Mathematics

Abstract—In this paper we study the decay behaviour for the entries of functions of skew symmetric matrices arising in the applications. Thanks to an algorithm proposed in [4] for computing the exponential of tridiagonal skew symmetric matrices we will be able to study the decay behaviour of more general functions of matrices and to define a banded version of such function of matrices. Numerical tests showing this kind of behaviour are reported for several kind of functions.

Krylov subspace methods for approximating the action of the matrix exponential exp (A) on a vector v are analyzed with A Hermitian and negative semidefinite. Our approach is based on approximating the exponential with the commonly employed diagonal Padé and Chebyshev rational functions, which yield a system of equations with a polynomial coefficient matrix. We derive optimality properties and error bounds for the convergence of a Galerkin-type approximation and of a computationally feasible and extensively used alternative.

Abstract Given a large square real matrix A and a rectangular tall matrix Q, many application problems require the approximation of the operation. Under certain hypotheses on A, the matrix preserves the orthogonality characteristics of Q; this property is particularly attractive when the associated application problem requires some geometric constraints to be satisfied. For small size problems numerical methods have been devised to approximate while maintaining the structure properties.

Abstract The aim of this paper is to outline the numerical solution of a reaction—diffusion system describing the evolution of an epidemic in an isolated habitat. The model we consider is described by two weakly coupled semi-linear parabolic equations and we introduce a finite difference scheme for its numerical solution. We study the behaviour of the exact solution by means of the numerical scheme. We show the positivity, the decrease and the decay to extinction of the numerical solution.

In this paper we show some symmetry properties of Lyapunov exponents of a dynamical system when the linearized problem evolves on a quadratic group, XTHX= H, with H orthogonal. It is well understood that in this case the exponents are symmetric with respect to the origin. Here, we give lower bounds on the number of Lyapunov exponents which are 0 and show that some Lyapunov exponents may have even multiplicity.

Abstract We give a constructive argument to establish existence of asmooth singular value decomposition (SVD) for a generic C k symplectic function X. We rely on the explicit structure of the polar factorization of X in order to justify the form of the SVD. Our construction gives a new algorithm to find the SVD of X, which we have used to approximate the Lyapunov exponents of a Hamiltonian differential system. Algorithmic details and an example are given. Keywords: symplectic group, singular values, Lyapunov exponents

In this work we give a constructive argument to establish existence of a smooth singular value decomposition (SVD) for a Ck function X in the Lorentz group. We rely on the explicit structure of the polar factorization of X in order to justify the form of the SVD. Our construction gives a simple algorithm to find the SVD of X, which we have used to approximate the Lyapunov exponents of a differential system whose fundamental matrix solution evolves on the Lorentz group. Algorithmic details and examples are given.

This paper is concerned with the numerical solution of an implicit matrix differential system of the form YT (Y) ̇-F (t, Y)= 0 Y^ T ̇ YF (t, Y)= 0, where Y (t) is an× n real matrix which may converge to a singular matrix. We propose a hybrid numerical technique based on an implicit second order Runge Kutta scheme which derives a particular algebraic Riccati equation and via its solution approximates the solutions of the differential problem at hand.

Résumé/Abstract On propose un schéma aux différences finies pour la résolution numérique du problème suivant:∂ u (a, t, x)∂ a+∂ u (a, t, x)/∂ t= div (c (x)⊇ u (a, t, x))− m (a) u (a, t, x), t≥ 0, 0≤ a< w, x∈ Ω, u (a, 0, x)= uo (a, x), 0≤ a< w, x∈ Ω, u (0, t, x)=∫ 0 w β (a) u (a, t, x) da t≥ 0, x∈ Ω; u (a, t, x)= 0 ou γu (a, t, x)/γν= 0 t≥ 0, 0≤ a< w, x∈∂ Ω

In this paper we consider ODEs whose solutions satisfy exponential monotonic quadratic forms. We show that the quadratic preserving Gauss–Legendre–Runge–Kutta methods do not preserve this qualitative feature, while certain Lie group preserving methods, as Crouch–Grossman methods and methods based on projection techniques are suitable integrators for such a kind of differential problems. We also show that these differential problems may be solved via associated differential systems with quadratic invariants.

Abstract: Discontinuous dynamical systems with sliding modes are often used in Control Theory to model differential equations with discontinuous control. Filippov and Utkin (see 2, 7) have proposed two different approaches to define the solution of these dynamical systems. In case of linear systems, these two approaches are equivalent, but in case of nonlinear systems, the ways to extend the vector field on the sliding surface is generally different.

In recent years there has been a growing interest in the dynamics of matrix differential systems on a smooth manifold. Research effort extends to both theory and numerical methods, particularly on the manifolds of orthogonal and symplectic matrices. This paper concerns dynamical systems on the manifold \calOB(n) of square oblique rotation matrices, a constraint appearing in some minimization problems and in multivariate data analysis.

Abstract In this paper, we study stability properties of a numerical method applied to linear second order perturbed boundary-value problems with boundary or interior layers. The method consists of a three-point difference scheme with variable stepsize, the stability of which may be investigated by studying that of an LU factorization for the coefficient matrix of a suitable tridiagonal system.

This note deals with the numerical solution of the matrix differential system Y′=[B (t, Y), Y], Y (0)= Y0, t⩾ 0, where Y0 is a real constant symmetric matrix, B maps symmetric into skew-symmetric matrices, and [B (t, Y), Y] is the Lie bracket commutator of B (t, Y) and Y, ie [B (t, Y), Y]= B (t, Y) Y− YB (t, Y). The unique solution of (1) is isospectral, that is the matrix Y (t) preserves the eigenvalues of Y0 and is symmetric for all t (see [1, 5]).

Abstract In this paper we consider the issue of sliding motion in Filippov systems on the intersection of two or more surfaces. To this end, we propose an extension of the Filippov sliding vector field on manifolds of co-dimension p, with p≥ 2. Our model passes through the use of a multivalued sign function reformulation. To justify our proposal, we will restrict to cases where the sliding manifold is attractive.

In recent years several numerical methods have been developed to integrate matrix differential systems of ODEs whose solutions remain on a certain Lie group throughout the evolution. In this paper some results, derived for the orthogonal group in by Diele et al.(1998), will be extended to the class of quadratic groups including the symplectic and Lorentz group. We will show how this approach also applies to ODEs on the Stiefel manifold and the orthogonal factorization of the Lorentz group will be derived.