矩阵代数及其在时滞动力系统中的应用

项目名称： 矩阵代数及其在时滞动力系统中的应用

项目编号： No.10871056

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 郑宝东

作者单位： 哈尔滨工业大学

项目金额： 27万元

中文摘要： 本项目主要工作如下:1)给出了直接利用系数判断实系数多项式有一对或两对模等于1的共轭复根，其余所有根的模都小于1的代数判据。2）利用矩阵理论给出了具有Z_2对称性的二次多项式系统的代数刻画，得到了二次多项式系统Z_2对称的判别条件。3）研究了具有对称性的耦合振子模型的集体行为。根据对称群理论，成功地将Golubisky的等变分支理论用于时滞耦合系统，研究了耦合系统在群作用下的不变性，给出了系统等变的性质。得到了描述耦合振子振动模式的多重周期解分支和斑图形式：镜像反射波、驻波和离散波。4）研究了具有二面体群D_3对称结构的耦合映射，证明了广义中心子空间在群作用下的不变性，从而展示了耦合离散映射的非平凡的集体行为如多重分支和混沌等。本课题亦首先开展了具有二面体群对称的自变量分段连续的耦合振子的等变分支问题。得到了对称分支的多种形式。5)应用若当代数的基本理论研究互补问题的解的存在唯一性。给出了若当代数上李雅普诺夫函数的一些静态性质以及它们的相互关系。提出了一类带有两个参数的互补函数，并且给出了这类函数具有的性质。基于这类函数给出了光滑化牛顿法来求解非线性互补问题。

中文关键词： Jury判据;Jordan 代数；对称群；动力系统；不变量

英文摘要： In this project we manly have done the followings: 1）Algebraic criteria are established to determine whether or not a real coefficient polynomial has one or two pairs of conjugate complex roots whose moduli are equal to 1 and the other roots have moduli less than 1 directly from its coefficients. 2）For the quadratic polynomial systems, using the matrix theorem, we have presented the general algebraic description of Z2-symmetric system and obtained the matrix criterion of Z2-equivariant. 3) We considered the collective behavior of coupled oscillators. Based on the Lie symmetric groups theorem, we successful applyed Gulubisky' equivariant bifurcation theorem to the delayed coupled symmetric system, and obtained certain invariant of the dynamical systems under the symmetric group action. Furthermore，we obtained some important results about the spontaneous bifurcations of multiple branches of periodic solutions and their spatio-temporal patterns: mirror-reflecting waves, standing waves, and discrete waves which describe the oscillatory mode of each oscillator . 4) We investigated a coupled maps with the structure can be represented by a dihedral group D3. We show that the generalized center subspace is invariant under the action of the symmetry group. Hence, the system exhibited nontrivial collective behavior, such as multiple bifurcations and chaos. We also first discussed the Symmetry bifurcations of a coupled oscillators with piecewise continuous arguments (EPCA). The system is equivariant under dihedral group. This causes several types of symmetrical bifurcations. 5）On the basis of Jordan algebra theory the uniqueness and existence of the complementarity problem are discussed. Some statice properties and those relations of Lyapunov function over Jordan algebra are given. Complementarity functions with two parameters and associated properties are presented.Nonlinear complementarity problems are solved by smoothing Newton method on account of complementarity functions.

英文关键词： Jury criterion; Jordan algebra;symmetric group;dynamic system;invariant