Quadratic matrix polynomials with a parameter

In this paper we examine matrix polynomials of the form L(λ) = Aλ² + εBλ + C in which ε is a parameter and A, B, C are positive definite. This arises in a natural way in the study of damped vibrating systems. The main results are concerned with the generic case in which det L(λ) has at least 2n − 1 distinct zeros for all ε ϵ [0, ∞). The values of ε at which there is a multiple zero of det L(λ) are of major interest in this analysis. The dependence of first degree factors of L(λ) on ε is also discussed.     

Multiple solutions to damped Hamiltonian systems with impulsive effects

In this paper, we aim to prove the existence of solutions for impulsive Hamiltonian systems with first derivative. Two theorems of the existence of triple solutions are obtained via variational methods and three-critical-points theorems.     

Quadratic integrals of linear Hamiltonian systems

Momentum mapping of an autonomous, real linear Hamiltonian system is determined by its set of quadratic integrals. Such a system can be identified with an element of the real symplectic algebra and its quadratic integrals correspond to the centralizer of this element inside the symplectic algebra. In this paper, using a new set of normal forms for the elements of the real symplectic algebra, we compute their centralizers explicitly.Research supported in part by the National Science Foundation under NSF-MCS 8205355.     

Hamiltonian weakly damped autonomous systems exhibiting periodic attractors

The dynamic local stability of autonomous Hamiltonian, weakly damped, lumped-mass (discrete) systems is reconsidered. For such potential(conservative) systems conditions for the existence of limit cycles are discussed by studying the effect of the damping matrix on the Jacobian eigenvalues. New findings that contradict existing results are presented. Thus, undamped stable symmetric systems with the inclusion of slight damping may experience: (a) a double zero eigenvalue bifurcation, a degenerate Hopf bifurcation and a generic (usual) Hopf bifurcation, and (b) a limit cycle (dynamic) mode of instability prior to the static (divergence) mode of instability (failure of Zieglers kinetic criterion). A variety of numerical examples verified by a nonlinear analysis confirm the validity of the theoretical findings presented herein. Received: January 3, 2003; revised: July 14, 2003 and February 17, 2004     

Fractional Hamiltonian systems with positive semi-definite matrix

We study the existence of solutions for the following fractional Hamiltonian systems $$ \left\{ \begin{array}{ll} - _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla W(t,u(t))=0,\\[0.1cm] u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n), \end{array} \right. ~~~~~~~~~~~~~~~~~(FHS)_\lambda $$ where $\alpha\in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^n$, $\lambda>0$ is a parameter, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric matrix, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$. Assuming that $L(t)$ is a positive semi-definite symmetric matrix, that is, $L(t)\equiv 0$ is allowed to occur in some finite interval $T$ of $\mathbb{R}$, $W(t,u)$ satisfies some superquadratic conditions weaker than Ambrosetti-Rabinowitz condition, we show that (FHS)$_\lambda$ has a solution which vanishes on $\mathbb{R}\setminus T$ as $\lambda \to \infty$, and converges to some $\tilde{u}\in H^{\alpha}(\R, \R^n)$. Here, $\tilde{u}\in E_{0}^{\alpha}$ is a solution of the Dirichlet BVP for fractional systems on the finite interval $T$. Our results are new and improve recent results in the literature even in the case $\alpha =1$.     

Factorization of selfadjoint quadratic matrix polynomials with real spectrum

Factorization theorems, and properties of sets of eigenvectors, are established for regular selfadjoint quatratic matrix polynomials L(λ) whose leading coefficeint is indefinite or possibly singular, and for which all eigenvalues are real of definite type. The two linear factors obtained have spectra which are just the eigenvalues of L(λ) of positive and negative types, respectively.     

Quadratic decomposition of orthogonal polynomials: a matrix approach

We consider a scalar orthogonal polynomial set associated with a regular functional. Its quadratic decomposition allows us to consider 2×2 matrix orthogonal polynomials whose coefficients are the components of the mentioned decomposition. The connection between the recurrence relations is given.Partially supported by an Acción Integrada Hispano-Portuguesa, no. 28 B/91.Realized in part during a stay in Paris, Lab. Analyse Numérique, Univ. Pierre et Marie Curie — CNRS, granted by MEC(Spain)/MRT(France) FR91.     

Symplectic schemes for highly oscillatory Hamiltonian systems with varying fast frequencies

We derive symplectic integrators for a class of highly oscillatory Hamiltonian systems. Our approach is based upon a two-scale expansion of the solution to the Hamilton–Jacobi equation associated to the original dynamics. This Note presents an extension of the approach previously introduced in Le Bris and Legoll (2007, 2010) [10], [11] to the case where the fast frequencies of the system, instead of being constant, explicitly depend on the slow degrees of freedom.     

Conservation properties of numerical integrators for highly oscillatory Hamiltonian systems

^** Email: David.Cohen{at}math.unige.ch. Present address: Mathematisches Institut, Universität Tübingen, D-72076 Tübingen, Germany (cohen{at}na.uni-tuebingen.de) Modulated Fourier expansion is used to show long-time near-conservationof the total and oscillatory energies of numerical methods forHamiltonian systems with highly oscillatory solutions. The numericalmethods considered are an extension of the trigonometric methods.A brief discussion of conservation properties in the continuousproblem and in the multi-frequency case is also given.     

Oscillatory behavior of linear matrix Hamiltonian systems

We establish some new oscillation criteria for the matrix linear Hamiltonian system X ′ = A (t)X + B (t)Y, Y ′ = C (t)X –A *(t)Y by using a new function class X and monotone functionals on a suitable matrix space. In doing so, many existing results are generalized and improved. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

线性矩阵哈密顿系统的振动性

研究了线性矩阵 Hamilton系统X′=A( t) X + B( t) YY′=C( t) X -A*( t) Y　　 t≥ 0的振动性 .其中 A( t) ,B( t) ,C( t) ,X,Y为实 n× n矩阵值函数 ,B,C为对称矩阵 ,B正定 .借助于正线性泛函 ,采用加权平均法 ,得到了该系统的非平凡预备解的振动性 .这些结果推广、改进了许多已知的结果     

Oscillation theorems for self-adjoint matrix Hamiltonian systems

By using generalized Riccati technique, linear positive functional and the weighted averages technique, some new oscillation criteria for self-adjoint Hamiltonian matrix system

(E)     

Essential spectrum of singular discrete linear Hamiltonian systems

The paper is concerned with the essential spectral points of singular discrete linear Hamiltonian systems. Several sufficient conditions for a real point to be in the essential spectrum are obtained in terms of the number of linearly independent square‐summable solutions of the corresponding homogeneous linear system, and a sufficient and necessary condition for a real point to be in the essential spectrum is given in terms of the number of linearly independent square‐summable solutions of the corresponding nonhomogeneous linear system. As a direct consequence, the corresponding results for singular higher‐order symmetric vector difference expressions are given.     

New oscillation criteria for linear matrix Hamiltonian systems

Some new oscillation criteria are established for the matrix linear Hamiltonian system X′=A(t)X+B(t)Y, Y′=C(t)X−A∗(t)Y under the hypothesis: A(t), B(t)=B∗(t)>0, and C(t)=C∗(t) are n×n real continuous matrix functions on the interval [t₀,∞), (−∞<t₀). These results are sharper than some previous results even for self-adjoint second order matrix differential systems.     

Multiplicity of the spectrum of representations connected with integrable G-invariant Hamiltonian systems

Functional Analysis and Its Applications -