open fuzzy cubature rule with application to nonlinear fuzzy volterra integral equations in two dimensions abstract in this paper we propose an open fuzzy cubature formula and as application we construct an iterative numerical method for solving nonlinear fuzzy volterra integral equations in two dimensions the convergence of the method is proved by providing the error estimate and it is tested on two numerical experiments

approximating the solution of nonlinear hammerstein fuzzy integral equations abstract in this paper we prove the convergence of the method of successive approximations used to approximate the solution of nonlinear hammerstein fuzzy integral equations and define the notion of numerical stability of the algorithm with respect to the choice of the first iteration this numerical stability is proved and finally some numerical experiments confirm the theoretical results and illustrate the accuracy of the method

the henstock stieltjes integral for fuzzy number valued functions in this paper we firstly define and discuss the henstock stieltjes integral for fuzzy number valued functions which is an extension of the usual fuzzy riemann stieltjes integral in addition several necessary and sufficient conditions of the integrability for fuzzy number valued functions are given by means of the henstock stieltjes integral of real valued functions and henstock integral of fuzzy number valued functions secondly the continuity and the differentiability of the primitive for the fuzzy henstock stieltjes integral are discussed we find that there exists a fuzzy number valued function which is fuzzy henstock stieltjes integrable but whose primitive is not a differentiable almost everywhere thirdly we introduce some quadrature rules for the fuzzy henstock stieltjes integral by giving error bounds for the mappings of bounded variation and of lipschitz type we also consider the generalization of classical quadrature rules such as midpoint type trapezoidal and simpson s quadrature finally we propose the concept of weak equi integrability for sequences of fuzzy henstock stieltjes integrable functions under this concept we prove two convergence theorems for sequences of the fuzzy henstock stieltjes integrable functions at the same time the formula of integration by parts is also studied

a solution to the continuous time rm h _ infty fixed interval smoother problem the minimum variance fixed interval smoother is a state space realization of the wiener solution generalized for time varying problems it involves forward and adjoint wiener hopf factor inverses in which the gains are obtained by solving a riccati equation this technical note introduces a continuous time h smoother having the structure of the minimum variance version in which the gains are obtained by solving a riccati equation that possesses an indefinite quadratic term it is shown that the smoother exhibits an increase in mean square error the error is bounded and the upper error bound is greater than that for the h filter

embedding problem of fuzzy number space part ii abstract using a concrete structure into which we embed the fuzzy number space e1 several necessary and sufficient conditions of fuzzy set valued functions are given by means of abstract function theory

brief paper state estimation for linear systems with state equality constraints this paper deals with the state estimation problem for linear systems with linear state equality constraints using noisy measurements which are available from the observable system we construct the optimal estimate which also satisfies linear equality constraints for this purpose after reviewing modeling problems in linear stochastic systems with state equality constraints we formulate a projected system representation by using the constrained kalman filter for the projected system and comparing its filter riccati equation with those of the unconstrained and the projected kalman filters we clearly show without using optimality that the constrained estimator outperforms the other filters for estimating the constrained system state finally a numerical example is presented which demonstrates performance differences among those filters

receding horizon estimation for discrete time linear systems the problem of estimating the state of a discrete time linear system can be addressed by minimizing an estimation cost function dependent on a batch of recent measure and input vectors this problem has been solved by introducing a receding horizon objective function that includes also a weighted penalty term related to the prediction of the state for such an estimator convergence results and unbiasedness properties have been proved the issues concerning the design of this filter are discussed in terms of the choice of the free parameters in the cost function the performance of the proposed receding horizon filter is evaluated and compared with other techniques by means of a numerical example

a smoothly constrained kalman filter this paper presents the smoothly constrained kalman filter sckf for nonlinear constraints a constraint is any relation that exists between the state variables constraints can be treated as perfect observations but linearization errors can prevent the estimate from converging to the true value therefore the sckf iteratively applies nonlinear constraints as nearly perfect observations or equivalently weakened constraints integration of new measurements is interlaced with these iterations which reduces linearization errors and hence improves convergence compared to other iterative methods the weakening is achieved by artificially increasing the variance of the nonlinear constraint the paper explains how to choose the weakening values and when to start and stop the iterative application of the constraint

ostrowski type inequalities for fuzzy valued functions and its applications in quadrature theory abstract this study provides a new characterization of the switching points for generalized hukuhara differentiable fuzzy valued functions new results on generalized hukuhara differential and integral calculus for fuzzy valued functions are developed also some ostrowski type inequalities for fuzzy valued functions are obtained with which is formulated a new quadrature rules to deal with integral of fuzzy valued functions and show that our results are better than previous ones moreover numerical examples are provided in order to illustrate the applicability of the mathematical tools developed herein

one sided fuzzy numbers and applications to integral equations from epidemiology in this paper we present the algebraic and topological structure of one sided fuzzy numbers and introduce the notion of side preserving fuzzy number valued function we define the dual decomposition of a commutative monoid and prove that the sets of left sided fuzzy numbers and right sided fuzzy numbers realize a dual decomposition of the additive monoid of all fuzzy numbers we prove the usefulness of right sided fuzzy numbers in epidemiology presenting an application of these fuzzy numbers to a mathematical model for the spread of infectious diseases with a rate of contact that varies seasonally

controlled convergence theorems for infinite dimension henstock integrals of fuzzy valued functions based on weak equi integrability abstract in this paper we give the concept of weak henstock equi integrability for a sequence of fuzzy number valued functions under this notion we investigate a new version of the henstock s lemma of fuzzy number valued functions thus it is possible to discuss the controlled convergence theorems of fuzzy henstock integral in sense of vitali covering moreover we prove that a uniform version of sklyarenko s and lusin s integrability condition of fuzzy henstock integrals together with pointwise convergence of a sequence of integrable functions is sufficient for a convergence theorem of fuzzy henstock integrals as the applications of the controlled convergence theorem we discuss the existence theorems of generalized solution for a class of discontinuous fuzzy differential equations

kalman filtering and riccati equations for descriptor systems a general formulation of a discrete time filtering problem for descriptor systems is considered it is shown that the nature of descriptor systems leads directly to the need to examine singular estimation problems using a dual approach to estimation the authors derive a so called 3 block form for the optimal filter and a corresponding 3 block riccati equation for a general class of time varying descriptor models which need not represent a well posed system in that the dynamics may be either over or under constrained specializing in the time invariant case they examine the asymptotic properties of the 3 block filter and in particular analyze in detail the resulting 3 block algebraic riccati equation the noncausal nature of discrete time descriptor dynamics implies that future dynamics may provide some information about the present state a modified form for the descriptor kalman filter that takes this information into account is presented

chaotic pulse trains the authors study a third order nonlinear ordinary differential equation whose solutions under certain specific conditions are individual pulses these correspond to homoclinic orbits in the phase space of the equation and the possible pulse types are studied in some detail sufficiently close to the conditions under which a homoclinic orbit exists the solutions take the form of trains of well separated pulses a measure of closeness to homoclinic conditions provides a small parameter for the development of an asymptotic solution consisting of superposed isolated pulses the solvability condition in the resulting singular perturbation theory is a timing map relating successive pulse spacings this map of the real line onto itself together with the known form of the homoclinic orbit provides a concise and accurate solution of the equation

asymptotic optimality of the minimum variance fixed interval smoother this correspondence investigates the asymptotic performance of the discrete time and continuous time time varying minimum variance fixed interval smoothers comparison theorems are generalized to provide sufficient conditions for the monotonic convergence of the underlying riccati equations under these conditions the energy of the estimation errors asymptotically approach a lower bound and attain lscr2 l2 stability

traveling waves of calcium in a pancreatic acinar cells model construction and bifurcation analysis abstract we construct and study a model for intracellular calcium wave propagation with particular attention to pancreatic acinar cells the model is based on a model of the inositol trisphosphate ip3 receptor which assumes that calcium modulates the binding affinity of ip3 to the receptor two versions of the model one simpler than the other are studied numerically in both versions solitary waves in the excitable regime arise via homoclinic bifurcations in the traveling wave equations as the background concentration of ip3 is increased the wave speed increases and for some values of the ip3 concentration the initial pulse gives rise to secondary pulses that travel in both directions this can give rise to irregular spatio temporal behavior or to trains of pulses in the simpler model these secondary waves are related to the presence of a t point a heteroclinic cycle and an associated spiral of homoclinic orbits which terminate the branch of homoclinic orbits

finite aspect ratio taylor couette flow shil nikov dynamics of 2 tori abstract the nonlinear dynamics of the flow in a short annular container driven by the rotation of the inner cylinder is studied using direct numerical simulations of the three dimensional navier stokes equations the basic state is s o 2 z 2 symmetric for aspect ratios between 3 6 and 4 4 we have located three codimension two bifurcations a cusp a double hopf and a fold hopf bifurcation all these local bifurcations are z 2 invariant the breaking of z 2 symmetry involves very complex shil nikov type dynamics not directly connected to any of the three codimension two bifurcations but associated with five unstable limit cycles and a wealth of heteroclinic connections between them period adding cascades both direct and reverse of 2 tori have been found narrow regions of chaotic dynamics are interspersed within these quasiperiodic solutions

short communication optimal fuzzy quadrature formula for classes of fuzzy number valued functions of lipschitz type in this paper we found out the optimal fuzzy quadrature formulas with given knots and the optimal fuzzy quadrature formulas among all fuzzy quadrature formulas for classes of fuzzy number valued functions of lipschitz type we also obtain the exact values of the errors of these optimal fuzzy quadrature formulas

numerical solution of two dimensional nonlinear hammerstein fuzzy integral equations based on optimal fuzzy quadrature formula in this paper our aim is to provide an efficient iterative method of successive approximations to approximate solution of linear and nonlinear two dimensional hammerstein fuzzy integral equations by defining and developing an optimal quadrature formula for classes of two dimensional fuzzy number valued functions of lipschitz type after the introduction of the optimal formula we prove the convergence of the method of successive approximations used to approximate the solution of two dimensional hammerstein fuzzy integral equations and investigate the numerical stability of the presented method with respect to the choice of the first iteration finally some illustrative numerical experiments confirm the theoretical results and demonstrate the accuracy of the method

numerical solution of nonlinear fuzzy fredholm integral equations using iterative method in this paper we propose an iterative procedure based on quadrature formula to solve nonlinear fuzzy fredholm integral equations of the second kind moreover the error estimation of the proposed method in terms of uniform and partial modulus of continuity is given finally illustrative examples are included in order to demonstrate the accuracy and the convergence of the proposed method

experimental unfolding of the nonlinear dynamics of a cable mass suspended system around a divergence hopf bifurcation systematic experimental investigation of the finite amplitude dynamics of a multiple internally resonant suspended cable mass subjected to anti phase support motion at primary resonance is accomplished upon getting hints from a basic system configuration assumed as reference setup about the multiple bifurcation event possibly governing transition to complex dynamics an improved experimental apparatus is used to make it technically accessible results obtained by varying three control parameters namely the frequency and amplitude of excitation and the temperature of a thermostatic chamber embedding the experimental system allow us to characterize in depth various occurring classes of motion in terms of time and spatial complexity to describe peculiar and or persistent features of transition to nonregular dynamics and to trace them back to a canonical scenario from bifurcation theory variable response paths are detected via bifurcation diagrams and spectra of singular values of measurement results and overall behaviour charts are built in the excitation parameter space considering the temperature as a controllable parameter shows to be fundamental for i indirectly setting cable material properties to values for which the conjectured codimension 2 bifurcation becomes apparent ii qualitatively referring the experimental unfolding of regular and nonregular cable dynamics to the theoretical unfolding of the divergence hopf bifurcation normal form and iii determining system response not only in the strict neighbourhood of the organizing divergence hopf bifurcation but also in the ensuing postcritical regions where the dependence of material damping on temperature affects secondary bifurcations to low dimensional homoclinic chaos

switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in odes the paper provides full algorithmic details on switching to the continuation of all possible codim 1 cycle bifurcations from generic codim 2 equilibrium bifurcation points in n dimensional odes we discuss the implementation and the performance of the algorithm in several examples including an extended lorenz 84 model and a laser system

merging of resonance tongues abstract this paper considers the resonance behavior found near a saddle node hopf bifurcation in a flow the global structure of resonance tongues is examined and it is shown that in the absence of certain symmetries different resonance tongues can merge with one another the position and shape of the resonance tongues are seen to be related to the presence of homoclinic and heteroclinic bifurcations of two saddle foci the implications of these results for the normal form analysis of the saddle node hopf bifurcation are discussed

a new bernoulli wavelet method for accurate solutions of nonlinear fuzzy hammerstein volterra delay integral equations in this article bernoulli wavelet method has been developed to solve nonlinear fuzzy hammerstein volterra integral equations with constant delay this type of integral equation has a particular case the fuzzy variant of a mathematical model from epidemiology bernoulli wavelets have been generated by dilation and translation of bernoulli polynomials the properties of bernoulli wavelets and bernoulli polynomials are first presented the present wavelet method reduces these integral equations to a system of nonlinear algebraic equations and again these algebraic systems have been solved numerically by newton s method convergence analysis of the present method has been discussed in this article also the results obtained by present wavelet method have been compared with that of by b spline wavelet method some illustrative examples have been demonstrated to show the applicability and accuracy of the present method

optimal and robust noncausal filter formulations the paper describes an optimal minimum variance noncausal filter or fixed interval smoother the optimal solution involves a cascade of a kalman predictor and an adjoint kalman predictor a robust smoother involving h sub spl infin predictors is also described filter asymptotes are developed for output estimation and input estimation problems which yield bounds on the spectrum of the estimation error these bounds lead to a priori estimates for the scalar spl gamma in the h sub spl infin filter and smoother design the results of simulation studies are presented which demonstrate that optimal robust and extended kalman smoothers can provide performance benefits

dynamics in the 1 2 spatial resonance with broken reflection symmetry the dynamics near the 1 2 spatial resonance in systems with broken o 2 symmetry are described the system is assumed to remain so 2 invariant and attention is focused on the behavior that replaces the structurally stable heteroclinic cycles present in the o 2 symmetric system the symmetry breaking destroys these cycles and introduces a variety of new bifurcations into the system some which are associated with complex behavior the possible scenarios with increasing symmetry breaking are described and possible applications of the results are discussed

fuzzy valued mappings with finite variation fuzzy valued measures and fuzzy valued lebesgue stieltjes integrals the fuzzy valued mapping with finite variation the distribution function of a fuzzy valued measure and the fuzzy valued lebesgue stieltjes integral are defined the correlative properties with them are established finally the fuzzy valued stochastic lebesgue stieltjes integral is introduced

a riccati equation approach to the design of stabilizing controllers and observers for a class of uncertain linear systems this paper presents a procedure for designing a full state observer and feedback control law which will stabilize a given uncertain linear system the uncertain linear systems under consideration are described by state equations which depend on uncertain parameters these uncertain parameters may be time varying their values however are constrained to lie within known compact bounding sets the design procedure involves solving two algebraic riccati equations a feature of the design procedure presented is the fact that it reduces to the standard lqg design procedure if the system contains no uncertain parameters

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

on numerical solution of two dimensional nonlinear urysohn fuzzy integral equations based on fuzzy haar wavelets in the present paper using fuzzy haar wavelets we introduce an iterative method for numerical solving two dimensional nonlinear urysohn fuzzy integral equations 2dnufie we define a useful metric to prove the convergence and error estimate of the method moreover we investigate the numerical stability of the presented numerical successive approximations finally numerical experiments show the accuracy of the method

restricted kalman filtering revisited we propose a more compact and general derivation of results concerning the estimation of linear state space models with linear restrictions in the state vector the resulting methodological contributions are that the restricted kalman filtering is valid regardless of the type of linear restriction being considered and that linear restrictions can be carried out by any type of state smoothing

robust filtering for discrete time systems with bounded noise and parametric uncertainty this note presents a new approach to finite horizon guaranteed state prediction for discrete time systems affected by bounded noise and unknown but bounded parameter uncertainty our framework handles possibly nonlinear dependence of the state space matrices on the uncertain parameters the main result is that a minimal confidence ellipsoid for the state consistent with the measured output and the uncertainty description may be recursively computed in polynomial time using interior point methods for convex optimization with n states l uncertain parameters appearing linearly in the state space matrices with rank one matrix coefficients the worst case complexity grows as o l n l sup 3 5 with unstructured uncertainty in all system matrices the worst case complexity reduces to o n sup 3 5

evaluate fuzzy riemann integrals using the monte carlo method techniques for using the monte carlo method to evaluate fuzzy riemann integrals and improper fuzzy riemann integrals are proposed in this paper owing to the level set of the improper fuzzy riemann integral being the closed interval whose end points are the classical improper riemann integrals it is possible to invoke the monte carlo method to approximate the end points of the level closed intervals we develop the strong law of large numbers for fuzzy random variables in order to give the techniques proposed for evaluating the improper fuzzy riemann integrals using the monte carlo approach more theoretical support the membership function of the improper fuzzy riemann integral can be transformed into mathematical programming problems therefore we can obtain the membership value by solving the mathematical programming problems using the commercial optimizer

application of simpson quadrature rule and iterative method for solving nonlinear fuzzy delay integral equations abstract in this study firstly we present an iterative method by using simpson quadrature formula for solving nonlinear fuzzy delay integral equations in addition the convergence analysis of the proposed method is investigated also we prove numerical stability analysis of the method by means of choosing the perturbation in the first iteration finally numerical examples to illustrate applicability of this method are presented

iterative method for numerical solution of two dimensional nonlinear fuzzy integral equations in this paper we prove the convergence of the method of successive approximations used to approximate the solution of two dimensional nonlinear fuzzy fredholm integral equations of the second kind also we present an iterative procedure based on quadrature formula to solve such equations the error estimation of the proposed method is given in terms of uniform and partial modulus of continuity finally some numerical experiments confirm the theoretical results and illustrate the accuracy of the method

diversity in interaction strength promotes rich dynamical behaviours in a three species ecological system abstract asymmetrical interactions are thought to have an important influence on multi species competitive systems and understanding this ecological process is a challenging task because of uncertain roles that asymmetrical interactions can have in maintaining species biodiversity to address this problem we employ a simple ecological model for multiple species with asymmetrical competitive strength and we analyse this model using the techniques from dynamical systems and bifurcation analysis in particular we examine the stability properties of steady states and also its bifurcation structures under parameter variation we demonstrate that there exists threshold values for asymmetrical competitive strength which correspond to transcritical and supercritical hopf bifurcations the occurrence of these bifurcations give rise to numerous dynamics in the model e g multi species coexistence species exclusion alternative stable state communities and oscillatory dynamics among multiple species additionally we observe the conditions under which co dimension two gavrilov guckenheimer bifurcation occurs as a result of the interaction between transcritical and hopf bifurcations which can lead to complex dynamics we also investigate how diversity in interaction strength between multiple competing species causes counterintuitive observations on one hand asymmetrical interactions promote the maintenance of biodiversity through multi species coexistence state and also coexistence via stable limit cycles corresponding to oscillatory dynamics on the other hand we also observe that the occurrence of stable limit cycles can lead to destabilisation of multi species communities

the entwined wiggling of homoclinic curves emerging from saddle node hopf instabilities abstract an analysis is presented of the homoclinic bifurcations occurring in a generic unfolding of a saddle node hopf singularity also known as a gavrilov guckenheimer point specifically an explanation is given of previously numerically observed oscillations of loci of homoclinic orbits to two different saddle focus equilibria these oscillations occur within an exponentially thin wedge of parameter space that emerges from the codimension two point the frequency of oscillation tends to zero as the codimension two point is approached earlier theory by gaspard showed that homoclinic orbits must exist inside the parameter wedge this result is here extended to give the frequency and amplitude of the oscillations of the homoclinic loci within the wedge it is also shown how the two loci are related to each other and in the case of only cubic perturbations of the normal form that they are precisely out of phase the analysis is shown to agree with numerical results on perturbed normal forms and in two model systems arising in applications to atmospheric dynamics and to calcium wave propagation

on kalman filtering with nonlinear equality constraints the state space description of some physical systems possess nonlinear equality constraints between some state variables in this paper we consider the problem of applying a kalman filter type estimator in the presence of such constraints we categorize previous approaches into pseudo observation and projection methods and identify two types of constraints those that act on the entire distribution and those that act on the mean of the distribution we argue that the pseudo observation approach enforces neither type of constraint and that the projection method enforces the first type of constraint only we propose a new method that utilizes the projection method twice once to constrain the entire distribution and once to constrain the statistics of the distribution we illustrate these algorithms in a tracking system that uses unit quaternions to encode orientation

numerical solutions of the nonlinear fuzzy hammerstein volterra delay integral equations in this paper an iterative numerical method that solves nonlinear fuzzy hammerstein volterra integral equations with constant delay is developed by using the error estimates a practical stopping criterion of the algorithm is obtained the convergence and the numerical stability of the method is proved and its accuracy is illustrated on three numerical examples the study of this integral equation is important because it has as a particular case the fuzzy variant of a mathematical model from epidemiology

the numerical calculus of expectations of fuzzy random variables in this paper the concept of the fuzzy henstock integral on infinite interval is proposed in order to calculate the expectation of fuzzy random variables two calculating methods are proposed one is to calculate directly by the fuzzy henstock integral proposed in this paper including quadrature rules and the error estimates such as the midpoint type rule trapezoidal type rule simpson s formula d fine quadrature rules and their error estimates another is to calculate by using the equivalent characteristic of fuzzy henstock integrability whose membership function could be obtained by solving a nonlinear programming problem

ill posedness for fuzzy fredholm integral equations of the first kind and regularization methods abstract in this study the concept of ill posedness for fuzzy fredholm integral equations of the first kind was first proposed an efficient regularization method based on classic tikhonov regularization scheme to determine the approximate solution of the fuzzy fredholm integral equations of the first kind is presented to support our investigation some examples are illustrated

local birth of homoclinic chaos abstract local generation of chaotic behaviours associated with a sil nikov homoclinic orbit is studied near the codimension two bifurcation at the double stationary oscillatory instability of eigenvalues 0 we develop a perturbative calculation of the sil nikov homoclinic orbit and of its bifurcation locus starting from a conservative vector field flat nonanalytic behaviours due to non axisymmetric perturbations of the normal form are discussed and we show that the codimension one locus of sil nikov homoclinicity is unavoidable near bicriticality under certain conditions the results are applied to rossler s model quasiperiodic regimes born at the double instability are also studied and illustrated in the rossler model

bounded constrained filtering for gps ins integration this paper considers estimation problems where inequality constraints are imposed on the outputs of linear systems and can be modeled by nonlinear functions in this case censoring functions can be designed to constrain measurements for use by filters and smoothers it is established that the filter and smoother output estimates are unbiased provided that the underlying probability density functions are even and the censoring functions are odd the bounded real lemma is employed to ensure that the output estimates satisfy a performance criterion a global positioning system gps and inertial navigation system ins integration application is discussed in which a developed solution exhibits improved performance during gps outages when a priori information is used to constrain the altitude and velocity measurements

quadrature rules for integrals of fuzzy number valued functions in this paper we introduce some quadrature rules for the henstock integral of fuzzy number valued mappings by giving error bounds for mappings of bounded variation and of lipschitz type we also consider generalizations of classical quadrature rules such as midpoint type trapezoidal and three point type quadrature finally we study fine quadrature rules and we present some numerical applications

fuzzy spline interpolation with optimal property in parametric form in this paper we propose a new fuzzy interpolation procedure that uses cubic splines with an optimal property obtained in parametric form this fuzzy interpolation operator is also a linear operator of uniform approximation that preserves the monotonicity moreover by integrating this fuzzy cubic spline we obtain the trapezoidal quadrature rule for fuzzy henstock integrable functions

on henstock integral of fuzzy number valued functions i abstract in this paper we define and discuss the fh integral for fuzzy number valued functions using a concrete structure into which we embed the fuzzy number space e 1 several necessary and sufficient conditions of integrability for fuzzy number valued functions are given by means of abstract function theory