Analysis of stochastic bifurcation and chaos in stochastic Duffing-van der Pol system via Chebyshev polynomial approximation

The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing-van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.

Real scalar field scattering with polynomial approximation around Schwarzschild–de Sitter black-hole

As one of the fitting methods, the polynomial approximation is effective to process sophisticated problem. In this paper, we employ this approach to handle the scattering of scalar field around the Schwarzschild-de Sitter blackhole. The complicated relationship between tortoise coordinate and radial coordinate is replaced by the approximate polynomial. The Schroedinger-like equation, the real boundary conditions and the polynomial approximation construct a full Sturm Liouville type problem. Then this boundary value problem can be solved numerically for two limiting cases： the first one is the Nariai black-hole whose horizons are close to each other, the second one is the black-hole with the horizons widely separated. Compared with previous results （Brevik and Tian）, the field near the event horizon and cosmological horizon can have a better description.

SHAPE-PRESERVING BIVARIATE POLYNOMIAL APPROXIMATION IN C([-1,1]×[-1,1])

In this paper we construct bivariate polynomials attached to a bivariate function, that approximate with Jackson-type rate involving a bivariate Ditzian-Totik ωξ-modulus of smoothness and preserve some natural kinds of bivariate monotonicity and convexity of function.The result extends that in univariate case-of D. Leviatan in [5-6], improves that in bivariate case of the author in [3] and in some special cases, that in bivariate case of G. Anastassiou in [1].

Parametric Problems in Power System Analysis:Recent Applications of Polynomial Approximation Based on Galerkin Method

In power systems, there are many uncertainty factors such as power outputs of distributed generations and fluctuations of loads. It is very beneficial to power system analysis to acquire an explicit function describing the relationship between these factors(namely parameters) and power system states(or performances). This problem, termed as parametric problem(PP) in this paper, can be solved by Galerkin method,which is a powerful and mathematically rigorous method aiming to seek an accurate explicit approximate function by projection techniques. This paper provides a review of the applications of polynomial approximation based on Galerkin method in power system PPs as well as stochastic problems. First, the fundamentals of polynomial approximation and Galerkin method are introduced. Then, the process of solving three types of typical PPs by polynomial approximation based on Galerkin method is elaborated. Finally, some application examples as well as several potential applications of power system PPs solved by Galerkin method are presented, namely the probabilistic power flow, approximation of static voltage stability region boundary, and parametric time-domain dynamic simulation.

K-Functional, Weighted Moduli of Smoothness, and Best Weighted Polynomial Approximation on a Simplex

An inverse theorem for the best weighted polynomial approximation of a function in Lwαp（S） is established. We also investigate Besov spaces generated by Freud weight and their connection with algebraic polynomial approximation in Lp（R）wλ, where wα is a Jacobi-type weight on S, 0【p≤∞, S is a simplex and Wλ is a Freud weight. For Ditzian-Totik K-functionals on LP（S）, 1≤P≤∞, we obtain a new equivalence expression.

Inequalities for the Extended Best Polynomial Approximation Operator in Orlicz Spaces

In this paper we pursue the study of the best approximation operator extended from L~Φ to L~φ, where φ denotes the derivative of the function Φ. We get pointwise convergence for the coefficients of the extended best approximation polynomials for a wide class of function f, closely related to the Calder′on–Zygmund class t_m^p(x) which had been introduced in 1961. We also obtain weak and strong type inequalities for a maximal operator related to the extended best polynomial approximation and a norm convergence result for the coefficients is derived. In most of these results, we have to consider Matuszewska–Orlicz indices for the function φ.

On the Choice of Design Points for Least Square Polynomial Approximations with Application to Uncertainty Quantification

In this work,we concern with the numerical comparison between different kinds of design points in least square(LS)approach on polynomial spaces.Such a topic is motivated by uncertainty quantification(UQ).Three kinds of design points are considered,which are the Sparse Grid(SG)points,the Monte Carlo(MC)points and the Quasi Monte Carlo(QMC)points.We focus on three aspects during the comparison:(i)the convergence properties;(ii)the stability,i.e.the properties of the resulting condition number of the design matrix;(iii)the robustness when numerical noises are present in function values.Several classical high dimensional functions together with a random ODE model are tested.It is shown numerically that(i)neither the MC sampling nor the QMC sampling introduce the low convergence rate,namely,the approach achieves high order convergence rate for all cases provided that the underlying functions admit certain regularity and enough design points are used;(ii)The use of SG points admits better convergence properties only for very low dimensional problems(say d≤2);(iii)The QMC points,being deterministic,seem to be a good choice for higher dimensional problems not only for better convergence properties but also in the stability point of view.

THE BERNSTEIN TYPE INEQUALITY AND SIMULTANEOUS APPROXIMATION BY INTERPOLATION POLYNOMIALS IN COMPLEX DOMAIN

In this paper we investigate simultaneous approximation for arbitrary system of nodes on smooth domain in complex plane. Some results which are better than those of known theorems are obtained.

ON SIMULTANEOUS APPROXIMATION BY LAGRANGE INTERPOLATING POLYNOMIALS

This paper considers to replace △_m(x)=(1-x^2)~2(1/2)/n +1/n^2 in the following result for simultaneous Lagrange interpolating approximation with (1-x^2)~2(1/2)/n: Let f∈C_(-1.1)~0 and r=[(q+2)/2],then |f^(k)(x)-P_^(k)(f,x)|=O(1)△_(n)^(a-k)(x)ω(f^(a),△(x))(‖L_n-‖+‖L_n‖),0≤k≤q, where P_n( f ,x)is the Lagrange interpolating polynomial of degree n+ 2r-1 of f on the nodes X_n U Y_n(see the definition of the text), and thus give a problem raised in [XiZh] a complete answer.

A DEVELOPMENT ON APPROXIMATION BY MONOTONE SEQUENCES OF POLYNOMIALS

Recently people proved that every f∈C[0, 1] can be uniformly approximated by polynomial sequences {P_n}, {Q_n} such for any x∈[0,1] and n=1,2,…that Q_n(x)<Q_(n+1)(x)<f(x)<P_(n+1)(x)<P_n(x). For example, Xie and Zhou showed that one can construct such monotone polynomial sequences which do achieve the best uniform approximation rate for a continuous func- tion. Actually they obtained a result as ‖P_n(x)-Q_n(x)‖≤42E_n (f), (1) which essentially improved a conclusion in Gal and Szabados. The present paper, by optimal procedure, improves this inequality to ‖[P_n(x)-Q_n(x)‖≤(18+ε)E_n(f), where εis any positive real number.

APPROXIMATION BY KANTOROVICH-BERNSTEIN POLYNOMIALS IN L_p(0

The purpose of the present paper is to evaluate the error of the approximation of the func- tion fL_1[0,1]by Kantorovich-Bernstein polynomials in L_p-metric(0<p<1).

UNIFORM APPROXIMATION BY COMBINATIONS OF BERNSTEIN POLYNOMIALS

We prove a pointwise characterization result for combinations of Bernstein polynomials. The main result of this paper includes an equivalence theorem of H. Berens and G. G. Lorentz as a special case.

ON CONSERVATIVE APPROXIMATION BY LINEAR POLYNOMIAL OPERATORS AN EXTENSION OF THE BERNSTEIN'S OPERATOR

In this work we slwly linear polynomial operators preserving some consecutive i-convexities and leaving in-verant the polynomtals up to a certain degree. First we study the existence of an incompatibility between the conservation of cenain i-cotivexities and the invariance of a space of polynomials. Interpolation properties are obtained and a theorem by Berens and DcVore about the Bernstein's operator ts extended. Finally, from these results a genera'ized Bernstein's operator is obtained.

ON APPROXIMATION OF A FUNCTION AND ITS DERIVATIVES BY A POLYNOMIAL AND ITS DERIVATIVES

Let g∈C^q[-1, 1] be such that g^((k))(±1)=0 for k=0,…,q. Let P_n be an algebraic polynomial of degree at most n, such that P_n^((k))(±1)=0 for k=0,…,[_2~ (q+1)]. Then P_n and its derivatives P_n^((k)) for k≤q well approximate g and its respective derivatives, provided only that P_n well approxi- mates g itself in the weighted norm ‖g(x)-P_n(x) (1-x^2)^(1/2)~q‖ This result is easily extended to an arbitrary f∈C^q[-1, 1], by subtracting from f the polynomial of minnimal degree which interpolates f^((0))…,f^((q)) at±1. As well as providing easy criteria for judging the simultaneous approximation properties of a given Polynomial to a given function, our results further explain the similarities and differences between algebraic polynomial approximation in C^q[-1, 1] and trigonometric polynomial approximation in the space of q times differentiable 2π-periodic functions. Our proofs are elementary and basic in character, permitting the construction of actual error estimates for simultaneous approximation proedures for small values of q.

Bifurcation Analysis of a Nonlinear Vibro-Impact System with an Uncertain Parameter via OPA Method

In this paper,the bifurcation properties of the vibro-impact systems with an uncertain parameter under the impulse and harmonic excitations are investigated.Firstly,by means of the orthogonal polynomial approximation(OPA)method,the nonlinear damping and stiffness are expanded into the linear combination of the state variable.The condition for the appearance of the vibro-impact phenomenon is to be transformed based on the calculation of themean value.Afterwards,the stochastic vibro-impact systemcan be turned into an equivalent high-dimensional deterministic non-smooth system.Two different Poincarésections are chosen to analyze the bifurcation properties and the impact numbers are identified for the periodic response.Consequently,the numerical results verify the effectiveness of the approximation method for analyzing the considered nonlinear system.Furthermore,the bifurcation properties of the system with an uncertain parameter are explored through the high-dimensional deterministic system.It can be found that the excitation frequency can induce period-doubling bifurcation and grazing bifurcation.Increasing the randomintensitymay result in a diffusion-based trajectory and the impact with the constraint plane,which induces the topological behavior of the non-smooth system to change drastically.It is also found that grazing bifurcation appears in advance with increasing of the random intensity.The stronger impulse force can result in the appearance of the diffusion phenomenon.

An approximation algorithm for parallel machine scheduling with simple linear deterioration

In this paper, a parallel machine scheduling problem was considered , where the processing time of a job is a simple linear function of its starting time. The objective is to minimize makespan. A fully polynomial time approximation scheme for the problem of scheduling n deteriorating jobs on two identical machines was worked out. Furthermore, the result was generalized to the case of a fixed number of machines.

COMONOTONE APPROXIMATION WITH INTERPOLATION AT THE ENDS ON AN INTERVAL

Let a function f E C[-1, 1], changes its monotonisity at the finite collection Y ：= {y1,……, ys} of s points Yi ∈ （-1, 1）. For each n 〉 N（Y）, we construct an algebraic polynomial Pn, of degree 〈 n, which is comonotone with f, that is changes its monotonisity at the same points yi as f, and ｜f（x） - Pn（x）｜ ≤ c（s）ω2 （f1 √1-x^2/n）,x∈ [-1,1] where N（Y） is a constant depending only on Y, c（s） is a constant depending only on s and ω2 （f, t） is the second modulus of smoothness of f.

Improved Approximation Schemes for Early Work Scheduling on Identical Parallel Machines with a Common Due Date

We study the early work scheduling problem on identical parallel machines in order to maximize the total early work,i.e.,the parts of non-preemptive jobs that are executed before a common due date.By preprocessing and constructing an auxiliary instance which has several good properties,for any desired accuracy,we propose an efficient polynomial time approximation scheme with running time O(f(1/ε)n),where n is the number of jobs and f(1/ε)is exponential in 1/ε,and a fully polynomial time approximation scheme with running time O(1/ε^(2m+1)+n)when the number of machines is fixed.

Time-domain identification of dynamic properties of layered soil by using extended Kalman filter and recorded seismic data

A novel time-domain identification technique is developed for the seismic response analysis of soil-structure interaction.A two-degree-of-freedom (2DOF) model with eight lumped parameters is adopted to model the frequency- dependent behavior of soils.For layered soil,the equivalent eight parameters of the 2DOF model are identified by the extended Kalman filter (EKF) method using recorded seismic data.The polynomial approximations for derivation of state estimators are applied in the EKF procedure.A realistic identification example is given for the layered-soil of a building site in Anchorage,Alaska in the United States.Results of the example demonstrate the feasibility and practicality of the proposed identification technique.The 2DOF soil model and the identification technique can be used for nonlinear response analysis of soil-structure interaction in the time-domain for layered or complex soil conditions.The identified parameters can be stored in a database tor use in other similar soil conditions,lfa universal database that covers information related to most soil conditions is developed in the thture,engineers could conveniently perform time history analyses of soil-structural interaction.

Stochastic period-doubling bifurcation analysis of stochastic Bonhoeffer-van der Pol system

In this paper, the Chebyshev polynomial approximation is applied to the problem of stochastic period-doubling bifurcation of a stochastic Bonhoeffer-van der Pol （BVP for short） system with a bounded random parameter. In the analysis, the stochastic BVP system is transformed by the Chebyshev polynomial approximation into an equivalent deterministic system, whose response can be readily obtained by conventional numerical methods. In this way we have explored plenty of stochastic period-doubling bifurcation phenomena of the stochastic BVP system. The numerical simulations show that the behaviour of the stochastic period-doubling bifurcation in the stochastic BVP system is by and large similar to that in the deterministic mean-parameter BVP system, but there are still some featured differences between them. For example, in the stochastic dynamic system the period-doubling bifurcation point diffuses into a critical interval and the location of the critical interval shifts with the variation of intensity of the random parameter. The obtained results show that Chebyshev polynomial approximation is an effective approach to dynamical problems in some typical nonlinear systems with a bounded random parameter of an arch-like probability density function.