Transport of phase space volume near isolated invariant sets

A measure preserving homeomorphismf determines a discrete dynamical system. Measurable sets in phase space are moved or transported byf. It is shown that the asymptotic rate of escape of phase space volume from neighborhoods which isolate the same invariant set is unique.     

Lunar landing trajectory design based on invariant manifold

The low-energy lunar landing trajectory design using the invariant manifolds of restricted three-body problem is studied.Considering angle between the ecliptic plane and lunar orbit plate the four-body problem of sun-earth-moon-spacecraft is divided into two three-body problems,the sun-earth-spacecraft in the ecliptic plane and the earth- moon-spacecraft in the lunar orbit plane.Using the orbit maneuver at the place where the two planes and the invariant manifolds intersect,a general method to design low energy lunar landing trajectory is given.It is found that this method can save the energy about 20% compared to the traditional Hohmann transfer trajectory,The mechanism that the method can save energy is investigated in the point of view of energy and the expression of the amount of energy saved is given.In addition,some rules of selecting parameters with respect to orbit design are provided.The method of energy analysis in the paper can be extended to energy analysis in deep space orbit design.     

Optimal control of quadratic functionals for affine nonlinear systems

In this paper we analyze the optimal control problem for a class of affine nonlinear systems under the assumption that the associated Lie algebra is nilpotent. The Lie brackets generated by the vector fields which define the nonlinear system represent a remarkable mathematical instrument for the control of affine systems. We determine the optimal control which corresponds to the nilpotent operator of the first order. In particular, we obtain the control that minimizes the energy of the given nonlinear system. Applications of this control to bilinear systems with first order nilpotent operator are considered.     

Bifurcation analysis for spherically symmetric systems using invariant theory

We study a degenerate steady state bifurcation problem with spherical symmetry. This singularity, with the five dimensional irreducible action ofO(3), has been studied by several authors for codimensions up to 2. We look at the case where the topological codimension is 3, theC -codimension is 5. We find a tertiary Hopf bifurcation and a heteroclinic orbit. Our analysis does not use any specific properties of the five dimensional representation and can in principle be used for higher representations as well. The computations are based on invariant theory and orbit space reduction.     

Stability and strict positive realness of convex polytopes of interval polynomials

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The invariant manifold of beam deformations

The subcentre invariant manifold of elasticity in a thin rod may be used to give a rigorous and appealing approach to deriving one-dimensional beam theories. Here I investigate the analytically simple case of the deformations of a perfectly uniform circular rod. Many, traditionally separate, conventional approximations are derived from within this one approach. Furthermore, I show that beam theories are convergent, at least for the circular rod, and obtain an accurate estimate of the limit of their validity. The approximate evolution equations derived by this invariant manifold approach are complete with appropriate initial conditions, forcing and, in at least one case, boundary conditions.     

Stability and strict positive realness of convex polytopes of interval polynomials

For an uncertain system described by convex combination of interval polynomials, its Hurwitz-stability can be guaranteed by certain subset composed of vertices and edges. Furthermore, the testing set does not increase when the order of given system increases. Foundation item: the National Natural Science Foundation of China (69925307) Biography: WANG Zhi-zhen (1974-)     

Necessary and sufficient stability conditions for invariant sets of nonlinear impulsive systems

Necessary and sufficient conditions for the uniform asymptotic stability of the invariant set of a nonlinear impulsive system are established __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 132–142, February 2008.     

晃荡液舱内水动冲击力的分段仿射模型

针对舱内晃荡液体与舱壁的相互作用,对舱内水动冲击力的等效力学模型进行了研究。基于混合系统理论,建立了强非线性液体晃荡的等效摆分段仿射模型,重点对矩形液舱的简化等效力学模型进行了分析。利用计算流体动力学软件Flow3D对矩形液舱内的强非线性液体晃荡进行了数值仿真。理论分析表明：分段仿射模型更符合刚性碰撞的假定,可以更有效地描述等效摆和舱壁碰撞时的速度跃变。仿真结果的对比表明：受到激励时,等效摆分段仿射模型所产生的力与Flow3D计算的结果比较接近,利用此模型可以恰当地描述强非线性液体晃荡。     

基于仿射算法的深部隧道衬砌力学状态区间分析

根据深部地层隧道结构物理、力学参数信息资料缺乏及各类不确定性模型的特征,揭示了区间模型于深部地层隧道工程不确定性分析的优越性。将限制条件少、适用范围广的链杆模式的矩阵位移理论与响应面思路结合,提出了衬砌结构力学状态参量解析式的搭建方法,并进一步推导出了基于仿射运算的响应面函数矩阵表达形式及其区间上、下界计算公式,构筑起基于矩阵表达的响应面函数区间仿射运算模型。将区间离散逐步逼近规则引入,建立了该模型的优化求解方法。利用该模型及方法分析了某衬砌墙脚轴力及拱顶弯矩分布区间,与蒙特卡洛法计算结果比较,显示本文模型及方法有效地克服了经典区间方法在复杂状态函数中的扩张和溢出这一致命缺陷,提高了准确性和计算效率,分析结果精确度满足工程使用要求。     

Constructing invariant tori for two weakly coupled van der Pol oscillators

An algorithm is developed for the construction of an invariant torus of a weakly coupled autonomous oscillator. The system is put into angular standard form. The determining equations are found by averaging and are solved for the approximate amplitudes of the torus. A perturbation series is then constructed about the approximate amplitudes with unknown coefficients as periodic functions of the angular variables. A sequence of solvable partial differential equations is developed for determining the coefficients. The algorithm is applied to a system of nonlinearly coupled van der Pol equations and the first order coefficients are generated in a straightforward manner. The approximation shows both good numerical accuracy and reproducibility of the periodicities of the van der Pol system. A comparitive analysis of integrating the van der Pol system with integrating the phase equations from the angular standard form on the approximate torus shows numerical errors of the order of the perturbation parameter =0.05 for integrations of up to 10,000 steps. Applying FFT to the numerical periodicities generated by integrating the van der Pol system near the tours reveals the same predominant frequencies found in the perturbation coefficients. Finally an expected rotation number is found by integrating the phase equations on the approximate torus.Contribution of the National Institute of Standards and Technology, a Federal agency.     

Finite-time stability with respect to a closed invariant set for a class of discontinuous systems

This paper discusses the problem of finite-time stability with respect to a closed, but not necessarily compact, invariant set for a class of nonlinear systems with discontinuous right-hand sides in the sense of the Filippov solutions. When the Lyapunov function is Lipschitz continuous and regular, the Lyapunov theorem on finite-time stability with respect to a closed invariant set is presented.     

The invariant manifold method and the controllability of nonlinear control system

The problem of controllability of nonlinear control system is a significant field which has an extensive prospect of application. A.M.Kovalev of Ukraine Academy of Science applied the oriented manifold method developed in dynamics of rigid body to nonlinear control system for the first time and obtained a series of efficient results. Based on Kovalev’s oriented manifold method, firstly, by invariant manifold method the problem of controllability of nonlinear control system was studied and the necessary condition of the controllability of a kind of affine nonlinear system was given out. Then the realization of the necessary condition was discussed. At last, the motion of a rigid body with two rotors was investigated and the necessary condition which is satisfied by this system was proved.     

Constructing Galerkin's approximations of invariant tori using MACSYMA

Invariant tori of solutions for nonlinearly coupled oscillators are generalizations of limit cycles in the phase plane. They are surfaces of aperiodic solutions of the coupled oscillators with the property that once a solution is on the surface it remains on the surface. Invariant tori satisfy a defining system of nonlinear partial differential equations. This case study shows that with the help of a symbolic manipulation package, such as MACSYMA, approximations to the invariant tori can be developed by using Galerkin's variational method. The resulting series must be manipulated efficiently, however, by using the Poisson series representation for multiply periodic functions, which makes maximum use of the list processing techniques of MACSYMA. Three cases are studied for the single van der Pol oscillator with forcing parameter =0.5, 1.0, 1.5, and three cases are studied for a pair of nonlinearly coupled van der Pol oscillators with forcing parameters =0.005, 0.5, 1.0. The approximate tori exhibit good agreement with direct numerical integrations of the systems.Contribution of the National Institute of Standards and Technology, a Federal agency. Not subject to copyright.     

惯性信息辅助的快速大视角图像匹配方法

大视角图像匹配算法的鲁棒性与实时性直接影响飞行器对远距离目标定位的性能。针对目前仿射不变图像匹配算法实时性较差的问题,提出一种惯性信息辅助的快速大视角图像匹配方法。该方法对现有的快速图像匹配算法进行改进,避免了构建高斯金字塔,提高了算法效率。然后利用机载惯性导航信息求解实时图与参考图之间的单应性矩阵,并对实时图进行模拟视角变换以此减小图像间视角差异,克服了现有的大视角图像匹配算法盲目多次的匹配计算,实现了大视角图像的快速匹配。实验结果表明,惯性信息辅助的大视角图像匹配算法与现有的快速仿射不变性匹配算法相比,匹配效率提高了至少2倍。     

Modeling and analysis of the coupled dynamics of machine degradation and repair processes using piecewise affine stochastic differential equations

This paper introduces an approach to model the coupled dynamics of recurring degradation and restoration processes that take place in real-world systems, such as manufacturing machines, in the form of non-linear (piecewise affine) differential equations. Unlike previous methods, interactions between degradation and repair dynamics that influence downtime distributions in such manufacturing systems can be explicitly considered and dependencies beyond correlations between the time between failures (TBF) and the time to repair (TTR) can be captured. The periodic solutions of the model capture the progressive evolution of long time-scale failure and repair patterns. The distribution of short time-scale failure–repair cycles can be captured by providing a class of random perturbations to certain model parameters. We provide sufficient conditions for the existence and stability of the resulting non-linear stochastic differential equation (n-SDE) model solutions that mimic the breakdown and repair patterns observed in many real-world manufacturing systems, namely, fairly regular (periodic) large breakdown and repair cycles, interspersed with highly right skewed distributions of short cycles. We also define the basin of attraction for the periodic orbit. The n-SDE model was parametrized using real-world data sets acquired from an automotive manufacturing assembly line segment, and the model solutions were compared with actual observations of TBF and TTR patterns, as well as the performance of the process. Our approach reduces the computation time by about 25% when compared to a discrete-event simulation model, which uses conventional TBF and TTR distributions, implemented on a commercial platform. Experimental investigations also suggest that the model can capture the correlations and non-linear coupled dynamics that exist in real-world operations among TBF and TTR, which are typically ignored in traditional approaches.     

Interval stochastic matrices: A combinatorial lemma and the computation of invariant measures of dynamical systems

The concept of an interval stochastic matrix is introduced. We prove a combinatorial theorem which describes the network flow associated with an interval matrix. The semi-invariant vectors of are characterized in terms of eigenvectors with unit eigenvalue of stochastic matrices . These results are then applied to the approximation and machine computation of invariant measures of dynamical systems.Funded under Australian Research Council Grant A 8913 2609.     

Regular partially invariant solutions of defect 1 of the equations of ideal magnetohydrodynamics

All irreducible regular partially invariant submodels with one noninvariant function for the equations of ideal magnetohydrodynamics are constructed. The submodels are completed to involution, and partially integrated. The submodels specify Ovsyannikov vortex type motion or motion with homogeneous deformation in some spatial directions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 5–15, March–April, 2009.     

The Full Study of Planar Quadratic Differential Systems Possessing a Line of Singularities at Infinity

In this article we make a full study of the class of non-degenerate real planar quadratic differential systems having all points at infinity (in the Poincaré compactification) as singularities. We prove that all such systems have invariant affine lines of total multiplicity 3, give all their configurations of invariant lines and show that all these systems are integrable via the method of Darboux having cubic polynomials as inverse integrating factors. After constructing the topologically distinct phase portraits in this class we give invariant necessary and sufficient conditions in terms of the 12 coefficients of the systems for the realization of each one of them and give representatives of the orbits under the action of the affine group and time rescaling. We construct the moduli space of this class for this action and give the corresponding bifurcation diagram. Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday     

The J2 invariant relative configuration of spaceborne SAR interferometer for digital elevation measurement

A 3-craft formation configuration is proposed to perform the digital elevation model （DEM） for the distributed spacebome interferometric synthetic aperture radar （InSAR）, and it is optimized by the modified ant colony algorithm to have the best compatibility with J2 invariant orbits created by differential correction algorithm. The configuration has succeeded in assigning the across-track baseline to vary periodically and with its mean value equal to the optimal baseline determined by the relative height measurement accuracy. The required relationship between crafts＇ magnitudes and phases is formulated for the general case of interferometry measure from non-orthographic and non-lateral view. The J2 invariant configurations created by differential correction algorithm are employed to investigate their compatibility with the required configuration. The colony algorithm is applied to search the optimal configuration holding the near-constant across-track baseline under the J2 perturbation, and the absolute height measurement accuracy is preferable as expected.