Lyapunov exponent calculation of a two-degree-of-freedom vibro-impact system with symmetrical rigid stops

A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole impact process, the Poincar'e map of the system is constructed. Using the Poincar'e map and the Gram-Schmidt orthonormalization, a method of calculating the spectrum of Lyapunov exponents of the above vibro-impact system is presented. Then the phase portraits of periodic and chaotic attractors for the system and the corresponding convergence diagrams of the spectrum of Lyapunov exponents are given out through the numerical simulations. To further identify the validity of the aforementioned computation method, the bifurcation diagram of the system with respect to the bifurcation parameter and the corresponding largest Lyapunov exponents are shown.     

Properties of some families of hypergeometric orthogonal polynomials in several variables

Limiting cases are studied of the Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials. We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable Wilson, continuous Hahn and Jacobi type polynomials, respectively. For each class of polynomials we provide systems of difference (or differential) equations, recurrence relations, and expressions for the (squared) norms of the polynomials in question.

     

Study on the orthonormalization eigenstates of α_q~3

Ｓｔｕｄｙｏｎｔｈｅｏｒｔｈｏｎｏｒｍａｌｉｚａｔｉｏｎｅｉｇｅｎｓｔａｔｅｓｏｆα_ｑ~３￥ＸＩＡＤｅｙｏｎｇ（ＨａｒｂｉｎＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｈａｒｂｉｎ１５００８０，Ｃｈｉｎａ）ＲＥＮＭｉｎ，ＭＡＡｉｑｕｎ（ＨａｒｂｉｎＣｏｌｌｅｇ...     

Using Selective Orthonormalization to Update the Analytic Center after Addition of Multiple Cuts

We study the issue of updating the analytic center after multiple cutting planes have been added through the analytic center of the current polytope. This is an important issue that arises at every stage of cutting-plane algorithms. If q n cuts are to be added, we show that we can use a selective orthonormalization procedure to modify the cuts before adding them; it is then easy to identify a direction for an affine step into the interior of the new polytope and the next analytic center is then found in O(qlog q) Newton steps. Further, we show that multiple cut variants with selective orthonormalization of standard interior-point cutting-plane algorithms have the same complexity as the original algorithms.This research was partially supported by ONR Grant N00014-94-1-0391, by NSF Grants CCR-9901822 and DMS-0317323, and by a grant from the Dutch NWO and Delft University of Technology for the 1997–98 academic year, while the senior author was visiting ITS/TWI/SSOR. The authors thanks two anonymous referees for careful reading of the paper and useful suggestions.     

Riesz Bases in Subspaces of L2 (R+ )

In a recent investigation [8] concerning the asymptotic behavior of Gram—Schmidt orthonormalization procedure applied to the nonnegative integer shifts of a given function, the problem of determining whether or not such functions form a Riesz system in arose. In this paper, we provide a sufficient condition to determine whether the nonnegative translates form a Riesz system on . This result is applied to identify a large class of functions for which very general translates enjoy the Riesz basis property in . August 5, 1998. Date revised: August 25, 1999. Date accepted: January 11, 2000.     

Applications of Commuting Difference Operators to Orthogonal Polynomials in Several Variables

Elementary properties of the Koornwinder--Macdonald multivariable Askey--Wilson polynomials are discussed. Studied are the orthogonality, the difference equations, the recurrence relations, and the orthonormalization constants for these polynomials. Essential in our approach are certain commuting difference operators simultaneously diagonalized by the polynomials.     

Influence of iterated Gram–Schmidt orthonormalization in the asymptotic numerical methodInfluence de l'orthonormalisation itérative de Gram–Schmidt dans la méthode asymptotique numérique

In this paper we discuss the influence of Gram–Schmidt orthonormalization for the computation of the rational representation in the asymptotic numerical method. Classical, modified and iterated Gram–Schmidt algorithms are compared using the ordinary and the mass scalar products. The accuracy of the proposed algorithms are tested on elastic shells. To cite this article: R. Jamai, N. Damil, C. R. Mecanique 331 (2003).