仿射联络空间切丛之间的映射─—纪念导师吴光磊教授

本文依据Ｄｏｍｂｒｏｗｓｋｉ在仿射联络空间切丛上引进的近复结构，证明了：两个仿射联络空间之间的光滑映射的切映射保持近复结构不变的充分必要条件是该映射为全测地映射．     

非定常的热传导一对流问题的特征混合元法

１．引言 设 　 Ｒ２是足够光滑的有界区域，考虑非定常的热传导－对流方程的初边值问题： 问题Ｉ．求ｕ＝（ｕ１，ｕ２），ｐ，Ｔ满足：其中ｕ是流体的速度向量，ｐ为压力，Ｔ是温度，ｖ＞０是运动粘性系数，λ＞０是Ｇｒｏｓｈｏｆｆ数，ｊ＝（０，１）是二维向量，ｘ＝（ｘ１，ｘ２）． 非定常的热传导一对流方程是大气动力学中的一个重要的方程，这个方程组也称为强迫耗散的非线性系统方程组，其较Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程多了一个未知函数温度场，它与速度和压力之间存在着复杂的非线性关系．从热动力学可知，任何运动都会产生…     

黎曼流形在正交联络下的全脐点子流形

本文研究了正交联络下子流形基本方程以及在全脐点子流形中的应用.利用Cartan的方法将挠率张量分解成三个部分,计算得到正交联络下的三个基本方程,并考虑一个特殊的正交联络,证明了其黎曼曲率会有类似于Levi-Civita联络下的性质.利用基本方程得到常曲率空间中的全脐点子流形的性质,推广了Levi-Civita联络下的相应结果.     

随机动态线性经济系统的稳定性

本文讨论随机动态线性经济系统：Ｙｔ＝ＡＹ＿（ｔ－１）＋ｂ十μ＿ｔ在矩阵Ａ为一般情形下的稳定性问题。并给出该系统稳定的充要条件。     

带一类第一积分的非完整系统的Szebehely问题

研究非完整系统动力学的一类逆问题·给出非完整系统的运动方程及其显式，考虑一类仅受齐次非完整约束的力学系统的Ｓｚｅｂｅｈｅｌｙ问题，研究已知一类第一积分的一般非完整系统的情形·最后举例说明其应用·     

非定常的热传导-对流问题的非线性Galerkin混合元法(Ⅱ):向后一步的Euler全离散化格式

1．引言本文的工作主要是讨论非定常的热传导一对流问题的向后一步的Euler全离散化的非线性Galerkin混合元解的存在性及其误差估计．该工作是对山中的同一问题研究的第二部分．在第一部分［1］，我们已经讨论了此问题的半离散化的情形．由于所研究的目标都是非定常的热传导一对流问题，其背景是相同的，在此将不重复了，请参考［1］．本文的安排如下，52先回顾非定常的热传导一对流问题的混合元解的经典性质．53回顾半离散化的非线性Galerkin混合元解的性质，并导出后续讨论需要的一些关于时间导数的估计．54讨论向后一步的Euler全离散化…     

论约束的力学性质

研究约束的力学性质是一个非常重要而又实际的问题。本文在研究非定常的完整约束的力学性质的基础上，进一步研究了非完整约束的力学性质。内容复盖了Ｖａｃｃｏ动力学和Ｃｈｅｔａｅｖ动力学各类运动分方程中约束反力的性质。     

函数空间多体挠性结构系统动力学、稳定性与控制的研究

利用现代数学方法,在函数空间中研究了一类无穷维系统动力学、稳定性与控制问题.首先提出并建立了具有阻尼、陀螺部件和约束阻尼的多拓扑结构多挠体分布参数系统动力学控制模型;其次给出并论证了多体挠性结构特征、系统分析结果--可控可观性充要条件、稳定性理论和系统的渐近性质.研究的结果扩充和发展了本领域关于多挠体系统动力学与控制的理论成果,具有重要的工程意义.     

引力、电磁和自旋的一种经典统一场论——Ⅰ.基础

基于Einstein关于物理强几何化的原理,本文提出了一种引力和电磁的经典统一场论。这种统一场论的数学基础是约化丛P[M,SU(2)],以复酉群SU(2)的实表示作为结构群。SU(2)丛给出运动学背景,而M上的广义Einstein方程确定了宇宙的动力学内容;如果在给定的主纤维丛上定义了仿射联络,它们就完全确定了空-时结构。这样确定的空-时结构自动地蕴含了电磁和内蕴自旋的存在。在引力和电磁(其中自旋是重要因素)的交互作用下,发现存在某种类似“磁荷和磁流”的东西。     

周期激励浅拱1∶2内共振参数平面定常运动分布

本文研究了已具有静变形的受周期激励作用下浅拱在１：２内共振条件下的分岔特性，进而按系统的运动形式将整个参数平面分成不同的区域，得到了物理参数平面上浅拱的定常运动分布情况，结合数值分析方法详细分析了系统在各个区域内特别是Ｈｏｐｆ分岔区域内系统的动力学特性，指出系统模态相互作用的规律及其通向混沌的过程·     

Nonholonomic LR Systems as Generalized Chaplygin Systems with an Invariant Measure and Flows on Homogeneous Spaces

We consider a class of dynamical systems on a compact Lie group G with a left-invariant metric and right-invariant nonholonomic constraints (so-called LR systems) and show that, under a generic condition on the constraints, such systems can be regarded as generalized Chaplygin systems on the principle bundle G \to Q = G/H, H being a Lie subgroup. In contrast to generic Chaplygin systems, the reductions of our LR systems onto the homogeneous space Q always possess an invariant measure. We study the case G = SO(n), when LR systems are ultidimensional generalizations of the Veselova problem of a nonholonomic rigid body motion which admit a reduction to systems with an invariant measure on the (co)tangent bundle of Stiefel varieties V(k, n) as the corresponding homogeneous spaces. For k = 1 and a special choice of the left-invariant metric on SO(n), we prove that after a time substitution the reduced system becomes an integrable Hamiltonian system describing a geodesic flow on the unit sphere S^n-1. This provides a first example of a nonholonomic system with more than two degrees of freedom for which the celebrated Chaplygin reducibility theorem is applicable for any dimension. In this case we also explicitly reconstruct the motion on the group SO(n).     

Obstructions to toric integrable geodesic flows in dimension 3

The geodesic flow of a Riemannian metric on a compact manifold Q is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle T ^* Q\Q. If the geodesic flow is toric integrable, the cosphere bundle admits the structure of a contact toric manifold. By comparing the Betti numbers of contact toric manifolds and cosphere bundles, we are able to provide necessary conditions for the geodesic flow on a compact, connected 3-dimensional Riemannian manifold to be toric integrable.Mathematics Subject Classifications (2000): primary 53D25; secondary 53D10     

New dynamical invariants on hyperbolic manifolds

The rotation measure is an asymptotic dynamical invariant assigned to a typical point of a flow in a fiber bundle over a hyperbolic manifold. The total mass of the rotation measure is the average speed of the orbit and its “direction” is the ergodic invariant probability measure of the hyperbolic geodesic flow which best captures the asymptotic dynamics of the given point. The rotation measure exists almost everywhere and is constant for an ergodic measure of the given flow and so it may be viewed as assigning an ergodic measure of the geodesic flow to one of the given flow. It generalizes the usual notion of homology rotation vector by encoding homotopy information.     

Are Hamiltonian flows geodesic flows?

When a Hamiltonian system has a ``Kinetic + Potential' structure, the resulting flow is locally a geodesic flow. But there may be singularities of the geodesic structure; so the local structure does not always imply that the flow is globally a geodesic flow. In order for a flow to be a geodesic flow, the underlying manifold must have the structure of a unit tangent bundle. We develop homological conditions for a manifold to have such a structure.

We apply these criteria to several classical examples: a particle in a potential well, the double spherical pendulum, the Kovalevskaya top, and the -body problem. We show that the flow of the reduced planar -body problem and the reduced spatial 3-body are never geodesic flows except when the angular momentum is zero and the energy is positive.

     

Characterization of Geodesic Flows on T 2 with and without Positive Topological Entropy

In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus T ² for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic behavior on an invariant set in the phase space of positive Hausdorff-dimension (horseshoe). We show that in the case of zero topological entropy the flow has properties similar to integrable systems. In particular, there exists a non-trivial continuous constant of motion which measures the direction of geodesics lifted onto the universal covering \mathbbR²{\mathbb{R}^{2}} . Furthermore, those geodesics travel in strips bounded by Euclidean lines. Moreover, we derive necessary and sufficient conditions for vanishing topological entropy involving intersection properties of single geodesics on T ².     

Negatively Invariant Sets and Entire Trajectories of Set-Valued Dynamical Systems

Strongly negatively invariant compact sets of set-valued autonomous and nonautonomous dynamical systems on a complete metric space, the latter formulated in terms of processes, are shown to contain a weakly positively invariant family and hence entire solutions. For completeness the strongly positively invariant case is also considered, where the obtained invariant family is strongly invariant. Both discrete and continuous time systems are treated. In the nonautonomous case, the various types of invariant families are in fact composed of subsets of the state space that are mapped onto each other by the set-valued process. A simple example shows the usefulness of the result for showing the occurrence of a bifurcation in a set-valued dynamical system.     

Relation between Different Types of Global Attractors of Set-Valued Nonautonomous Dynamical Systems

The article is devoted to the study of the relation between forward and pullback attractors of set-valued nonautonomous dynamical systems (cocycles). Here it is proved that every compact global forward attractor is also a pullback attractor of the set-valued nonautonomous dynamical system. The inverse statement, generally speaking, is not true, but we prove that every global pullback attractor of an α-condensing set-valued cocycle is always a local forward attractor. The obtained general results are applied while studying periodic and homogeneous systems. We give also a new criterion of the absolute asymptotic stability of nonstationary discrete linear inclusions. Dedicated to our friend Professor Enrico Primo Tomasini on the occasion of his 55th birthdayMathematics Subject Classifications (2000) Primary: 34C35, 34D20, 34D40, 34D45, 58F10,58F12, 58F39; secondary: 35B35, 35B40.     

Inverse problem of the calculus of variations on Lie groups

This article studies the inverse problem of the calculus of variations for the special case of the geodesic flow associated to the canonical symmetric bi-invariant connection of a Lie group. Necessary background on the differential geometric structure of the tangent bundle of a manifold as well as the Fröhlicher-Nijenhuis theory of derivations is introduced briefly. The first obstructions to the inverse problem are considered in general and then as they appear in the special case of the Lie group connection. Thereafter, higher order obstructions are studied in a way that is impossible in general. As a result a new algebraic condition on the variational multiplier is derived, that involves the Nijenhuis torsion of the Jacobi endomorphism. The Euclidean group of the plane is considered as a working example of the theory and it is shown that the geodesic system is variational by applying the Cartan-Kähler theorem. The same system is then reconsidered locally and a closed form solution for the variational multiplier is obtained. Finally some more examples are considered that point up the strengths and weaknesses of the theory.     

Attractors for discrete periodic dynamical systems

A mathematical framework is introduced to study attractors of discrete, nonautonomous dynamical systems which depend periodically on time. A structure theorem for such attractors is established which says that the attractor of a time-periodic dynamical system is the union of attractors of appropriate autonomous maps. If the nonautonomous system is a perturbation of an autonomous map, properties that the nonautonomous attractor inherits from the autonomous attractor are discussed. Examples from population biology are presented.