Geometrical Aspects in Hydrodynamics and Integrable Systems

The motion of fluid particles of an inviscid incompressible fluid on a bounded domain is formulated from a Lagrangian point of view. This is accomplished by observing that Euler's equation of motion is a geodesic equation on a group of volume-preserving diffeomorphisms with the metric defined by the kinetic energy. This formulation is based on Riemannian geometry and Lie group theory, first developed by Arnold (1966). Behaviors of the geodesics are characterized by Riemannian (sectional) curvatures, which are shown to be mostly negative (with some exceptions). This property is related to the mixing and ergodicity of the fluid motions. Free rotation of a rigid body fixed at a point gives a simplest example of the dynamical systems which are integrable and represented with such formulation. The same method is applied to the other integrable systems such as the vortex-filament equation or the KdV equation. In contrast to the hydrodynamic system, sectional curvatures are found to be mostly positive (with exceptions). Thus it is found that integrable systems are more stable in the behavior of geodesics than the hydrodynamic system governed by the Euler's equation of motion. Received 16 January 1997 and accepted 30 May 1997     

大型射电望远镜悬挂馈源舱-Stewart 平台系统耦合分析

针对大射电望远镜悬挂系统特殊的结构形式,采用分离舱索系统和Stewart平台,应用有限元和牛顿-欧拉方程混合方法,提出Stewart平台与悬索馈源舱系统的混合动力学耦合方程,在此基础上,应用最优控制理论,通过Ham ilton-Jacobi-Isaacs不等式,探讨了耦合方程的解耦问题,得出在一定条件下具有鲁棒性的Stewart平台所允许的最大质量。数值仿真验证了方法的有效性和合理性。     

Etude approchee d'un systeme a hysteresis

On the basis of a geometrical interpretation of the method of first harmonic of Haag—Krylov—Bogolioubov—Mitropolski, a new dynamical study of an oscillator with hysteresis is made. This consists in replacing the equation of motion by another approximate equation which conserves the nonlinearity of the original equation. This method is valid in the case of sinusoidal excitation, but it can be extended to the case of random excitation. In particular, in the case when this excitation is white noise the Fokker—Planck equation for the system can be written down with the help of certain assumptions which are physically realisable. Hence, all useful information concerning the response can be derived.     

基于黎曼几何的非线性振子动力学分析递推解析算法

基于黎曼几何和变分原理，推导了黎曼流形上非线性耗散动力系统的二阶微分动力学方程，并运用流形收缩的概念将动力学方程离散化，进而建立了相应的递推求解格式。选取3个自治非线性阻尼振子系统，分别采用递推解析算法和龙格库塔法求解微分动力学方程，并比较分析了不同的时间步长下两种算法的计算耗时。结果表明，与龙格库塔法相比，基于黎曼几何的递推算法不仅能得到每一时步的解析表达式，而且计算耗时短，计算效率高。基于黎曼流形的动力学方程递推算法为非线性动力学系统的解析求解提供了新思路。     

Connections and geodesic characteristic of equations of motion for constrained mechanical systems

I.IntroductionSinceEinsteinestablishedgeneralrelativityatthebiginningofthiscentury,differentialgeometry,especiallythemodernditTerentialgeometry,hasbeenextellsivelyappliedtomanyfieldsofphysics.Thestudyofregularholonomicmechanicalsystemsinthemodernsettingofdifferentialgeometryhasahistoryofmorethanthirtyyears.Andtheresearchtendstoperfectgraduallyt'~'l.Sinceearlyin1980'sthegeometrizationaboutconstrainedmechanicalsystemsandsingularmechanicalsystemshasbeenattachedimportanceextensivelyandsomeresult…     

Lie transformation method for dynamical systems having chaotic behavior

The paper persents recent developments in a singular perturbation method, known as the Lie transformation method for the analysis of nonlinear dynamical systems having chaotic behavior. A general approximate solution for a system of first-order differential equations having algebraic nonlinearities is introduced. Past applications to simple dynamical nonlinear models have shown that this method yields highly accurate solutions of the systems. In the present paper the capability of this method is extended to the analysis of dynamical systems having chaotic behavior: indeed, the presence of small divisors in the general expression of the solution suggests a modification of the method that is necessary in order to analyze nonlinear systems having chaotic behavior (indeed, even non-simple-harmonic behavior). For the case of Hamiltonian systems this is consistent with the KAM (Kolmogorov-Arnold-Moser) theory, which gives the limits of integrability for such systems; in contrast to the KAM theory, the present formulation is not limited to conservative systems. Applications to a classic aeroelastic problem (panel flutter) are also included.     

A new approach to studying the dynamics of a thin curved vortex

The paper studies the dynamics of a thin curved vortex in a potential flow of an ideal incompressible fluid. The flow is specified by a number of geometrical restrictions and does not satisfy the Biot–Savart law. The form of the derived equation of the vortex dynamics coincides with the form of the wellknown equation of local induction for selfinduced vortex motion. The parameters of the new equation are simultaneously flow parameters, and in this sense, they do not show uncertainty typical of classical equations. The coefficient of the new equation can take any specified values (not necessarily much greater than unity, as required according to the concept of local induction) and generally is a function of a natural filament parameter.     

A structured population model and a related functional differential equation: Global attractors and uniform persistence

A structured population model of a single population having two distinct life stages is considered. The model equations, consisting of a hyperbolic partial differential equation coupled to an ordinary differential equation, can be reduced to a single, scalar functional differential equation. This allows us to use the well-developed dynamical systems theory for functional differential equations in order to study the dynamical system generated by the more complicated coupled system. A precise relation is established between the dynamical systems generated by each system of equations and a correspondence between their respective global attractors is made. The two systems are topologically equivalent on their respective attractors. These relationships are used to determine sharp sufficient conditions for the uniform persistence of the population.     

Stochastic averaging of n-dimensional non-linear dynamical systems subject to non-Gaussian wide-band random excitations

The averaged generalized Fokker-Planck-Kolmogorov (GFPK) equation for response of n-dimensional (n-d) non-linear dynamical systems to non-Gaussian wide-band stationary random excitation is derived from the standard form of equation of motion. The explicit expressions for coefficients of the fourth-order approximation of the averaged GFPK equation are given in series form. Conditions for convergences of these series are pointed out. The averaged GFPK equation is then reduced to that for 1-d dynamical systems derived by Stratonovich and compared with the closed form of GFPK equation for n-d dynamical systems subject to Poisson white noise derived by Di Paola and Falsone. Finally, this averaged GFPK equation is further reduced to that for quasi linear system subject to non-Gaussian wide-band stationary random excitation. Stationary probability density for quasi linear system subject to filtered Poisson white noise is obtained. Theoretical results for an example are confirmed by using Monte-Carlo simulation for different parameter values.     

Random attractors

In this paper, we generalize the notion of an attractor for the stochastic dynamical system introduced in [7]. We prove that the stochastic attractor satisfies most of the properties satisfied by the usual attractor in the theory of deterministic dynamical systems. We also show that our results apply to the stochastic Navier-Stokes equation, the white noise-driven Burgers equation, and a nonlinear stochastic wave equation.     

Bifurcation from Discrete Rotating Waves

Discrete rotating waves are periodic solutions that have discrete spatiotemporal symmetries in addition to their purely spatial symmetries. We present a systematic approach to the study of local bifurcation from discrete rotating waves. The approach centers around the analysis of diffeomorphisms that are equivariant with respect to distinct group actions in the domain and the range. Our results are valid for dynamical systems with finite symmetry group, and more generally, for bifurcations from isolated discrete rotating waves in dynamical systems with compact symmetry group.     

迭代图胞映射方法的快速生成实现

在迭代图胞映射方法的框架下,基于摄动微分多项式的思想讨论了常微分方程的快速求解,将所得结果与迭代图胞映射方法有机结合,有效地解决了迭代图胞映射动力系统的快速生成问题,克服了微分方程动力系统生成迭代图胞映射系统过程中耗时较多、效率低下的不足,大大提高了计算效率。通过对典型非线性系统——杜芬方程的应用分析,证实了该方法的有...     

Approximate symmetries and conservation laws of the geodesic equations for the Schwarzschild metric

Approximate symmetries have been defined in the context of differential equations and systems of differential equations. They give approximately, conserved quantities for Lagrangian systems. In this paper, the exact and the approximate symmetries of the system of geodesic equations for the Schwarzschild metric, and in particular for the radial equation of motion, are studied. It is noted that there is an ambiguity in the formulation of approximate symmetries that needs to be clarified by consideration of the Lagrangian for the system of equations. The significance of approximate symmetries in this context is discussed.     

Dynamic Systems Analysis and Control Based on a Configuration Manifold Model

This paper presents a configuration manifold model for the analysis of dynamic systems and the development of control algorithms from both geometrical and topological points of view. The fundamental theory of surfaces and differential manifolds endowed with Riemannian metrics is overviewed. The concepts of configuration manifolds (C-manifolds) and their immersions and embeddings are then introduced and applied to dynamic systems modeling. An explicit form of the smooth embedding for a given dynamic system with its C-manifold is derived. In an open serial-chain robotic system, a topological equivalence, i.e. a homeomorphism, is found and shown to be useful for dynamic model reduction. With topology being viewed as the structure of geometry, we discover and prove that the kinematics of a dynamic system determines its topology so that the kinematics is virtually a structure of the system's dynamics. This key point of view is further extended to the development of an adaptive control strategy. A computer simulation study is finally performed to verify the proposed model and adaptive control scheme.     

三类随机系统广义概率密度演化方程的解析解

近年来逐步发展的概率密度演化方法理论为随机动力系统的分析与控制研究提供了新的途径.过去若干年来,已经发展了一系列数值方法如有限差分法、无网格法用于求解广义概率密度演化方程.但是,针对典型随机系统,关于这一方程解析解尚比较缺乏.本文以李群方法为工具,研究给出了Van der Pol振子、Riccati方程和Helmholtz振子3类典型随机非线性系统的广义概率密度演化方程解析解.这些结果,不仅可以作为检验求解广义概率密度演化方程的数值方法结果正确性的判别依据,也为概率密度演化理论的进一步深入研究提供了若干分析实例.      

IMPLEMENTATION OF VORTEX FILAMENT METHODS ON PARALLEL MACHINES WITH DISTRIBUTED ADAPTIVE DATA STRUCTURE

This paper addressed the implementation of vortex filament methods on parallel machines with distributed memory to simulate a three-dimensionally evolving jet. Vortical structure developments due to Kelvin–Helmholtz instability of the axially perturbed jet are also examined. The implementation is conducted in a single-programme multiple-data (SPMD) environment and the parallelism is focused on issues of data distribution, efficient support of parallel I/O and overlapping of communications with computations. In addition, since the number of segment markers in a filament is dynamically growing according to the requirement of numerical accuracy, a novel packet-oriented data structure is proposed not only to partition filament segment markers among distributed processors but also to support dynamical load balancing at run time. This work is the first to apply packet-oriented structures to implement a parallel vortex filament method. Experimental results indicate performance improvement from 1·5 to 2·6 times over static schemes on nCUBE2, DEC Alpha and IBM SP2 by incorporating the proposed scheme with packet-oriented structures. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 939–951, 1997.     

An Integrable Model of Nonstationary Rotationally Symmetrical Motion of Ideal Incompressible Liquid

We consider a partially invariant solution of the Eulerequations with respect to a six-parameter Lie group admitted by thissystem where the vertical component of velocity is a function of thevertical coordinate and time only while two other components andpressure do not depend on the polar angle in a cylindrical coordinatesystem. The analysis of the corresponding overdetermined system leads totheir special (but nontrivial) dependence of the polar radius. Afterthis, the nonlinear factor-system for invariants of the group is reducedto a system of ordinary differential equations by introduction ofLagrangian coordinates. As a result, we obtain a wide class of new exactsolutions which describes vortex motions of an ideal incompressibleliquid including motions with singularities.     

Optimal bounded control for minimizing the response of quasi-integrable Hamiltonian systems

A procedure for designing optimal bounded control to minimize the response of quasi-integrable Hamiltonian systems is proposed based on the stochastic averaging method for quasi-integrable Hamiltonian systems and the stochastic dynamical programming principle. The equations of motion of a controlled quasi-integrable Hamiltonian system are first reduced to a set of partially completed averaged Itô stochastic differential equations by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, the dynamical programming equation for the control problems of minimizing the response of the averaged system is formulated based on the dynamical programming principle. The optimal control law is derived from the dynamical programming equation and control constraints without solving the dynamical programming equation. The response of optimally controlled systems is predicted through solving the Fokker-Planck-Kolmogrov equation associated with fully completed averaged Itô equations. Finally, two examples are worked out in detail to illustrate the application and effectiveness of the proposed control strategy.     

Optimal Bounded Control of First-Passage Failure of Quasi-Integrable Hamiltonian Systems with Wide-Band Random Excitation

The optimal bounded control of quasi-integrable Hamiltonian systems with wide-band random excitation for minimizing their first-passage failure is investigated. First, a stochastic averaging method for multi-degrees-of-freedom (MDOF) strongly nonlinear quasi-integrable Hamiltonian systems with wide-band stationary random excitations using generalized harmonic functions is proposed. Then, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximizinig reliability and maximizing mean first-passage time are formulated based on the averaged Itô equations by applying the dynamical programming principle. The optimal control law is derived from the dynamical programming equations and control constraints. The relationship between the dynamical programming equations and the backward Kolmogorov equation for the conditional reliability function and the Pontryagin equation for the conditional mean first-passage time of optimally controlled system is discussed. Finally, the conditional reliability function, the conditional probability density and mean of first-passage time of an optimally controlled system are obtained by solving the backward Kolmogorov equation and Pontryagin equation. The application of the proposed procedure and effectiveness of control strategy are illustrated with an example.