Nonlinear Vibration of a Pendulum in a Viscous Stream

Zusammenfassung In dieser Arbeit wird eine Regelgleichung, die den Gang eines Stabpendels in einem Reibungsstrom von veränderlicher Geschwindigkeit beschreibt, entwickelt. Nichtlineare Effekte, die durch Reibungskräfte verursacht sind, und grosse Stabamplituden sind in der Diskussion eingeschlossen. Wir setzen voraus, dass die Reibungskräfte sowohl am Stab als auch am eigentlichen Pendelkörper wirken. Die Methode erster Ordnung von Krylov und Bogoliuboff ergibt eine approximative allgemeine Lösung der Bewegungsgleichung. Wir wenden die Lösung auf den Spezialfall eines Pendels in einem gleichmässigen Reibungsstrom an.     

基于滑模方法的倒立摆台车系统的非线性控制

以倒立摆台车系统为研究对象,将系统状态分成两组,构造出一种双层滑动面,给出一种基于滑模控制的非线性控制律,并设计了控制器参数．然后采用Lyapunov方法,证明了各级滑动平面的稳定性,实现了台车在水平方向位置及摆角的渐近跟踪．     

新三维自治混沌系统动力学性质研究

采用动力系统理论分析和计算机数值仿真相结合的方法,研究了一类新三维自治混沌系统的非线性动力学行为,如平衡点及其稳定性、不变集、混沌吸引子、吸引域等,从而展示了该混沌系统的丰富的动力学特性并且用matlab给出了相应的计算机模拟.创新点在于同时考虑了该混沌系统的最终界和全局吸引集,并且对于这个混沌系统的任意正参数,分别得到了该混沌系统最终界的一个参数族数学表达式和全局指数吸引集的一个参数族数学表达式,最后利用交集的思想分别得到了混沌系统最终界和全局吸引集的一个较小的数学表达式.混沌系统有望在实际保密通信中得到应用.     

Dynamics of Semi-Discretizations of the Defocusing Nonlinear Schrodinger Equation

It has been demonstrated that the nonlinear Schrödinger(NLS) equation is sensitive to discretizations. In the focusingcase this is due to the homoclinic structure associated withthe NLS equation. In this paper we show that various numericalschemes for the defocusing case are also prone to instabilities,although not as severe as those of the focusing equation. Anintegrable discretization due to Ablowitz and Ladik does notsuffer from the same instabilities. However, it is shown thatit develops a focusing singularity if a threshold conditionis exceeded. Numerical examples illustrating the phenomena pertainingto the defocusing equation are given.     

Nonlinear Dynamics of Pipes with Pulsatile Flow

The dynamic behaviour of pipes conveying flowing fluid can be significantly influenced by the periodically changing flow. The fluid is considered to be incompressible, frictionless and its velocity relative to the pipe is harmonic function of time of which magnitude does not depend on spatial coordinates. The pipe is modelled as an inextensible continuum. The investigations were performed using three bifurcation parameters: the frequency, the perturbation amplitude and the mean flow velocity. In super‐critical case the nonlinear dynamics was shown in bifurcation diagrams.     

Nonlinear Dynamics Equation in p-Adic String Theory

We investigate nonlinear pseudodifferential equations with infinitely many derivatives. These are equations of a new class, and they originally appeared in p-adic string theory. Their investigation is of interest in mathematical physics and its applications, in particular, in string theory and cosmology. We undertake a systematic mathematical investigation of the properties of these equations and prove the main uniqueness theorem for the solution in an algebra of generalized functions. We discuss boundary problems for bounded solutions and prove the existence theorem for spatially homogeneous solutions for odd p. For even p, we prove the absence of a continuous nonnegative solution interpolating between two vacuums and indicate the possible existence of discontinuous solutions. We also consider the multidimensional equation and discuss soliton and q-brane solutions.     

Dynamics of Nonlinear Random Walks on Complex Networks

Journal of Nonlinear Science - In this paper, we study the dynamics of nonlinear random walks. While typical random walks on networks consist of standard Markov chains whose static transition...     

Nonlinear Dynamics on the Plane and Integrable Hierarchies of Infinitesimal Deformations

A class of nonlinear problems on the plane, described by nonlinear inhomogeneous     -equations, is considered. It is shown that the corresponding dynamics, generated by deformations of inhomogeneous terms (sources), is described by Hamilton–Jacobi-type equations associated with hierarchies of dispersionless integrable systems. These hierarchies are constructed by applying the quasiclassical     -dressing method.     

一类非线性微分代数系统的零动态

控制系统的零动态是系统一种内部动态品质,其行为与系统的许多性质相联系,如系统的稳定性,反馈镇定与输出跟踪等.针对一类非线性微分代数系统,提出了输出零子流形和零动态的概念.利用M-导数方法,探讨了此类系统的输出零子流形的性质,并给出了此类系统的输出零子流形和零动态算法,也讨论了该算法的一些性质.最后,给出一个例子说明如何利用系统的零动态来讨论系统的反馈稳定化问题.     

Nonreflecting Boundary Conditions for Nonlinear Problems of Fluid Dynamics

External problems of fluid dynamics are modeled on artificially bounded regions. A method is proposed for applying nonreflecting boundary conditions to the linearized Euler equations obtained from the original nonlinear problem. The nonreflecting conditions are modified to allow for viscosity and nonhomogeneity of unperturbed background flow in linear Euler, Navier–Stokes, and quasi-fluid-dynamic models. One-dimensional test problems are computed in the linear and nonlinear cases.     

Dissipative Dynamics for a Class of Nonlinear Pseudo-Differential Equations

We study a class of nonlinear evolutionary equations generated by an elliptic pseudo-differential operator, and with nonlinearity of the form G(u _x ) where cη² ≤ G(η) ≤ Cη² for large |η|. For the evolution in spaces of periodic functions with zero mean we demonstrate existence of a universal absorbing set and compact attractor. Furthermore, we show that the attractor is of a finite Hausdorf dimension. The dissipation mechanism for the class of equations studied in the paper is akin to the nonlinear saturation in the Kuramoto-Sivashinsky equation. A similar generalization of the Kuramoto-Sivashinsky equation was studied by Nicolaenko et al. under the assumption of a purely quadratic nonlinearity and reflection invariance of both: the equation and solutions.      

Nonlinear Dynamics of Non-uniform Current-Vortex Sheets in Magnetohydrodynamic Flows

A theoretical model is proposed to describe fully nonlinear dynamics of interfaces in two-dimensional MHD flows based on an idea of non-uniform current-vortex sheet. Application of vortex sheet model to MHD flows has a crucial difficulty because of non-conservative nature of magnetic tension. However, it is shown that when a magnetic field is initially parallel to an interface, the concept of vortex sheet can be extended to MHD flows (current-vortex sheet). Two-dimensional MHD flows are then described only by a one-dimensional Lagrange parameter on the sheet. It is also shown that bulk magnetic field and velocity can be calculated from their values on the sheet. The model is tested by MHD Richtmyer–Meshkov instability with sinusoidal vortex sheet strength. Two-dimensional ideal MHD simulations show that the nonlinear dynamics of a shocked interface with density stratification agrees fairly well with that for its corresponding potential flow. Numerical solutions of the model reproduce properly the results of the ideal MHD simulations, such as the roll-up of spike, exponential growth of magnetic field, and its saturation and oscillation. Nonlinear evolution of the interface is found to be determined by the Alfvén and Atwood numbers. Some of their dependence on the sheet dynamics and magnetic field amplification are discussed. It is shown by the model that the magnetic field amplification occurs locally associated with the nonlinear dynamics of the current-vortex sheet. We expect that our model can be applicable to a wide variety of MHD shear flows.     

径向基网络的非线性插值与3D点云曲面重构技术

研究了径向基网络插值算法与3D曲面重构方法,分别从研究价值、插值理论、仿真等等方面做了详细分析,研究结果表明RBF-Network在逼近一维复杂的非线性函数时具有收敛速度快、精度高、泛化能力更强等优点.但在3D重构方面,基于RBF的单位分解法重构效果更好.     

Global Dynamics below the Ground State Energy for the Klein-Gordon-Zakharov System in the 3D Radial Case

We consider the global dynamics below the ground state energy for the Klein-Gordon-Zakharov system in the 3D radial case; and obtain the dichotomy between scattering and finite time blow up.     

Observing Pattern Dynamics in Nonlinear Optical Systems Using the Video-sampling Method

We present a method of synchronizing a shutter for a video camera that allows an instantaneous slow-motion visualization of fast repetitive processes with a time scaling factor of up to 1,000,000 or more. The method uses standard, unmodified CCIR or EIA video equipment, including video cassette recorders for low-cost recording. A number of examples showing spatio-temporal dynamics in nonlinear optical systems is shown, including magneto-optical beam-splitting, drifting hexagons and phase synchronization of coupled solid-state lasers.     

Some Explicit Resonating Waves in Weakly Nonlinear Gas Dynamics

Equations governing leading order wave amplitudes of resonating almost periodic wave trains in weakly nonlinear acoustics have been obtained by Majda and Rosales [Stud. Appl. Math. 71:149–179 (1984)]. These equations consist of a pair of Burgers equations coupled through an integral term with a known kernel. Numerical experiments reported by Majda, Rosales, and Schonbek have suggested the existence of smooth solutions of this system whose components consist of traveling waves moving in opposite directions. For the simplest cosine kernel, explicit formulae are given here for such resonating wave solutions. There is a wave of maximum amplitude with a “peak.” For more general kernels, small amplitude resonating waves are constructed via bifurcation.