solutions an asymptotic expansion for u is constructed that involves boundary and corner layer functions by perturbing this asymptotic expansion we obtain certain sub and super solutions and thus show the existence of a solution u that is close to the constructed asymptotic expansion the polygonal boundary forces the study of the nonlinear autonomous elliptic equation dz f z 0 posed in an infinite sector and then well posedness of the corresponding linearized problem

maximum norm error analysis of a 2d singularly perturbed semilinear reaction diffusion problem a semilinear reaction diffusion equation with multiple solutions is considered in a smooth two dimensional domain its diffusion parameter e 2 is arbitrarily small which induces boundary layers constructing discrete sub and super solutions we prove existence and investigate the accuracy of multiple discrete solutions on layer adapted meshes of bakhvalov and shishkin types it is shown that one gets second order convergence with in the case of the shishkin mesh a logarithmic factor in the discrete maximum norm uniformly in e for 0 is the maximum side length of mesh elements while the number of mesh nodes does not exceed ch 2 numerical experiments are performed to support the theoretical results

global stability of a higher order rational recursive sequence abstract in this paper we study the qualitative behavior of solutions of the class of delay difference equation x n 1 x n k 1 x n 2 k 1 a bx n k 1 n 0 1 2 where the initial conditions x 2 k 1 x 1 x 0 are positive k 1 2 and the parameters a b are positive our concentration is on invariant intervals and the global stability of the above mentioned equation we obtain sufficient conditions for the global attractivity of all positive solutions about the zero and positive equilibrium points with basins that depend on specific conditions posed on the coefficients furthermore the oscillation and the character of semicycles about the positive equilibrium are thoroughly discussed it is also illustrated that for some special case of parameters the solution will be either a period 2 solution or will converge to the equilibrium point or will have unbounded solutions

dynamics of a non linear difference equation abstract our aim in this paper is to investigate the dynamics of difference equation y n 1 y n py n k y n q n 0 1 2 where the initial conditions y k y 1 y0 are non negative k 1 2 3 and the parameters p and q are non negative we study characteristics such as periodicity invariant interval the character of semicycles the global stability and the boundedness of positive solutions of the above mentioned difference equation in particular our results solve the open problem introduced by kulenovic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002