复杂双摆的KAM定理

本文建立了复杂双摆的Ｈａｍｉｌｔｏｎｉａｎ运动方程,并将重力作为扰动,利用ＫＡＭ理论研究了复杂双摆的运动规律,研究结果表明,复杂双摆运动规律与传统双摆运动规律灰似,当重力能量与总能量相比很小时,ＫＡＭ不变闭曲一存在表明无重力系统的“总动量”守恒的特点。     

具有退化性的辛映射的KAM定理

本文讨论辛映射的扰动问题．利用修改的ＡｒｎｏｌｄＫＡＭ迭代格式，给出一个不变环面的保持性定理．由于可积映射具有退化性，这推广了文［６］的工作．     

反应扩散方程组的弱不变区域

反应扩散方程组的弱不变区域郑斯宁（大连理工大学应用数学系，大连１１６０２４）ＷＥＡＫＬＹＩＮＶＡＲＩＡＮＴＲＥＧＩＯＮＳＦＯＲＲＥＡＣＴＩＯＮ－ＤＩＦＦＵＳＩＯＮＳＹＳＴＥＭＳ￥ＺＨＥＮＧＳＩＮＩＮＧ（ＤｅｐａｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅ...     

实幂等矩阵的伴随矩（英）

ＡｎＡｄｊｏｉｎｔＭａｔｒｉｘｏｆａＲｅａｌＩｄｅｍｐｏｔｅｎｔＭａｔｒｉｘＪｉｎＢａｉＫｉｍ（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＷｅｓｔＶｉｒｇｉｎｉａＵｎｉｖｅｒｓｉｔｙＭｏｒｇａｎｔｏｗｎ，ＷＶ２６５０６，ＵＳＡ）ＨｅｅＳｉｋＫｉｍ（Ｄｅｐｔ．ｏｆＭａ...     

Fibonacci迹映的不变锥面

该文利用「１」中关于可反转映射的ＫＡＭ定理,证明一定条件下Ｒ＾３上的可反转映射在不动点附近存在不变锥面,并验证Ｆｉｂｏｎａｃｃｉ迹映射中这种不变锥面。     

A SIMPLE PROOf OF KUPKA-SMALE THEOREN

ＡＳＩＭＰＬＥＰＲＯＯｆＯＦＫＵＰＫＡ－ＳＭＡＬＥＴＨＥＯＲＥＮ林振声福州大学，邮编：３５０００２ＡＳＩＭＰＬＥＰＲＯＯｆＯＦＫＵＰＫＡ－ＳＭＡＬＥＴＨＥＯＲＥＮ￥ＬｉｎＺｈｅｎｓｈｅｎｇ（ＦｕｚｈｏｕＵｎｉｖｅｒｓｉｔｙ，３５０００２，）Ａｂｓｔｒ...     

三角域上Meyer－KOnig－Zeller算子

定义了一种三角域上Ｍｅｙｅｒ－Ｋ［ＡＫｏＤ］ｎｉｇ－Ｚｅｌｌｅｒ算子，并研究其在Ｃ度量下的逼近性质。     

HANKEL OPERATORS AND HANKEL ALGEBRAS

ＨＡＮＫＥＬＯＰＥＲＡＴＯＲＳＡＮＤＨＡＮＫＥＬＡＬＧＥＢＲＡＳＹＡＮＳＨＡＯＺＯＮＧＣＨＥＮＸＩＡＯＭＡＮＧＵＯＫＵＮＹＵＭａｎｕｓｃｒｉｐｔｒｅｃｅｉｖｅｄＪｕｎｅ２３，１９９５．ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃｓ，Ｆｕｄａ...     

COMPLETE EXTREMAL SURFACES OF MIXED TYPE IN 3-DIMENSIONAL MINKOWSKI SPACE

ＣＯＭＰＬＥＴＥＥＸＴＲＥＭＡＬＳＵＲＦＡＣＥＳＯＦＭＩＸＥＤＴＹＰＥＩＮ３－ＤＩＭＥＮＳＩＯＮＡＬＭＩＮＫＯＷＳＫＩＳＰＡＣＥ￥ＧＵＣＨＡＯＨＡＯ（ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃｓ，ＦｕｄａｎＵｎｉｖｅｒｓｉｔｙ，Ｓｈａｎｇｈａｉ２００...     

Fibonacci迹映射的不变锥面

该文利用［１］中关于可反转映射的ＫＡＭ定理,证明一定条件下Ｒ~３上的可反转映射在下动点附近存在不变锥面．并验证Ｆｉｂｏｎａｃｃｉ迹映射中存在这种不变锥面．     

圆型限制性三体问题中双恒星系统的存在性

在圆型限制性三体问题的研究中，研究较多的是在其中一体附近的运动。本文研究远离两个大质量恒星的行星的运行情况，在此范围内，使用KAM理论研究大振幅轨道的存在性，说明了星系中双恒星系统的存在性，而这一现象已被天文学家所观察到。     

径向质量偏心的自由转子陀螺的漂移运动

讨论均匀重力场中具有微小径向质量偏心的刚体定点运动,建立用状态变量描述的刚体动力学方程．对于刚体高速自旋情形应用平均法求出其近似解析积分,用以分析径向质量偏心对自由转子陀螺进动特性的影响．对径向质量偏心引起陀螺漂移现象给出物理解释,并导出陀螺常值漂移的简明的解析公式,与数值计算结果符合．     

Active vibration control of unbalanced flexible rotor–shaft systems parametrically excited due to base motion

Rotor vibrations caused by large time-varying base motion are of considerable importance as there are a good number of rotors, e.g., the ship and aircraft turbine rotors, which are often subject to excitations, as the rotor base, i.e. the vehicle, undergoes large time varying linear and angular displacements as a result of different maneuvers. Due to such motions of the base, the equations of vibratory motion of a flexible rotor–shaft relative to the base (which forms a non-inertial reference frame) contains terms due to Coriolis effect as well as inertial excitations (generally asynchronous to rotor spin) generated by different system parameters. Such equations of motion are linear but time-varying in nature, invoking the possibility of parametric instability under certain frequency–amplitude combinations of the base motion. An investigation of active vibration control of an unbalanced rotor–shaft system on moving bases is attempted in this work with electromagnetic control force provided by an actuator consisting of four electromagnetic exciters, placed on the stator in a suitable plane around the rotor–shaft. The actuator does not levitate the rotor or facilitate any bearing action, which is provided by the conventional suspension system. The equations of motion of the rotor–shaft continuum are first written with respect to the non-inertial reference frame (the moving base in this case) including the effect of rotor internal damping. A conventional model for the electromagnetic exciter is used. Numerical simulations performed on the flexible rotor–shaft modelled using beam finite elements shows that the control action is successful in avoiding the parametric instability, postponing the instability due to internal material damping and reducing the rotor response relative to the rigid base significantly, with sufficiently low demand of control current in comparison with the bias current in the actuator coils.     

Dynamics of a rotor partially filled with a viscous incompressible fluid

A rotor partially filled with a viscous incompressible fluid is modeled as planar system. Its structural part, i. e. the rotor, is assumed to be rigid, circular, elastically supported and running with a prescribed time-dependent angular velocity. Both parts, structure and fluid, interact via the no-slip condition and the pressure. The point of departure for the mathematical formulation of the fluid filling is the Navier-Stokes equation, which is complemented by an additional equation for the evolution of its free inner boundary. Further, rotor and fluid are subjected to volume forces, namely gravitation. Trial functions are chosen for the fluid velocity field, the pressure field and the moving boundary, which fulfill the incompressibility constraint as well as the boundary conditions. Inserting these trial functions into the partial differential equations of the fluid motion, and applying the method of weighted residuals yields equations with time derivatives only. Finally, in combination with the rotor equations, a nonlinear system of 12 differential-algebraic equations results, which sufficiently describes solutions near the circular symmetric state and which may indicate the loss of its stability. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the impulsive dynamics of M-blocks

This paper is concerned with the motion of a cubic rigid body (cube) with a rotor, caused by a sudden brake of the rotor, which imparts its angular momentum to the body. This produces an impulsive reaction of the support, leading to a jump or rolling from one face to another. Such dynamics was demonstrated by researchers from Massachusetts Institute of Technology at the IEEE/RSJ International Conference on Intelligent Robots and Systems in Tokio in November 2013. The robot, called by them M-block, is 4 cm in size and uses an internal flywheel mechanism rotating at 20 000 rev/min. Initially the cube rests on a horizontal plane. When the brake is set, the relative rotation slows down, and its energy is imparted to the case. The subsequent motion is illustrated in a clip [13]. Here the general approach to the analysis of dynamics of M-cube is proposed, including equations of impulsive motion and methods of their solution. Some particular cases are studied in details.     

Non-periodic and chaotic response of three-multilobe air bearing system

The dynamic response of a three-multilobe air bearing (TMAB) system is investigated for various values of the rotor mass and bearing number using a hybrid numerical scheme consisting of the differential transformation method (DTM) and the finite difference method (FDM). The validity of the numerical scheme is demonstrated by comparing the results obtained for the rotor center orbit under typical operating conditions with those obtained from the traditional FDM approach and a perturbation method, respectively. The dynamic behavior of the rotor center is then investigated for rotor mass values in the range of 1.0 ≤ m_r ≤ 16.0 kg and bearing number values in the range of 1.0 ≤ Λ ≤ 5.0. The phase trajectories, power spectra, bifurcation diagrams, Poincaré maps and maximum Lyapunov exponents show that the TMAB system exhibits a complex dynamic behavior consisting of periodic, quasi-periodic and chaotic motion at certain values of the rotor mass and bearing number. In general, the numerical results obtained in this study provide a useful insight into the dynamic response of TMAB systems. In particular, the results indicate the operating conditions which should be avoided in order to achieve a desirable periodic motion of the system.     

The equilibria and stability of a dynamically symmetrical satellite with a two-degree-of-freedom powered gyroscope

For a dynamically symmetrical satellite carrying a two-degree-of-freedom powered gyroscope, all the relative equilibria in a circular orbit are found as a function of the angular momentum of the rotor and the angle between the precession axis of the gyroscope and the plane of dynamical symmetry. The case with no spring on the axis of the gyroscope frame and the case with a spring whose stiffness satisfies definite conditions are considered. The secular stability of the equilibria is investigated. For a system with dissipation in the axis of the gyroscope frame, the Barbashin–Krasovskii theorem is used to perform a detailed analysis, which enables the character of the Lyapunov stability of all the equilibria to be determined, with the exception of a few points. The results of a numerical solution of the problem of the optimal values of the system parameters, for which asymptotically stable equilibria are obtained with maximum speed, are presented.     

Persistence of Invariant Tori on Submanifolds in Hamiltonian Systems

Summary. Generalizing the degenerate KAM theorem under the Rüssmann nondegeneracy and the isoenergetic KAM theorem, we employ a quasilinear iterative scheme to study the persistence and frequency preservation of invariant tori on a smooth submanifold for a real analytic, nearly integrable Hamiltonian system. Under a nondegenerate condition of Rüssmann type on the submanifold, we shall show the following: (a) the majority of the unperturbed tori on the submanifold will persist; (b) the perturbed toral frequencies can be partially preserved according to the maximal degeneracy of the Hessian of the unperturbed system and be fully preserved if the Hessian is nondegenerate; (c) the Hamiltonian admits normal forms near the perturbed tori of arbitrarily prescribed high order. Under a subisoenergetic nondegenerate condition on an energy surface, we shall show that the majority of unperturbed tori give rise to invariant tori of the perturbed system of the same energy which preserve the ratio of certain components of the respective frequencies.     

Arnol′d Diffusion in a Pendulum Lattice

The main model studied in this paper is a lattice of pendula with a nearest‐neighbor coupling. If the coupling is weak, then the system is near‐integrable and KAM tori fill most of the phase space. For all KAM trajectories the energy of each pendulum stays within a narrow band for all time. Still, we show that for an arbitrarily weak coupling of a certain localized type, the neighboring pendula can exchange energy. In fact, the energy can be transferred between the pendula in any prescribed way. © 2014 Wiley Periodicals, Inc.     

Some Aspects of Conservative and Dissipative KAM Theorems

We discuss some aspects of conservative and dissipative KAM theorems, with particular reference to a comparison between the main assumptions needed to develop KAM theory in the two settings. After analyzing the qualitative behavior of a paradigmatic model (the standard mapping), we study the existence of quasi?Cperiodic tori in the two frameworks, paying special attention to the occurrence of small divisors and to the non?Cdegeneracy (twist) condition in the conservative and in the dissipative case. These conditions are the main requirements for the applicability of KAM theorem, which is then stated for invariant tori as well as for invariant attractors. We proceed to discuss a criterion for the determination of the breakdown threshold of invariant tori and invariant attractors through approximating periodic orbits. These results can be applied to a wide set of physical problems; concrete applications to Celestial Mechanics are discussed with particular reference to the rotational and orbital motion of celestial bodies.