Random excitation of a vibratory system with autoparametric interaction

The paper is concerned with the broad band random excitation of a two degree of freedom vibratory system with non-linear coupling of autoparametric type. A general equation for the evolution of the moments of any order of the response co-ordinates is derived by using stochastic calculus and found to represent an infinite hierarchy set. Consideration is given to the determination of the mean square stability boundary for unimodal response with no transverse motion of the coupled system. Two approximate solutions are obtained. These are first of all a solution based on a Gaussian closure technique applied to the system moment equations which allows the stability condition to be determined from the eigenvalues of a four by four matrix, and secondly a perturbation solution which leads to a simple analytical expression for the stability boundary. The two methods give results in close agreement for low values of system damping, but which differ appreciably at high damping levels. Finally, results are obtained from an investigation of the response regions of a laboratory model excited from a random noise generator. The experimental results are found to give excellent correlation with the predicted instability boundaries in the close neighbourhood of internal resonance but show a distinct indication of a wider instability region than predicted by both analytical methods.     

Dynamics of a nonlinear parametrically excited partial differential equation

We investigate a parametrically excited nonlinear Mathieu equation with damping and limited spatial dependence, using both perturbation theory and numerical integration. The perturbation results predict that, for parameters which lie near the 2:1 resonance tongue of instability corresponding to a single mode of shape cos nx, the resonant mode achieves a stable periodic motion, while all the other modes are predicted to decay to zero. By numerically integrating the p.d.e. as well as a 3-mode o.d.e. truncation, the predictions of perturbation theory are shown to represent an oversimplified picture of the dynamics. In particular it is shown that steady states exist which involve many modes. The dependence of steady state behavior on parameter values and initial conditions is investigated numerically. (c) 1999 American Institute of Physics.     

Frequency modulation indicator, Arnold’s web and diffusion in the Stark-Quadratic-Zeeman problem

We notice that the fundamental frequencies of a slightly perturbed integrable Hamiltonian system are not time-constant inside a resonance but frequency modulated, as is evident from pendulum models and wavelet analysis. Exploiting an intrinsic imprecision inherent to the numerical frequency analysis algorithm itself, hence transforming a drawback into an opportunity, we define the Frequency Modulation Indicator, a very sensitive tool in detecting where fundamental frequencies are modulated, localizing so the resonances without having to resort, as in other methods, to the integration of variational equations. For the Kepler problem, the space of the orbits with a fixed energy has the topology of the product of two 2-spheres. The perturbation Hamiltonian, averaged over the mean anomaly, has surely a maximum and a minimum, to which correspond two periodic orbits in physical space. Studying the neighbourhood of these two elliptic stable points, we are able to define adapted action-angle variables, for example, the usual but “SO(4)-rotated” Delaunay variables. The procedure, implemented in the program KEPLER, is performed transparently for the user, providing a general scheme suited for generic perturbation. The method is then applied to the Stark-Quadratic-Zeeman problem, displaying very clearly the Arnold web of the resonances. Sectioning transversely one of the resonance strips so highlighted and performing a numerical frequency analysis, one is able to locate with great precision the thin stochastic layer surrounding a separatrix. Another very long (10⁸ revolutions) frequency analysis on an orbit starting here reveals, as expected, a well defined pattern, which ensures that the integration errors do not eject the point out of the layer, and moreover a very slow drift in the frequency values, clearly due to Arnold diffusion.     

PASTOCHASTIC INSTABILITY IN HOT ELECTRON PLASMA

Secular perturbation theory is used to describe the hot electron motion in an inhomogeneous magnetic field. A group of precessional modes is investigated. A criterion for determining the stochastic instability is obtained. It is found that higher precessional frequencies have more stabilizing effects.     

Some new systems that generate a uniform stochastic web

Evidence is given that many classes of periodically kicked Hamiltonian system with 1.5 degree of freedom generate infinite, uniform stochastic webs. The kick term in the Hamiltonian or the equation of motion need not be purely sinusoidal or some small perturbation of a sinusoidal function. For the resonance condition q=4 the structure of the web can be different from a square lattice; However, remarkably symmetric patterns of chaos are still present throughout the whole phase space. Examples are given for the square wave function and sawtooth function in the kick term of the equation of motion. The sensitive dependence on initial conditions of those systems is investigated.     

Muscle‘s Motion in an Overdamped Regime

Based on the stochastic inclined rods model proposed by H.Matsuura et al., we study the motion of actin myosin system in an overdamped regime.Our model is composed of an inclined spring (rod),a myosin head and a myosin filament.The results of calculation show that the model can convert the random motion to one-directional motion,and the myosin head works as a resonator of random noise,which absorbs the energy through a stochastic resonance.The results show that the inclined rod and the intermolecular potential are very important for the system to move.     

偶极场扰动与冷却效应对同步运动的影响

在同步加速器特别是冷却存储环中,偶极场扰动和冷却效应将会对粒子的Betatron振荡和同步振荡产生影响。 叙述了偶极场扰动对粒子同步运动的影响与高频相位调制等效, 重点研究了偶极场扰动与冷却效应同时存在的情况对粒子的同步运动产生的影响。 在经典力学框架内把粒子的同步运动方程化为广义的摆方程, 然后利用Melnikov方法对系统的稳定性进行了分析, 讨论了系统进入Smale马蹄混沌的物理意义,并导出了系统稳定的临界条件。根据稳定性判据, 给出了系统稳定性所需要的偶极场扰动的高频相位调制幅度和冷却系数的限制条件。 结果表明, 系统的稳定性与它的参数有关, 只需适当调节这些参数, 混沌便可原则上控制或避免。 The Betatron oscillation and the synchrotron oscillation for the particles are effected by the dipole field perturbation and the cooling effect. Both effects are considered in the paper and the synchrotron motion equation of the particle in the synchroton is reduced to the general pendulum equation in the classical mechanics frame. The stabilities of the system caused by dipole field perturbation and cooling are analyzed by using Melnikov method. The thresholds entered a Smale horseshoe chaos are derived in detail and the stability conditions for the system caused by dipole field perturbation and cooling are dicussed. The results show that the critical condition or the stability thresholds are related to the system parameters. The chaos or an instability can be avoided or controlled in principle by regulating some parameters of the system.     

Duffing系统随机相位抑制混沌与随机共振并存现象的机理研究

针对随机相位作用的Duffing混沌系统, 研究了随机相位强度变化时系统混沌动力学的演化行为及伴随的随机共振现象. 结合Lyapunov指数、庞加莱截面、相图、时间历程图、功率谱等工具, 发现当噪声强度增大时, 系统存在从混沌状态转化为有序状态的过程, 即存在噪声抑制混沌的现象, 且在这一过程中, 系统亦存在随机共振现象, 而且随机共振曲线上最优的噪声强度恰为噪声抑制混沌的参数临界点. 通过含随机相位周期力的平均效应分析并结合系统的分岔图, 探讨了噪声对混沌运动演化的作用机理, 解释了在此过程中随机共振的形成机理, 论证了噪声抑制混沌与随机共振的相互关系.     

Signal amplification in NbN superconducting resonators via stochastic resonance

We exploit nonlinearity in NbN superconducting stripline resonators, which originates from local thermal instability, for studying stochastic resonance. As the resonators are driven into instability, small amplitude modulation (AM) signals are amplified with the aid of injected white noise. Simulation results based on the equations of motion for the system yield a good agreement with the experimental data both in the frequency and time domains.     

过阻尼搓板势系统的随机共振

研究在周期信号和高斯白噪声共同作用下过阻尼搓板势系统的随机共振.由于用直接模拟法研究随机系统所用时间较多,考虑用半解析的方法对系统的随机共振现象进行研究.在弱周期信号极限下,结合线性响应理论和扰动展开法提出一种计算系统线性响应的矩方法.在此基础上,利用Floquet理论和非扰动展开法将矩方法扩展到系统非线性响应的计算.通过直接数值模拟结果和矩方法所得结果的比较展示了矩方法的有效性并采用均方差作为量化指标给出其适用的参数范围.研究结果表明,以系统的功率谱放大因子作为量化指标,发现在适当的参数条件下,系统的共振曲线有一个单峰出现,说明过阻尼搓板势系统存在随机共振现象.而且在一定范围内调节偏置参数时,共振曲线的峰值随偏置参数的增大而增大;在调节驱动幅值时,随机共振效应随驱动幅值的增大而增强.     

Master equations for time correlation functions of a quantum system interacting with stochastic perturbations and applications to emission and absorption line shapes

The stochastic and quantum dynamics of open quantum systems interacting with stochastic perturbations in considered. The master equations for one time and multi-time correlation functions of such a system are derived to all orders in the interaction with the stochastic perturbations. The importance of the non-markovian character of such equations in the study of various problems in optical resonance is discussed. The simplified form of the non-markovian master equations in Born approximation is also given. It is shown that such non-markovian master equations in Born approximation are exact if there is only one random perturbation, of the telegraphic signal type, acting on the system. The master equations for the linear response functions of an open system interacting with stochastic perturbations are also derived. The non-markovian master equations for multitime correlations are used to study the behaviour of two level atoms interacting with fluctuating laser fields. Both amplitude and phase fluctuations are taken into account. Explicit results are presented for the spectrum of resonance fluorescence, absorption spectrum, photon antibunching effects etc. The calculations are done for arbitrary values of the relaxation parameters and intial conditions. In general the fluorescence spectrum is found to be asymmetric for off resonant fields.     

Thermal roughening of a thin film: A new type of roughening transition

It is found that an m = n = 1 mode with amplitude exceeding a certain threshold can lead to stochastic motion of energetic ions in tokamaks, the large orbit width particles (potatoes) being most easily affected. An n = 1 mode can redistribute particles also in the absence of stochasticity but only when the perturbation is quickly switched on/off, e.g., during a sawtooth crash. In the latter case, the perturbation results in a regular motion of particles around a certain helical orbit, at which a resonance driven by the mode but having no amplitude threshold takes place.     

Parametric resonance of truncated conical shells rotating at periodically varying angular speed

Parametric resonance of a truncated conical shell rotating at periodically varying angular speed is studied in this paper. Based upon the Love?s thin shell theory and generalized differential quadrature (GDQ) method, the equations of motion of a rotating conical shell are derived. The time-dependent rotating speed is assumed to be a small and sinusoidal perturbation superimposed upon a constant speed. Considering the periodically rotating speed, the conical shell system is a parametric excited system of the Mathieu–Hill type. The improved Hill?s method is utilized for parametric instability analysis. Both the primary and combination instability regions for various natural modes and boundary conditions are obtained numerically. The effects of relative amplitude and constant part of periodically rotating speed and cone angle on the instability regions are discussed in detail. It is shown that for the natural mode with lower circumferential wavenumber, only the primary instability regions exist. With the increasing circumferential wavenumber, the instability widths are reduced significantly and the combination instability region might appear. The results for different boundary conditions are substantially similar. Increasing the constant rotating speed (or cone angle) all lead to the movements of instability regions and the appearance of combination instability region. The former will cause the instability width increasing, while the latter will reduce the instability width. The variation of length-to-radius ratio only causes the movements of instability regions.     

Effect of in-plane inertia on buckling of imperfect plates with large deformations

The influence of in-plane inertia on non-linear dynamic buckling of rectangular initially-imperfect plates is studied. A regular, consistent perturbation technique is used, and both the equations of motion and the boundary conditions are perturbed, yielding a relatively simple procedure for solving an otherwise very involved non-linear problem. Both pulse and vibration buckling during parametric resonance are analyzed. It is shown that, in the former case, the in-plane inertia can be disregarded whereas in the latter it plays an important role for non-slender plates.     

二维不可压流体Kelvin-Helmholtz不稳定性的弱非线性研究

通过将扰动速度势展至三阶,提出了Kelvin-Helmholtz（KH）不稳定性的弱非线性理论.在模耦合过程中观察到一个重要的共振现象,共振使得模耦合过程变得相当复杂,单模扰动很快进入非线性区,产生大量高次谐波,共振加强了非线性作用.分析了单模扰动中二次和三次谐波产生效应,以及对基模指数增长的非线性校正.模拟结果支持了解析理论.利用该理论,分析了KH不稳定的非线性阈值问题. 关键词： Kelvin-Helmholtz不稳定性 弱非线性理论 非线性阈值     

Nearly linear mappings and their applications

A simple 2-dimensional mapping is considered, both analytically and numerically, for which all nonlinear effects are of the same order as the perturbations and of the same origin. Properties of the stochastic instability are investigated, taking the beam-beam interaction in a storage ring as an important particular example of a dynamic system that can be modelled with such a mapping. The special case of time-dependent mappings is discussed. It is shown that low-frequency time dependence sharply decreases the critical perturbation strength for the stochastic transition.     

Regular and chaotic dynamics of the dipole moment of square dipole arrays

Dipole lattices, which represent square dipole arrays, are investigated. Various types of equilibrium configurations of arrays are obtained, and conditions are shown under which these configurations are established. On the basis of parametric bifurcation diagrams, the main types of regular and chaotic oscillation regimes of the total dipole moment of a system are considered and their dependence on the amplitude, frequency, and polarization of an alternating field, as well as on the initial equilibrium configuration of arrays, is analyzed. Scenarios of the onset of chaotic regimes are demonstrated, including those that occur via the establishment and variation of quasiperiodic oscillations of the dipole moment of a system. The dynamic bistability state is revealed in which a stochastic resonance—an increase in the response of a system to a harmonic signal in the presence of noise—can be implemented.     

Symmetry of quantum phase space in a degenerate Hamiltonian system

The structure of the global "quantum phase space" is analyzed for the harmonic oscillator perturbed by a monochromatic wave in the limit when the perturbation amplitude is small. Usually, the phenomenon of quantum resonance was studied in nondegenerate [G. M. Zaslavsky, Chaos in Dynamic Systems (Harwood Academic, Chur, 1985)] and degenerate [Demikhovskii, Kamenev, and Luna-Acosta, Phys. Rev. E 52, 3351 (1995)] classically chaotic systems only in the particular regions of the classical phase space, such as the center of the resonance or near the separatrix. The system under consideration is degenerate, and even an infinitely small perturbation generates in the classical phase space an infinite number of the resonant cells which are arranged in the pattern with the axial symmetry of the order 2&mgr; (where &mgr; is the resonance number). We show analytically that the Husimi functions of all Floquet states (the quantum phase space) have the same symmetry as the classical phase space. This correspondence is demonstrated numerically for the Husimi functions of the Floquet states corresponding to the motion near the elliptic stable points (centers of the classical resonance cells). The derived results are valid in the resonance approximation when the perturbation amplitude is small enough, and the stochastic layers in the classical phase space are exponentially thin. The developed approach can be used for studying a global symmetry of more complicated quantum systems with chaotic behavior. (c) 2000 American Institute of Physics.     

Stochastic resonance and nonequilibrium dynamic phase transition of Ising spin system driven by a joint external field

The dynamic response and stochastic resonance of a kinetic Ising spin system （ISS） subject to the joint action of an external field of weak sinusoidal modulation and stochastic white-nolse are studied by solving the mean-field equation of motion based on Glauber dynamics. The periodically driven stochastic ISS shows that the characteristic stochastic resonance as well as nonequilibrium dynamic phase transition （NDPT） occurs when the frequency ω and amplitude h0 of driving field, the temperature t of the system and noise intensity D are all specifically in accordance with each other in quantity. There exist in the system two typical dynamic phases, referred to as dynamic disordered paramagnetic and ordered ferromagnetic phases respectively, corresponding to a zero- and a unit-dynamic order parameter. The NDPT boundary surface of the system which separates the dynamic paramagnetic phase from the dynamic ferromagnetic phase in the 3D parameter space of ho-t-D is also investigated. An interesting dynamical ferromagnetic phase with an intermediate order parameter of 0.66 is revealed for the first time in the ISS subject to the perturbation of a joint determinant and stochastic field. The intermediate order dynamical ferromagnetic phase is dynamically metastable in nature and owns a peculiar characteristic in its stability as well as the response to external driving field as compared with a fully order dynamic ferromagnetic phase.     

Disturbances of Rotational Motions for String Models of Hadrons and the Stability Problem

Slight disturbances of a classical rotational motion (uniform rotation of a system) are considered for a relativistic string with massive ends and for the q-q-q and Y baryon string models. It is shown that for a string with massive ends this motion is stable in the linear approximation and the slight perturbations are representable by a series each term of which describes a standing wave of certain frequency. These modes make it possible to simulate various excited states of hadrons. At the same time, for the q-q-q and Y baryon string models the instability of rotational motions has been proved: exponentially growing modes have been detected in their perturbation spectra.