Sequences of orbits and the boundaries of the basin of attraction for two double heteroclinic orbits

The boundaries of the basin of attraction are usually assumed to be rather elementary for Hamiltonian systems with autonomous perturbations. In the case of one saddle point, the sequences of orbits before capture are unique for each basin. However, we show that for two saddle points each with double heteroclinic orbits, there is an infinite number of different sequences of nearly homoclinic orbits before capture depending on the four heteroclinic parameters. The probabilities of capture are independent of the capture sequence.     

Global Saddles for Planar Maps

We study the dynamics of planar diffeomorphisms having a unique fixed point that is a hyperbolic local saddle. We obtain sufficient conditions under which the fixed point is a global saddle. We also address the special case of \(D_2\)-symmetric maps, for which we obtain a similar result for \(C^1\) homeomorphisms. Some applications to differential equations are also given.     

On Singular Limits of Mean-Field Equations

Mean-field equations arise as steady state versions of convection-diffusion systems where the convective field is determined by solution of a Poisson equation whose right-hand side is affine in the solutions of the convection-diffusion equations. In this paper we consider the repulsive coupling case for a system of two convection-diffusion equations. For general diffusivities we prove the existence of a unique solution of the mean-field equation by a variational analysis of a saddle point problem (usually without coercivity). Also we analyze the small-Debye-length limit and prove convergence to either the so-called charge-neutral case or to a double obstacle problem for the limiting potential depending on the data.     

ℋ︁‐LU factorization in preconditioners for augmented Lagrangian and grad‐div stabilized saddle point systems

The (mixed finite element) discretization of the linearized Navier–Stokes equations leads to a linear system of equations of saddle point type. The iterative solution of this linear system requires the construction of suitable preconditioners, especially in the case of high Reynolds numbers. In the past, a stabilizing approach has been suggested which does not change the exact solution but influences the accuracy of the discrete solution as well as the effectiveness of iterative solvers. This stabilization technique can be performed on the continuous side before the discretization, where it is known as ‘grad‐div’ (GD) stabilization, as well as on the discrete side where it is known as an ‘augmented Lagrangian’ (AL) technique (and does not change the discrete solution). In this paper, we study the applicability of ??‐LU factorizations to solve the arising subproblems in the different variants of stabilized saddle point systems. We consider both the saddle point systems that arise from the stabilization in the continuous as well as on the discrete setting. Recently, a modified AL preconditioner has been proposed for the system resulting from the discrete stabilization. We provide a straightforward generalization of this approach to the GD stabilization. We conclude the paper with numerical tests for a variety of problems to illustrate the behavior of the considered preconditioners as well as the suitability of ??‐LU factorization in the preconditioners. Copyright © 2010 John Wiley & Sons, Ltd.     

悬索桥结构分析中的索-鞍座单元

为在悬索桥结构的有限元分析中真实、简洁、高效地模拟索鞍,本文建立了一类新的单元。新单元包括索段的一端固定在与其接触部分为单一半径圆弧的索鞍上,另一端分别位于索鞍两侧的两节点“左索-鞍座单元”和“右索-鞍座单元”,以及索段两端点分别位于索鞍两侧,中间一点固定于鞍座上的三节点“索-鞍座单元”,后者的鞍槽可为两不同半径圆弧的组合。根据要求的成桥状态几何参数确定结构的无应力状态时,可利用前二者进行悬索桥的单跨分析。新单元通过自动调整索与鞍座的脱离点而处于平衡状态,从而简化了计算。单元算法的推导基于有限元分析的基本原理和弹性悬链线的精确解,并利用了处于平衡状态时索与鞍座之间的内力关系。新单元可象常规单元一样直接用于成桥状态或施工过程中悬索桥结构的有限元分析。设计的算例验证了新单元的正确性,并举例说明了新单元在悬索桥结构分析中的应用。     

Morris-Lecar系统中通道噪声诱导的自发性动作电位研究

在生物物理学中,越来越多的现象是由于分段确定性的动力系统与连续时间马氏过程之间的耦合作用而产生的.因为这种耦合性,相关的数学模型更适合取为随机混合系统而不是扩散过程(基于It?o随机微分方程).本文从理论上和数值上研究了在弱噪声条件下无鞍点状态的随机混合Morris-Lecar系统中,由通道噪声诱导的自发性放电现象.一个动作电位的初始阶段可视为噪声诱导的逃逸事件,其最优路径和拟势可由辅助Hamilton系统给出.由于系统不存在鞍点,因此可选择虚拟分界线(ghost separatrix)为阈值,研究噪声诱导的自静息态的逃逸事件.通过计算在阈值处的拟势,便可发现其值有一个明显的最小值,其作用类似于鞍点.通过改进的Monte Carlo模拟方法,计算了历程概率分布,其结果对初始阶段和兴奋阶段的理论解均给出了验证.此外,基于前人将拟势等高线作为阈值的另一种选择,我们对两种阈值取法的优劣性进行了比较.最后,本文研究了钠离子和钾离子通道噪声的不同组合对最优路径和拟势的影响.结果表明:钾离子通道噪声在自发性放电过程中起主导作用,且两种噪声强度存在一个最优比例能使总的噪声强度达到最小.     

Wada分形吸引域边界上的混沌鞍

应用广义胞映射图论（Generalized Cell Mapping Digraph)方法，数值地研究Thompson的逃逸方程在最佳逃逸点附近的分岔。发现了嵌入在Wada分形吸引域边界上的混沌鞍，混沌鞍是状态空间不稳定（非吸引）的混沌不变集合。Wada分形吸引域边界是具有Wada性质的边界，即吸引域边界上的任意点也同时是至少两个其它吸引域的边界点，称为Wada域边界。我们证明Wada域边界上的混沌鞍导致局部鞍结分岔具有全局不确定性结局，研究了Wada域边界上混沌鞍的形成与演化，证明最终的逃逸分岔是混沌吸引子碰撞混沌鞍的边界激变。     

Two preconditioners for saddle point problems in fluid flows

In this paper two preconditioners for the saddle point problem are analysed: one based on the augmented Lagrangian approach and another involving artificial compressibility. Eigenvalue analysis shows that with these preconditioners small condition numbers can be achieved for the preconditioned saddle point matrix. The preconditioners are compared with commonly used preconditioners from literature for the Stokes and Oseen equation and an ocean flow problem. The numerical results confirm the analysis: the preconditioners are a good alternative to existing ones in fluid flow problems. Copyright © 2006 John Wiley & Sons, Ltd.     

含J2摄动的有限时间航天器追逃博弈问题研究

航天器追逃博弈是航天器在轨捕获任务的一个重要问题,具有极高的军民两用双重价值．针对有限时间且考虑J2摄动的航天器追逃博弈问题,本文提出了一种精确的求解方法．该方法的核心思想是将有限时间的航天器追逃博弈问题建模为有限时间二人零和对策,则博弈中两航天器的最优控制策略可以转化为有限时间二人零和对策的鞍点解．在鞍点解的求解过程中,本文首先基于考虑J2摄动的非线性动力学方程,将两航天器动力学方程和始末边值条件与鞍点解必要条件结合得到两点边值问题,然后提出一种结合遗传算法和配点法的混合算法求解该两点边值问题以得到精确的鞍点解．本文利用数值仿真对所提方法的有效性进行了验证．结果表明：(i) 在航天器追逃博弈过程中,J2摄动对两航天器的最优控制策略具有较大影响;(ii) 所提方法能够精确求解出两航天器在有限时间的追逃博弈过程中的最优控制策略．     

Line-of-Sight Guidance Control Using Video Images

A mathematical formulation of the line-of-sight control problem is proposed for the case when this line is directed at a target. An operator situated on a moving platform controls the line of sight using the data received from video images. Some functionals determining the quality of control by the operator are introduced. It is proved that, in the case of plane motion of the platform and an infinitely distant target, the problem has a saddle point.     

Undercompressive shocks for nonstrictly hyperbolic conservation laws

We study 2×2 systems of hyperbolic conservation laws near an umbilic point. These systems have Undercompressive shock wave solutions, i.e., solutions whose viscous profiles are represented by saddle connections in an associated family of planar vector fields. Previous studies near umbilic points have assumed that the flux function is a quadratic polynomial, in which case saddle connections lie on invariant lines. We drop this assumption and study saddle connections using Golubitsky-Schaeffer equilibrium bifurcation theory and the Melnikov integral, which detects the breaking of heteroclinic orbits. The resulting information is used to construct solutions of Riemann problems.     

Global Dynamics of a Rapidly Forced Cart and Pendulum

In this paper, we study emergent behaviors elicited by applying open-loop, high-frequency oscillatory forcing to nonlinear control systems. First, we study hovering motions, which are periodic orbits associated with stable fixed points of the averaged system which are not fixed points of the forced system. We use the method of successive approximations to establish the existence of hovering motions, as well as compute analytical approximations of their locations, for the cart and pendulum on an inclined plane. Moreover, when small-amplitude dissipation is added, we show that the hovering motions are asymptotically stable. We compare the results for all of the local analysis with results of simulating Poincaré maps. Second, we perform a complete global analysis on this cart and pendulum system. Toward this end, the same iteration scheme we use to establish the existence of the hovering periodic orbits also yields the existence of periodic orbits near saddle equilibria of the averaged system. These latter periodic orbits are shown to be saddle periodic orbits, and in turn they have stable and unstable manifolds that form homoclinic tangles. A quantitative global analysis of these tangles is carried out. Three distinguished limiting cases are analyzed. Melnikov theory is applied in one case, and an extension of a recent result about exponentially small splitting of separatrices is developed and applied in another case. Finally, the influence of small damping is studied. This global analysis is useful in the design of open-loop control laws.     

New phenomena concerning a screw dislocation interacting with two imperfect interfaces

Dislocation mobility and stability in nanocrystals and electronic materials are influenced by the material composition and interface conditions. Its mobility and stability then affect the mechanical behaviors of the composites. In this paper, we first address, in detail, the problem of a screw dislocation located in an annular coating layer which is imperfectly bonded to the inner circular inhomogeneity and to the outer unbounded matrix. Both the inhomogeneity-coating interface and coating-matrix interface are modeled by a linear spring with vanishing thickness to account for the possible damage occurring on the interface. An analytic solution in series form is derived by means of complex variable method, with all the unknown constants being determined explicitly. The solution is then applied to the study of the dislocation mobility and stability due to its interaction with the two imperfect interfaces. The most interesting finding is that when the middle coating layer is more compliant than both the inner inhomogeneity and the outer unbounded matrix and when the interface rigidity parameters for the two imperfect interfaces are greater than certain values, one stable and two unstable equilibrium positions can exist for the dislocation. Furthermore, under certain conditions an equilibrium position, which can be either stable or unstable (i.e., a saddle point), can exist, which has never been observed in previous studies. Results for a screw dislocation interacting with two parallel straight imperfect interfaces are also presented as the limiting case where the radius of the inner inhomogeneity approaches infinity while the thickness of the coating layer is fixed.     

Stability of transonic two-phase flow

The nature of a singular point in the stability of one-dimensional transonic flow of a vapor-drop mixture in a channel of variable cross section is considered within the framework of a two-lquid hydrodynamical model. It is shown that the singular point in the case of any lags of the drops preserves the nature of a saddle inherent to homogeneous gas flow, shifting only towards the divergent part of the channel if the content of condensed phase is not too high. Here the transition of subsonic two-phase flow into supersonic flow is stable and the predominance of drop agglomeration over fragmentation and the positive curvature of the channel profile are stabilizing factors. The saddle nature of the singularity is possible only if the lag of the drops is not too high in the case of flows with a higher content of condensed phase. In the opposite case, the point at which the speed of sound is attained loses the nature of a saddle point.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 163–171, January–February, 1976.     

Resonant waves in elastic structured media: Dynamic homogenisation versus Green’s functions

We address an important issue of dynamic homogenisation in vector elasticity for a doubly periodic mass-spring elastic lattice. The notion of logarithmically growing resonant waves is used in the analysis of star-shaped wave forms induced by an oscillating point force. We note that the dispersion surfaces for Floquet–Bloch waves in the elastic lattice may contain critical points of the saddle type. Based on the local quadratic approximations of a dispersion surface, where the radian frequency is considered as a function of wave vector components, we deduce properties of a transient asymptotic solution associated with the contribution of the point source to the wave form. The notion of local Green’s functions is used to describe localised wave forms corresponding to the resonant frequency. The special feature of the problem is that, at the same resonant frequency, the Taylor quadratic approximations for different groups of the critical points on the dispersion surfaces (and hence different Floquet–Bloch vectors) are different. Thus, it is shown that for the vector case of micro-structured elastic medium there is no uniformly defined dynamic homogenisation procedure for a given resonant frequency. Instead, the continuous approximation of the wave field can be obtained through the asymptotic analysis of the lattice Green’s functions, presented in this paper.     

Apparent transition behavior of widely-used turbulence models

The Spalart–Allmaras and the Menter k–ω SST turbulence models are shown to have the undesirable characteristic that, for fully turbulent external flow computations, a transition region can occur whose extent varies with grid density. Extremely fine two-dimensional grids over the front portion of an airfoil are used to demonstrate the effect. As the grid density is increased, the laminar region near the nose becomes larger. In the Spalart–Allmaras model this behavior is due to convergence to a laminar-behavior fixed point that occurs in practice when freestream turbulence is below some threshold. It is the result of a feature purposefully added to the original model in conjunction with a special trip function. This degenerate fixed point can also cause non-uniqueness regarding where transition initiates on a given grid. Consistent fully turbulent results can easily be achieved by either using a freestream turbulence level higher than the threshold or by making a simple change to one of the model constants. Near the area where turbulence initiates, the SST model exhibits sensitivity to numerical resolution, but its solutions are unique on a given grid. Inconsistent apparent transition behavior with grid refinement in this case does not stem from the presence of a degenerate fixed point. A nullcline analysis is used to visualize the local behavior of the model.     

Re-examined principles of thrust generation by a track on soft ground

Dr. Bekker’s first book Theory of land locomotion offers in fact two different concepts of thrust generation on soft ground with respect to the slip: (a) as the push of grousers causing horizontal soil “distortion” and (b) as the shear force in the failure plane linked with the shear deformation. Bekker preferred the second concept and backed it up by the unique shear-ring measuring technique. To clear up the matter, the author decided to re-examine the thrust generation by a track plate experimentally in field conditions. The tests have shown that the initial stage of thrust generation in compressible ground is always horizontal soil compression by grousers, which divides the soil under a track into separate blocks initially at rest. This compression increases at least to the transition point, when a block is sheared off simultaneously at the bottom and in both lateral planes and starts sliding along the channel formed by the preceding grouser. The analysis of these measurements enabled to define the compressive displacement of the face of the soil block (travel of the grouser) appurtenant to the mentioned transition point, useful to define the thrust–slip curve. The case may also be described by the conventional shear stress–shear displacement relationship, imagined to take place in the bottom failure plane, however, namely the “shear displacement” is rather an unusual quantity.     

Bifurcational instability of an atomic lattice

In a recent article N.H. Macmillan and A. Kelly (1972) have confirmed on the basis of a linear eigenvalue analysis that a mechanically stressed perfect crystal can exhibit a bifurcational instability at stresses ranging to 20 per cent below that of the limiting maximum of the primary stress-strain curve. The question thus arises as to whether the branching point is in a non-linear sense either stable or unstable. In the former case, perfect and slightly imperfect crystals would be capable of sustaining stresses over and above the eigenvalue critical stress. In the unstable case, however, this eigenvalue stress would represent the ultimate strength of a perfect solid, while an imperfect crystal would fail at a limiting stress substantially below the eigenvalue.At 20 per cent below the limit point such a branching point is essentially distinct, and the non-linear stability analysis needed to answer this question is provided by a recently established general branching theory for discrete conservative systems. Often, however, the two critical equilibrium states are much nearer than this, and the branching theory is here suitably extended to cover the case of near-compound instabilities.An illustrative study of a close-packed crystal under uniaxial tension is next presented. A kinematically-admissible displacement field is employed and a bifurcation point is located on the primary equilibrium path just before the limiting maximum, the eigenvector being associated with a transverse shearing strain. Under these conditions a corresponding small transverse shearing stress would represent an ‘imperfection’, and the non-linear branching problem is next studied using the new general theory. This shows (in excellent quantitative agreement with an ad hoc numerical solution) that the branching point is non-linearly unstable with a quite severe imperfection-sensitivity which manifests itself as a sharp cusp on the failure-stress locus.     

Some implications of vibration testing of continuous systems for resonant frequencies

The objective of this study is to demonstrate, both analytically and experimentally, that certain resonant frequencies of complex continuous systems can be suppressed when common laboratory testing techniques are employed. A taut string is used as the physical system for demonstrating the phenomenon of resonant-frequency suppression. Four steady-state sinusoidal excitation conditions are considered. The first two test conditions consist of dynamic loads which result when the ends of the string move vertically with equal amplitudes which are exactly in phase [case (i)] or out of phase [case (ii)]. The second two test conditions consist of a single-point force [case (iii)] and displacement input [case (iv)] located somewhere along the length of the string. The analytical solution shows that theeven multiples of the fundamental resonant frequency are suppressed for case (i), theodd multiples are suppressed for case (ii), and those resonant frequencies are suppressed for case (iii) which make the excitation point a nodal point, while for case (iv) those resonant frequencies are suppressed which do not have a node corresponding to the excitation point. The experimental results verify the general predictions of resonant-frequency suppression given by the analysis. While this study is basically phenomenological and qualitative in nature, the results clearly indicate that (i) the most common means of vibration testing using shakers, i.e., condition i, will suppress certain resonant frequencies for the physical system used in this study as well as other similar systems; (ii) the practice of applying a concentrated excitation force or input motion will suppress certain resonant frequencies which depend on nodal points corresponding to the concentrated excitation point; and (iii) extreme caution must be exercised if all significant resonant frequencies are to be obtained for this simple system as well as more complex systems by vibration tests in the laboratory.