非完整非保守系统的积分因子和守恒律

通过寻找积分因子来建立非完整非保守系统的守恒律，讨论了存在守恒定律的必要条件，并举例说明其应用。     

关于拉格朗日方程循环积分的讨论

作为《理论力学》课程的教学内容之一,教材中一般都给出了拉格朗日方程存在循环积分的充分条件.若拉格朗日函数中不显含某些广义坐标(称为循环坐标),则对于保守系统,对应于循环坐标的广义动量守恒.给出了非保守动力学系统的拉格朗日方程存在循环积分的条件,并用教材中常用的算例讨论了广义动量守恒的物理含义.结果表明用拉格朗日方程的循环积分比用动量定理和动量矩定理更容易找到系统的守恒量.     

完整力学系统的共形不变性与守恒量

将Birkhoff方程的共形不变性和共形因子的概念拓展到完整力学系统,研究一般完整力学系统在无限小变换下的共形不变性与守恒量.给出了一般完整力学系统的共形不变性的定义和确定方程;研究了系统的Noether对称性与共形不变性之间的关系,研究表明,当Noether对称变换的生成元和非势广义力满足一定条件时,变换也是共形不变的,给出了相应的共形因子表达式,得到了一般完整力学系统的共形不变性直接导致的Noether守恒量;研究了系统的Lie对称性与共形不变性之间的关系,给出了与Lie对称性相应的无限小变换共形不变的充分必要条件,得到了一般完整力学系统的共形不变性直接导致的Lutzky守恒量.文中还举例说明结果的应用.     

分数阶拟周期Mathieu方程的动力学分析

分数阶微积分有着诸多优异的特点, 目前在动力学领域主要用来提高非线性系统振动特性研究的准确性. 本文在拟周期Mathieu方程的基础上, 引入分数阶微积分理论, 研究了分数阶微分项参数对方程稳定性的影响. 首先, 采用摄动法得到方程稳定区和非稳定区分界线(即过渡曲线)近似表达式, 利用数值方法验证了解析结果的准确性, 图像显示两者吻合较好. 随后, 通过归纳总结不同情况下的过渡曲线近似表达式, 发现在系统中分数阶微分项以等效线性刚度和等效线性阻尼的方式存在. 根据这一特点, 得到了系统等效线性阻尼和等效线性刚度的一般形式, 并且定义了非稳定区域厚度. 最后, 通过数值仿真直观地分析了分数阶微分项参数对方程稳定区域大小和过渡曲线位置的影响. 结果发现, 分数阶微分项不仅具有阻尼特性还具有刚度特性, 并且以等效线性刚度和等效线性阻尼的方式影响着方程稳定区域大小和过渡曲线位置. 合理选择分数阶微分项参数可以使其呈现不同程度的刚度特性或阻尼特性, 方程稳定区域的大小和过渡曲线的位置也因此产生了不同程度的变化.      

基于分数阶模型的非完整系统的Mei对称性及其守恒量

为了进一步揭示非完整系统的对称性和守恒量之间的内在关系,提出并研究基于分数阶模型的非完整系统的Mei对称性及其守恒量．首先,根据分数阶d’Alembert-Lagrange原理建立基于分数阶模型的非完整系统的动力学方程．其次,根据动力学方程中的动力学函数经无限小变换后仍满足原方程的不变性,建立分数阶模型下非完整系统的Mei对称性定理,给出Mei守恒量．再次,讨论了几个特例：分数阶Hamilton系统、经典非完整系统和受非完整约束的分数阶Lagrange系统的Mei对称性定理．文末举例说明结果的应用．     

非完整系统正则形式的ЧАПЛЫТИН方程的一种降阶方法

文中根据能量积分进一步研究了非完整系统正则形式的ЧАПЛЫГИН方程的降阶问题,得到了处理这类系统的一般积分方法.给出的两个例子表明,该方法比文[3,4]更具优越性.     

受匀速转动约束的变质量非完整力学系统的Appell方程

本文建立受匀速转动约束的变质量非完整力学系统l...     

一种基于微分求积的空间分数阶扩散方程求解方法

通过微分求积建立求解变系数空间分数阶扩散方程的一种有效直接数值方法。基于Reciprocal Multiquadric和Thin-Plate Spline径向基函数推导两种逼近分数阶导数的微分求积公式,将所考虑的模型问题转化成易求解的常微分方程组,并采用Crank-Nicolson格式进行离散。给出5个数值算例,计算结果表明,只要径向基函数的形状参数选择恰当,本文方法在精度和效率上均优于一些现有算法。     

任意阶非完整约束系统的Appell型方程

本文从万有D'Alembert原理出发,得到任意阶非完整约束系统的Appell型方程,在此基础上可给出本结果的推论.     

任意阶非完整约束系统的Appell型方程

本文从万有D'Alembert原理出发,得到任意阶非完整约束系统的Appell型方程,在此基础上可给出本结果的推论.     

Dynamical Stability of Viscoelastic Column with Fractional Derivative Constitutive Relation

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Spectral Properties of Dynamical Systems, Model Reduction and Decompositions

In this paper we discuss two issues related to model reduction of deterministic or stochastic processes. The first is the relationship of the spectral properties of the dynamics on the attractor of the original, high-dimensional dynamical system with the properties and possibilities for model reduction. We review some elements of the spectral theory of dynamical systems. We apply this theory to obtain a decomposition of the process that utilizes spectral properties of the linear Koopman operator associated with the asymptotic dynamics on the attractor. This allows us to extract the almost periodic part of the evolving process. The remainder of the process has continuous spectrum. The second topic we discuss is that of model validation, where the original, possibly high-dimensional dynamics and the dynamics of the reduced model – that can be deterministic or stochastic – are compared in some norm. Using the “statistical Takens theorem” proven in (Mezić, I. and Banaszuk, A. Physica D, 2004) we argue that comparison of average energy contained in the finite-dimensional projection is one in the hierarchy of functionals of the field that need to be checked in order to assess the accuracy of the projection.     

Invariant Manifolds for Random Dynamical Systems with Slow and Fast Variables

We consider random dynamical systems with slow and fast variables driven by two independent metric dynamical systems modeling stochastic noise. We establish the existence of a random inertial manifold eliminating the fast variables. If the scaling parameter tends to zero, the inertial manifold tends to another manifold which is called the slow manifold. We achieve our results by means of a fixed point technique based on a random graph transform. To apply this technique we need an asymptotic gap condition.      

Motion Control by ZV Shaper Synthesis Extended for Fractional Systems and Its Application to CRONE Control

Shaping command input or preshaping is used for reducing system oscillation in motion control. Desired systems inputs are altered so that the system finishes the requested move without residual oscillation. This technique, developed by N.C. Singer and W.P. Seering, is used for example in the aerospace field, in particular in flexible structure control. This paper presents the study of ZV shaper for explicit fractional derivative systems (generalized derivative systems). A robustness study of ZV shaper is then presented and applied to improve second generation CRONE control response time. Results from simulation and from a DC motor bench are also given.     

Applications of Nonlinear Dynamical Systems Theory in Developmental Psychology: Motor and Cognitive Development

Applications of nonlinear dynamical systems theory to psychology have led to recent advances in understanding neuromotor development and advances in theories of cognitive development. This article reviews published findings associated with a specific coherent and influential application from which a theory of adaptive, self-organized cognition has been derived and related to a theory of developmental dynamics of the neuromotor system. The review focuses on implications of the two theories for quantifying developmental phenomena, and suggests a method for quantifying the cognitive theory.     

刚柔耦合约束多体系统的动力分析

本文提出了描述柔性多体系统的牵连坐标系统。该系统由惯性参考系,牵连坐标系,物体坐标系及单元坐标系组成,实现了对刚体平动,刚体转动及弹性运动的连续分解,最大限度地消除了由于刚体大角度转动导致的非线性特性。以有限元法为基础,应用拉格朗日方程建立了在该坐标下的刚柔耦合约束多体系统的动力学控制方程。该方程具有耦合程度小、易于推导、编程及求解等优点,为大规模约束多体系统的动力分析提供了新的途径。本文还讨论了平面铰链约束的约束形式及约束方程,最后给出了一个典型多体系统的数值算例。     

The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors

We consider stochastic differential equations in d-dimensional Euclidean space driven by an m-dimensional Wiener process, determined by the drift vector field f₀ and the diffusion vector fields f₁,...,f_m, and investigate the existence of global random attractors for the associated flows . For this purpose is decomposed into a stationary diffeomorphism given by the stochastic differential equation on the space of smooth flows on R^d driven by m independent stationary Ornstein Uhlenbeck processes z¹,...,z^m and the vector fields f₁,...,f_m, and a flow generated by the nonautonomous ordinary differential equation given by the vector field (_t/x)^–1[f₀(_t)+ _i=1 ¹ f_i(_t)z _t ⁱ ]. In this setting, attractors of are canonically related with attractors of . For , the problem of existence of attractors is then considered as a perturbation problem. Conditions on the vector fields are derived under which a Lyapunov function for the deterministic differential equation determined by the vector field f₀ is still a Lyapunov function for , yielding an attractor this way. The criterion is finally tested in various prominent examples.