一类空间连续系统稳定性与分岔的数值研究

空间连续系统的非线性动力学研究，由于其工程背景与复杂性，近年来越来越受到重视。局部狭窄圆管内的流体流动即为空间连续系统。本文采用有限差分方法，将由偏微分方程组描述的空间连续系统约化为由常微分方程组描述的高维离散动力系统。求得了动力系统的平衡解及判断其稳定性的最大Ｌｙａｐｕｎｏｖ指数，求得了动力系统的前三个Ｌｙａｐｕｎｏｖ指数，以此作为系统是否出现分岔的判别条件。     

不依赖数据库的最优动力系统建模理论及其应用

在Wu提出的流动数据库分析及建立低维动力系统的优化理论基础上,进一步提出不依赖于数据库、直接由偏微分方程构造最优低维动力系统的方法。以二维热对流问题为例求得五维最优低维动力系统,研究其动力学特性,并与Lorenz模型进行比较。结果表明：该方法无需依赖数据库即可提取真实反映复杂流动动力学特征的最优基,建立在最优基之上的动力系统更充分地描述了问题的复杂动力学行为,并揭示了一些新的动力学特征。     

非线性时滞动力系统的研究进展

具有时滞的动力系统广泛存在于各工程领域．本文从动力学角度对时滞动力系统的研究进展作一综述,内容包括时滞动力系统的特点、研究方法、动力学热点问题的研究进展等．由于时滞动力系统的演化趋势不仅依赖于系统的当前状态,还依赖于系统过去某一时刻或若干时刻的状态,其运动方程要用泛国微分方程来描述,解空间是无穷维的．即使系统中的时滞非常小,在许多情况下也不能忽略不计．对于非线性时滞常微分方程,目前的研究思路基本上与常微分方程系统理论相平行．主要研究方法可分为时域法和频域法,前者包括Ｔａｙｌｏｒ级数法,中心流形法,Ｐｏｉｎｃａｒｅ映射法等,后者包括Ｎｙｑｕｉｓｔ法等．目前对这类系统的动力学研究主要集中在稳定性、Ｈｏｐｆ分岔、混沌等方面．研究表明：时滞动力系统具有非常丰富和复杂的动力学行为,如单变量的一维非线性时滞动力系统可发生混沌现象,与用常微分方程描述的系统有本质性差别．另一方面,人们可巧妙地利用时滞来控制动力系统的行为,如时滞反馈控制是控制混饨的主要方法之一．最后,本文展望了存在的一些问题以及近期值得关注的研究．     

时间尺度上Herglotz变分原理及其Noether定理

Herglotz变分原理的作用量是由微分方程定义的,不仅可以描述所有经典变分原理能够描述的动力学过程,还可以对经典变分原理不能适用的非保守系统或耗散系统进行变分描述．时间尺度上微积分理论提供了一种可同时研究离散系统和连续系统的有效方法．本文结合Herglotz变分原理和时间尺度微积分理论来研究时间尺度上的Herglotz变分原理及其Noether定理．首先,给出时间尺度上Lagrange系统的Herglotz变分原理．其次,根据Herglotz变分原理和Dubois-Reymond引理,推导出时间尺度上Lagrange系统的Herglotz变分问题的运动微分方程．再次,基于时间尺度上Hamilton-Herglotz作用量在群的无限小变换下的不变性,给出Noether对称性的定义并导出其Noether等式．最后,建立了时间尺度上Lagrange系统的Herglotz变分问题的Noether定理,给出了连续和离散两种情况下基于Herglotz变分问题的Noether守恒量．文末举例说明结果的应用．     

外部激励Stuart-Landau模型的数值研究

将非线性常微分方程组周期解的求解看作一个边值问题 ,运用Newton迭代构造求解这组方程的数值方法。利用上述方法求得了激励Stuart Landau方程的周期解 ,研究了圆柱振动对圆柱后Karman涡街的抑制现象 ,和振动的频率锁定现象 ,证明了激励Stuart Landau方程描写钝体尾迹动力系统的有效性     

悬索桥非共振情况下的振动

本文在连续膜假设条件下,建立了新的能描述吊索变形和松弛影响的悬索桥横向振动非线性偏微分方程组.该方程组的不等式定解条件反映出吊索松弛与否情况.在假设吊索不松弛的条件下,对上述方程组进行简化后得到一组只含双侧约束的非线性偏微分方程组.此方程组的定解条件是用等式表示的双侧约束条件.通过Galerkin方法把双侧约束的偏微分方程组离散为时域上非线性常微分方程组.用多尺度法求得了非线性常微分方程组非共振情况下的一次近似解析解.通过比较数值解和解析解发现,解析解有良好的精度.同时数值和解析的结果指出,在非共振情况下悬索桥的加劲梁和主缆的振幅都是有限值并正比于激励的幅值.     

含高阶余项的非线性动力系统数值计算法

建立了一种求解非线性动力系统高精度数值计算的新方法,重构了等价的非线性动力系统方程,该方程考虑了非线性函数的任意高阶项,并给出了该方程的Duhamel积分表达式,在时间步长内用Newton-Raphson法进行数值迭代求解,该方法能连续满足微分方程而不只是在离散的步长端点满足方程,从而打破了传统的Euler型有限差分法。计算实例表明,该方法计算精度高于传统的Runge-Kutta,Newmark-β和Wilson-θ等方法。     

二次非线性粘弹性圆板的2/1⊕3/1超谐解

计及材料的非线性弹性和粘性性质 ,研究了圆板在简谐载荷作用下的 21 31超谐解 ,导出了相应的非线性动力方程。提出一类强非线性动力系统的叠加 叠代谐波平衡法。将描述动力系统的二阶常微分方程 ,化为基本解为未知函数的基本微分方程 ;及分岔解为未知函数的增量微分方程。通过叠加 迭代谐波平衡法得出了圆板的 21 31超谐解。对叠加迭代谐波平衡法和数值积分法进行了比较 ,两者结果吻合很好。并且讨论了 21 31超谐解的渐近稳定性     

小参数法在悬垂索共振中的应用

当面外横向振动和面内横向振动频率的比接近1:2时,悬索会出现面内和面外耦合共振现象。为了研究悬索这种复杂独特的非线性特性,利用多尺度法对谐波激励的悬索动力学方程进行求解,得到对应于不同阶小量的偏微分方程组,其中二阶小量偏微分方程中的久期项不为0;采用提出的小参数法可以得到由久期项引起的悬索振动形态,解决久期项频率与系统频率相同但不能直接求解的问题;为了证明小参数法的准确性,采用Galerkin方法离散悬垂索的运动方程,然后利用多尺度法求解离散的运动方程,得到采用基函数描述的由久期项引起的连续系统的振动形态,与小参数法结论一致。     

一维连续体释放和回收过程的构形计算

一维连续体的释放和回收过程由时变的动力学方程描述。将一维连续体离散为有限单元,建立其时变自由度的高维离散动力学模型。通过重新划分单元,重置系统质量、阻尼和刚度矩阵,以及位移和荷载向量,并基于改进的有限差分法,提出了一维连续体释放和回收过程的一种构形计算方法。以柔性索的面内运动为例,计算了其释放和回收过程的动力学构形,实...     

Numerical approximation of a class of discontinuous systems of fractional order

In this paper we investigate the possibility to formulate an implicit multistep numerical method for fractional differential equations, as a discrete dynamical system to model a class of discontinuous dynamical systems of fractional order. For this purpose, the problem is continuously transformed into a set-valued problem, to which the approximate selection theorem for a class of differential inclusions applies. Next, following the way presented in the book of Stewart and Humphries (Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996) for the case of continuous differential equations, we prove that a variant of Adams?CBashforth?CMoulton method for fractional differential equations can be considered as defining a discrete dynamical system, approximating the underlying discontinuous fractional system. For this purpose, the existence and uniqueness of solutions are investigated. One example is presented.     

Pullback Attractors in Dissipative Nonautonomous Differential Equations Under Discretization

The effects of discretization on the nonautonomous pullback attractors of skew-product flows generated by a class of dissipative differential equations, are investigated, It is assumed that the vector, field of the differential equations varies in time due to the input of an autonomous dynamical system acting on a compact metric space. In particular, it is shown that the corresponding discrete time skew-product system generated by a one-step numerical scheme with variable timesteps also has a pullback attractor, the component subsets of which converge upper semicontinuously to their counterparts of the pullback attractor of the original continuous time system.     

Reduced order state‐space models from the pulse responses of a linearized CFD scheme

This paper describes a method for obtaining a time continuous reduced order model (ROM) from a system of time continuous linear differential equations. These equations are first put into a time discrete form using a finite difference approximation. The unit sample responses of the discrete system are calculated for each system input and these provide the Markov parameters of the system. An eigenvalue realization algorithm (ERA) is used to construct a discrete ROM. This ROM is then used to obtain a continuous ROM of the original continuous system. The focus of this paper is on the application of this method to the calculation of unsteady flows using the linearized Euler equations on moving meshes for aerofoils undergoing heave or linearized pitch motions. Applying a standard cell‐centre spatial discretization and taking account of mesh movement a continuous system of differential equations is obtained which are continuous in time. These are put into discrete time form using an implicit finite difference approximation. Results are presented demonstrating the efficiency of the system reduction method for this system. Copyright © 2003 John Wiley & Sons, Ltd.     

Reductibilite des systemes hereditaires

A necessary and sufficient condition for an hereditary system in Rⁿ to be reducible to a dynamical one (i.e. governed by ordinary differential equations) is given, which also enables one to show that irreducible systems exist. It is then shown that any hereditary system is nevertheless equivalent to a system governed by non-hereditary equations, if an internal field is introduced, which is governed by partial differential equations. Two examples, one from rheology, the other from electromagnetism, illustrate this last point.     

时间尺度上Lagrange系统的Hojman守恒量

利用对称性和守恒律, 可以简化动力学问题甚至求解力学系统的精确解, 更好地理解其动力学行为. 时间尺度分析将连续和离散动力学模型统一并拓展到时间尺度框架, 既避免了重复研究又可揭示两者之区别和联系. 因此, 通过对称性来探寻在时间尺度的框架下新的守恒定律很有必要. 本文首先建立了时间尺度上Lagrange方程, 利用时间尺度微积分性质导出了时间尺度上Lagrange系统的两个重要关系式; 其次, 依据微分方程在单参数Lie变换群下的不变性, 建立了时间尺度上Lie对称性的定义和确定方程; 最后, 建立了时间尺度上Lie对称性定理并利用上述关系式给出了证明, 得到了时间尺度上Lagrange系统的新守恒量. 当时间尺度取为实数集时, 该守恒量退化为著名的Hojman守恒量. 文末考察了一个两自由度时间尺度Lagrange系统, 在3种不同时间尺度情形下得到了该系统的Hojman守恒量, 数值计算结果验证了定理的正确性.      

Hamel框架下几何精确梁的离散动量守恒律

Hamel场变分积分子是一种研究场论的数值方法, 可以通过使用活动标架规避几何非线性带来的计算复杂度, 同时数值上具有良好的长时间数值表现和保能动量性质. 本文在一维场论框架下, 以几何精确梁为例, 从理论上探究Hamel场变分积分子的保动量性质. 具体内容包括: 利用活动标架法对几何精确梁建立动力学模型, 通过变分原理得到其动力学方程, 利用其动力学方程及Noether定理得到系统动量守恒律; 将几何精确梁模型离散化, 通过变分原理得到其Hamel场变分积分子, 利用Hamel场变分积分子和离散Noether定理得到离散动量守恒律, 并给出离散动量的一阶近似表达式; Hamel场变分积分子可在计算中利用系统对称性消除系统运动带来的非线性问题, 但此框架中离散对流速度、离散对流 应变及位形均不共点, 而这种错位导致离散动量中出现级数项, 本文对几何精确梁的离散动量与连续形式的关系及其应 用进行了讨论, 并通过算例验证了结论. 上述证明方法也同样适用一般经典场论场景下的Hamel场变分积分子. Hamel场变分积分子的动量守恒为进一步研究其保结构性质提供了参考依据.      

A reduced‐order approach for optimal control of fluids using proper orthogonal decomposition

In this article, a reduced‐order modeling approach, suitable for active control of fluid dynamical systems, based on proper orthogonal decomposition (POD) is presented. The rationale behind the reduced‐order modeling is that numerical simulation of Navier–Stokes equations is still too costly for the purpose of optimization and control of unsteady flows. The possibility of obtaining reduced‐order models that reduce the computational complexity associated with the Navier–Stokes equations is examined while capturing the essential dynamics by using the POD. The POD allows the extraction of a reduced set of basis functions, perhaps just a few, from a computational or experimental database through an eigenvalue analysis. The solution is then obtained as a linear combination of this reduced set of basis functions by means of Galerkin projection. This makes it attractive for optimal control and estimation of systems governed by partial differential equations (PDEs). It is used here in active control of fluid flows governed by the Navier–Stokes equations. In particular, flow over a backward‐facing step is considered. Reduced‐order models/low‐dimensional dynamical models for this system are obtained using POD basis functions (global) from the finite element discretizations of the Navier–Stokes equations. Their effectiveness in flow control applications is shown on a recirculation control problem using blowing on the channel boundary. Implementational issues are discussed and numerical experiments are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Identification of nonlinear dynamical system equations using dynamic mode decomposition under invariant quantity constraints

In this paper, an algorithm for identifying equations representing a continuous nonlinear dynamical system from a noise-free state and time-derivative state measurements is proposed. It is based on a variant of the extended dynamic mode decomposition. A particular attention is paid to guarantee that the physical invariant quantities stay constant along the integral curves. The numerical methodology is validated on a two-dimensional Lotka–Volterra system. For this case, the differential equations are perfectly retrieved from data measurements. Perspectives of extension to more complex systems are discussed.     

Path integral synthesis of lyapunov functionals for partial differential equations

A method is given for constructing Lyapunov Functionals for dynamical systems governed by partial differential equations. The functionals are obtained as path integrals in a suitably chosen state space of a generalized gradient operator, and the method may be viewed as an extension to infinite dimensional systems of the variable gradient technique. Some of the fundamental concepts underlying the formalism are reviewed, and examples of applications to some linear, non-linear and hybrid systems are given.     

扁球面网壳的混沌运动研究

在圆形三向网架非线性动力学基本方程的基础上,用拟壳法给出了圆底扁球面三向网壳的非线性动力学基本方程．在固定边界条件下,引入了异于等厚度壳的无量纲量,对基本方程和边界条件进行无量纲化,通过Galerkin作用得到了一个含二次、三次的非线性动力学方程．为求Melnikov函数,对一类非线性动力系统的自由振动方程进行了求解,得到了此类问题的准确解．在无激励情况下,讨论了稳定性问题．在外激励情况下,通过求Melnikov函数,给出了可能发生混沌运动的条件．通过数字仿真绘出了平面相图,证实了混沌运动的存在．