Chaotic dynamics of a whirling pendulum

A physical system is considered consisting of a rigid frame which is free to rotate about a vertical axis and to which is attached a planar simple pendulum. This system has “one and a half” degrees of freedom due to the fact that the frame and pendulum may freely rotate about the vertical axis, i.e., conservation of angular momentum holds for the “ideal”, or unperturbed, system. Using a Hamiltonian formulation we reduce the unperturbed equations of motion to a conservative planar system in which the constant angular momentum plays the role of a parameter. This system is shown to possess one or two sets of homoclinic motions depending on the level of the angular momentum. When this system is perturbed by external excitations and dissipative forces these homoclinic motions can break into homoclinic tangles providing the conditions for chaotic motions of the horseshoe type to exist. The criteria for this to occur can be formulated using a variation of Melnikov's method developed for slowly varying oscillators [1, 2]. For the present problem, the angular momentum becomes a slowly varying parameter upon addition of the disturbances. These ideas are used to rigorously prove the existence of chaotic motions for this system and to compute, to first order, global bifurcation parameter conditions. Since two types of homoclinic motions can occur, two different chaotic modes of motion can result and physical interpretations of these motions are given. In addition, a limiting case is considered in which the system becomes a single degree of freedom oscillator with parametric excitation.     

周期性外力驱动的混沌摆

为了简单直观地呈现混沌现象,研制了受周期驱动力驱动的混沌摆实验系统. 实验系统由力学机构、采集机构和数据处理分析仪组成. 力学机构是受周期外力驱动的复摆;采集机构采用无触点角度传感器,通过串口通信实时将数据传输给计算机;数据处理分析仪用虚拟仪器平台LabVIEW软件编程实现. 该系统可实时显示反映混沌基本特征的角度时序图、角速度时序图和相图.     

基于PASCO系统的混沌摆实验

基于PASCO系统研制了受周期外力驱动的混沌摆实验仪,调节混沌摆系统的参量可演示非线性动力学特征行为,描绘了无驱动及有驱动下的系统相位图,并分析了初值敏感性、奇异性(奇异吸引子)现象.应用混沌摆的动力学方程,进行Matlab仿真实验.实验结果表明:混沌摆实验系统动力学的性质依赖于振动频率值,系统驱动振幅必须大于一定阈值是混沌相出现的必要条件.     

Structure of the stochastic layer in a driven pendulum

An analysis of the stochastic layer in a driven pendulum is extended to the case when the separatrix map contains both single-and double-frequency harmonics. Resonance invariants of the first three orders are found for the double-frequency harmonic. Combined with the previously known single-frequency invariants, they can be used to obtain further information about the layer, in particular, to examine the neighborhoods of zeros of Melnikov integrals.     

Chaotic perturbations of KdV: I. Rational solutions

Rational pole-solutions of a perturbed KdV equation, describing nonlinear ion-acoustic plasma waves, become chaotic in time, when a small perturbation is periodically driven. A McGehee transformation blows up a degenerate stationary point at infinity and the Smale-Birkhoff Homoclinic. Theorem adapted to submanifolds in phase permits the use of the Melnikov method, with pole-solutions being homoclinic orbits.     

Suppression of Smale horseshoe structure via secondary perturbations in pendulum systems

We analyse the use of parametric and quasiperiodic modulations in suppressing horseshoe structure in the phase plane of perturbed pendulum systems. Taking the Froude pendulum as a typical system, four different modulation mechanisms are studied by deriving analytic expressions for the window of the strength of modulation giving suppression in each case. A comparison of the four cases from the point of view of flexibility and efficiency is also given.     

Chaos in a driven pendulum under asymmetric forcing

An analysis of the stochastic layer in a pendulum driven by an asymmetric high-frequency perturbation of fairly general form is continued. Analytical expressions are found for the amplitudes of secondary harmonics, and their contributions to the amplitude of the separatrix map responsible for onset of dynamical chaos are evaluated. Additional evidence is presented of the previously established fact that the secondary harmonics completely determine the stochasticl-ayer width when the primary frequencies lie in certain intervals. The mechanism of the onset of chaos in the vicinity of zeros of Melnikov integrals is shown to be substantially different as compared to the previously analyzed case of symmetric perturbation.     

Nonlinear evolution of perturbations in a thin fluid layer during wave formation

A mathematical model is presented for the state of a free surface of a thin fluid layer (a fluid film) in heat-mass-exchange processes of condensation and evaporation. The wave motion of a fluid film is studied under inhomogeneous surface tension. Nonlinear development of perturbations belonging to a continuous band of wave numbers on the surface of a thin fluid layer is investigated within the framework of a non-linear parabolic equation. It is shown that wave packets with carrier wave lying near the harmonic of maximum increment become self-ordered; as a result, a monochromatic wave is generated on the surface of the fluid film. When a wave packet is generated in the neighborhood of the neutral stability curve, one can observe a phenomenon of directed energy transfer to the waves in the neighborhood of the harmonic of maximum increment.     

Experimental and theoretical investigation of boundary layer perturbations on a low-aspect-ratio wing

Laminar-turbulent transition in a boundary layer of low-aspect-ratio wing was investigated. Experiments clarifying the flow structure, its mean and oscillatory characteristics were carried out accompanied by linear stability analysis of the wind tunnel data on the laminar flow velocity profiles. Theoretical results obtained in a parallel flow approximation are in a good agreement with the experimental data on disturbances evolution at the initial stage of transition to turbulence. The study was supported by the Ministry of Education and Science of the Russian Federation (Grant No. RNP 2.1.1.471) and Russian Foundation for Basic Research (Grant No. 03-01-06145)     

外周期力驱动的倒摆混沌运动演示仪

设计并制作了受周期外力驱动的倒摆演示实验装置．利用自主开发的软件可自动演示和记录摆球运动状态的时间序列并存储实验数据．通过改变摆长、驱动电压，观察到该系统存在通过倍周期分岔通向混沌的过程．     

Chaotic motion in an oscillatory boundary layer

The chaotic time oscillations in an incompressible fluid driven into motion by a harmonic time-varying pressure gradient is examined. Special attention is given to centrifugal destabilization of the viscous boundary layer. The basic flow is shown to be linearly unstable. For increasing modulation amplitude, the flow exhibits chaotic oscillations. The energy exchange between subharmonics and superharmonics of the least-stable spanwise wave number is considered. The presence of subharmonic Fourier modes are shown to accelerate the transition to temporally chaotic motion. (c) 1996 American Institute of Physics.     

Deformation of limit cycle under perturbations

The robustness of limit cycles of nonlinear dynamical systems is investigated by adding a small random velocity field to the famous van der Pol (VDP) equation in its two-dimensional phase plane. Our numerical calculations show that a limit cycle does not change much under a weak random perturbation. Thus it confirms the conjecture that a limit cycle will make only a small deformation under an external perturbation. The idea can be used to understand the ac response of self-sustained oscillations in nonlinear dynamical systems.     

Dynamics of solitons under random perturbations

The dynamics of solitons is investigated in media with randomly inhomogeneous and fluctuating parameters. Some exact results of the theory of nonlinear stochastic waves are given. An analysis is made of various approximate approaches, e.g. of the mean field method and the Born approximation. Special attention is paid to the perturbation technique based on the inverse scattering transform and to the construction of the most adequate stochastic perturbation theory for solitons. The described formalism is used to investigate the evolution of nonlinear wave (soliton) parameters, and the statistical characteristics of radiation generated by solitons in fluctuating media are analysed also. The same approach makes it possible to take into account the simultaneous effect of random and regular (e.g., friction) perturbations on the dynamics of solitons. Examples are given of situations arising when one describes nonlinear waves in real physical systems.     

Turbulent mixing of small-obstacle-induced perturbations with the separated shear layer behind a backward-facing step

In the present paper, we consider one of the most efficient and simple methods to additionally intensify the exchange processes and heat transfer in the separated flow behind a backward-facing step. The method uses small obstacles installed upstream the step; such obstacle act as turbulators smaller in size than the main obstacle. As the turbulators, solid mini ribs, comb ribbings, and wall-detached mini ribs were used. Intensification of the turbulent mixing process behind the main obstacle occurs due to the introduction of small-obstacle-induced 2D and 3D perturbations into the separated shear layer behind the step. Results of a detailed experimental study of the distributions of pressure and heat transfer for different heights of the small intensifier and its positions with respect to the step are reported. The influence of intensifier shape on the thermal and dynamic characteristics of the flow has been analyzed. The distributions of pressure and heat-transfer coefficients were used to evaluate the effectiveness of the various mini obstacles and the limits of their action on the drag and heat transfer.     

不同控制参数下的弹簧摆

通过拉格朗日方程得出弹簧摆系统的动力学方程,并以频率比作为控制参数,利用Matlab软件对不同控制参数和初始摆角下的弹簧摆进行了数值模拟,从而直观地研究了弹簧摆的运动.     

Chaotic behaviour and limit cycle behaviour of anharmonic systems with periodic external perturbations

The chaotic behaviour and limit cycle behviaiour of the dynamical system $\text{x}\text{?}\text{+ α}\text{x}\text{?}\text{+}\text{d}\text{V(x)}\text{d}\text{x}\text{= ?}\text{cos}\text{(Ωt)}$ is investigated for various potentials V and parameters values α, $\text{?}$ and $\text{Ω}$.