Experimental evidence for chaotic behaviour of a parametrically forced pendulum

Experimental evidence is presented for chaotic type nonperiodic motions of a parametrically forced pendulum. A bifurcation diagram is measured directly, showing successive subharmonic bifurcations to ?/4, onset of a periodic motion and the appearance of periodic motions via intermittency. The experimentally determined threshold values of the amplitude of the driving force for the first period doublings and the onset of a periodic motion are found to be in good agreement with the theoretical predictions.     

Transition to instability in a kicked bose-einstein condensate

A periodically kicked ring of a Bose-Einstein condensate is considered as a nonlinear generalization of the quantum kicked rotor. For weak interactions between atoms, periodic motion (antiresonance) becomes quasiperiodic (quantum beating) but remains stable. There exists a critical strength of interactions beyond which quasiperiodic motion becomes chaotic, resulting in an instability of the condensate manifested by exponential growth in the number of noncondensed atoms. Similar behavior is observed for dynamically localized states (essentially quasiperiodic motions), where stability remains for weak interactions but is destroyed by strong interactions.     

CHAOTIC AND NON-LINEAR DYNAMIC ANALYSIS OF A TWO-AXIS RATE GYRO WITH FEEDBACK CONTROL MOUNTED ON A SPACE VEHICLE

An analysis is presented in this paper for a two-axis rate gyro subjected to linear feedback control mounted on a space vehicle, which is spinning with uncertain angular velocity ω_z(t) about its spin of the gyro. For the autonomous case in which ω_z(t) is steady, the stability analysis of the system is studied by Routh-Hurwitz theory. For the non-autonomous case in which ω_z(t) is sinusoidal function, this system is a strongly non-linear damped system subjected to parametric excitation. By varying the amplitude of sinusoidal motion, periodic and chaotic responses of this parametrically excited non-linear system are investigated using the numerical simulation. Some observations on symmetry-breaking bifurcations, period-doubling bifurcations, and chaotic behavior of the system are investigated by various numerical techniques such as phase portraits, Poincaré maps, average power spectra, and Lyapunov exponents. In addition, some discussions about chaotic motions of this system can be suppressed and changed into regular motions by a suitable constant motor torque are included.     

Ionisation of metastable 3P-state hydrogen atom by electron with exchange effects

Propagation of nonlinear shock waves for the generalised Oskolkov equation and dynamic motions of the perturbed Oskolkov equation are investigated. Employing the unified method, a collection of exact shock wave solutions for the generalised Oskolkov equations is presented. Collocation finite element method is applied to the generalised Oskolkov equation for checking the accuracy of the proposed method by two test problems including the motion of shock wave and evolution of waves with Gaussian and undular bore initial conditions. Considering an external periodic perturbation, the dynamic motions of the perturbed generalised Oskolkov equation are studied depending on the system parameters with the help of phase portrait and time series plot. The perturbed generalised Oskolkov equation exhibits period-3, quasiperiodic and chaotic motions for some special values of the system parameters, whereas the generalised Oskolkov equation presents shock waves in the absence of external periodic perturbation.     

Regular and chaotic motions in applied dynamics of a rigid body

Periodic and regular motions, having a predictable functioning mode, play an important role in many problems of dynamics. The achievements of mathematics and mechanics (beginning with Poincare) have made it possible to establish that such motion modes, generally speaking, are local and form "islands" of regularity in a "chaotic sea" of essentially unpredictable trajectories. The development of computer techniques together with theoretical investigations makes it possible to study the global structure of the phase space of many problems having applied significance. A review of a number of such problems, considered by the authors in the past four or five years, is given in this paper. These include orientation and rotation problems of artificial and natural celestial bodies and the problem of controlling the motion of a locomotion robot. The structure of phase space is investigated for these problems. The phase trajectories of the motion are constructed by a numerical implementation of the Poincare point map method. Distinctions are made between regular (or resonance), quasiregular (or conditionally periodic), and chaotic trajectories. The evolution of the phase picture as the parameters are varied is investigated. A large number of "phase portraits" gives a notion of the arrangement and size of the stability islands in the "sea" of chaotic motions, about the appearance and disappearance of these islands as the parameters are varied, etc. (c) 1996 American Institute of Physics.     

双δ激光脉冲作用下Paul阱中单离子的规则与混沌运动

考虑赝势近似下囚禁于Paul阱中的单离子与双δ脉冲型周期势相互作用系统的规则与混沌运动.应用积分方程方法得到系统的经典运动精确解,通过数值方法作出相空间轨道图和平均能量的时间演化曲线.结合分析与数值结果,发现两个有趣的结论.即在离子与单δ脉冲作用出现共振失稳的情形,在双δ脉冲作用下却出现了稳定的规则运动;离子随着双δ脉冲中两个脉冲之间的时间间隔减小而由规则运动转为混沌运动,其平均能量扩散的快慢与混沌运动的混乱程度相关.还研究了系统的共振失稳,发现通过 关键词： 双δ脉冲 囚禁离子 精确解 混沌     

CHAOS, CHAOS CONTROL AND SYNCHRONIZATION OF ELECTRO-MECHANICAL GYROSTAT SYSTEM

The dynamic behavior of electro-mechanical gyrostat system subjected to external disturbance is studied in this paper. By applying numerical results, phase diagrams, power spectrum, Period-T maps, and Lyapunov exponents are presented to observe periodic and chaotic motions. The effect of the parameters changed in the system can be found in the bifurcation and parametric diagrams. Several methods, the delayed feedback control, adaptive control algorithm (ACA) control are used to control chaos effectively. Anticontrol of chaos destroyed the periodic motions and replaced by chaotic motion effectively by adding constant motor torque and adding periodic motor torque. Finally, synchronization of chaos in the electro-mechanical gyrostat system is studied.     

3-D spatial chaos in the elastica and the spinning top: Kirchhoff analogy

The existence of spatially chaotic deformations in an elastica and the analogous motions of a free spinning rigid body, an extension of the problem originally examined by Kirchhoff are investigated. It is shown that a spatially periodic variation in cross sectional area of the elastica results in spatially complex deformation patterns. The governing equations for the elastica were numerically integrated and Poincare maps were created for a number of different initial conditions. In addition, three dimensional computer images of the twisted elastica were generated to illustrate periodic, quasiperiodic, and stochastic deformation patterns in space. These pictures clearly show the existence of spatially chaotic deformations with stunning complexity. This finding is relevant to a wide variety of fields in which coiled structures are important, from the modeling of DNA chains to video and audio tape dynamics to the design of deployable space structures.     

Ordering chaos by random shortcuts

In this Letter, the effects of random shortcuts in an array of coupled nonlinear chaotic pendulums and their ability to control the dynamical behavior of the system are investigated. We show that random shortcuts can induce periodic synchronized spatiotemporal motions, even though all oscillators are chaotic when uncoupled. This process exhibits a nonmonotonic dependence on the density of shortcuts. Specifically, there is an optimal amount of random shortcuts, which can induce the most ordered motion characterized by the largest order parameter that is introduced to measure the spatiotemporal order. Our results imply that topological randomness can tame spatiotemporal chaos.     

Experiments on periodic and chaotic motions of a parametrically forced pendulum

An experimental study of periodic and chaotic type aperiodic motions of a parametrically harmonically excited pendulum is presented. It is shown that a characteristic route to chaos is the period-doubling cascade, which for the parametrically excited pendulum occurs with increasing driving amplitude and decreasing damping force, respectively. The coexistence of different periodic solutions as well as periodic and chaotic solutions is demonstrated and various transitions between them are studied. The pendulum is found to exhibit a transient chaotic behaviour in a wide range of driving force amplitudes. The transition from metastable chaos to sustained chaotic behaviour is investigated.     

Chaotic advection in a 2-D mixed convection flow

Two-dimensional numerical simulations of particle advection in a channel flow with spatially periodic heating have been carried out. The velocity field is found to be periodic above a critical Rayleigh number of around 18 000 and a Reynolds number of 10. Particle motion becomes chaotic in the lower half plane almost immediately after this critical value is surpassed, as characterized by the power spectral density and Poincare section of the flow. As the Rayleigh number is increased further, particle motion in the entire domain becomes chaotic. (c) 1995 American Institute of Physics.     

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.     

Nonlinear dynamics of higher-dimensional system for an axially accelerating viscoelastic beam with in-plane and out-of-plane vibrations

In this paper, the bifurcations and chaotic motions of higher-dimensional nonlinear systems are investigated for the nonplanar nonlinear vibrations of an axially accelerating moving viscoelastic beam. The Kelvin viscoelastic model is chosen to describe the viscoelastic property of the beam material. Firstly, the nonlinear governing equations of nonplanar motion for an axially accelerating moving viscoelastic beam are established by using the generalized Hamilton’s principle for the first time. Then, based on the Galerkin’s discretization, the governing equations of nonplanar motion are simplified to a six-degree-of-freedom nonlinear system and a three-degree-of-freedom nonlinear system with parametric excitation, respectively. At last, numerical simulations, including the Poincare map, phase portrait and Lyapunov exponents are used to analyze the complex nonlinear dynamic behaviors of the axially accelerating moving viscoelastic beam. The bifurcation diagrams for the in-plane and out-of-plane displacements via the mean axial velocity, the amplitude of velocity fluctuation and the frequency of velocity fluctuation are respectively presented when other parameters are fixed. The Lyapunov exponents are calculated to identify the existence of the chaotic motions. From the numerical results, it is indicated that the periodic, quasi-periodic and chaotic motions occur for the nonplanar nonlinear vibrations of the axially accelerating moving viscoelastic beam. Observing the in-plane nonlinear vibrations of the axially accelerating moving viscoelastic beam from the numerical results, it is found that the nonlinear responses of the six-degree-of-freedom nonlinear system are much different from that of the three-degree-of-freedom nonlinear system when all parameters are same.     

Controlling chaos based on an adaptive nonlinear compensator mechanism

The control problems of chaotic systems are investigated in the presence of parametric uncertainty and persistent external disturbances based on nonlinear control theory. By using a designed nonlinear compensator mechanism, the system deterministic nonlinearity, parametric uncertainty and disturbance effect can be compensated effectively. The renowned chaotic Lorenz system subjected to parametric variations and external disturbances is studied as an illustrative example. From the Lyapunov stability theory, sufficient conditions for choosing control parameters to guarantee chaos control are derived. Several experiments are carried out, including parameter change experiments, set-point change experiments and disturbance experiments. Simulation results indicate that the chaotic motion can be regulated not only to steady states but also to any desired periodic orbits with great immunity to parametric variations and external disturbances.     

ADDING ONE BIFURCATION IN A ROTOR SYSTEM WITH CLEARANCES

A mathematical model of a rotor system with clearances is analysed by the application of modern nonlinear dynamic theory. From the bifurcation diagrams, it is discovered that the rotor system alternates between periodic and chaotic motions at a supercritical rotational speed, and after undergoing a chaotic region the periodic number of the motion will increase by one. At the same time, the periodic number is equal correspondingly to the integral multiple of the critical rotational speed. At the subcritical rotational speed, it is discovered that the chaotic bands among successive orders of superharmonic responses return to the period one through a reversed period-doubling bifurcation, as a result of a period-doubling bifurcation. It is shown that the increase of damping may reduce the width of the chaotic bands and the amplitude of the periodic response; the increase of nonlinear degree also leads to the reduction of chaotic bandwidth, but makes the amplitude of the subharmonic response increase. So it is suggested that proper damping and correct material selection by considering the dynamic characteristics of the rotor system may reduce the proportion of faults and enhance the dynamic characteristics when designing the rotor system. The working speed should not be selected at N times its natural frequency and should not be set in the chaotic bands among the successive orders of periodic motion for the same purpose.     

Schedule and complex motion of shuttle bus induced by periodic inflow of passengers

We have studied the dynamic behavior of a bus in the shuttle bus transportation with a periodic inflow. A bus schedule is closely related to the dynamics. We present the modified circle map model for the dynamics of the shuttle bus. The motion of the shuttle bus depends on the loading parameter and the inflow period. The shuttle bus displays the periodic, quasi-periodic, and chaotic motions with varying both loading parameter and inflow rate.     

有界噪声与谐和激励联合作用下Duffing-Rayleigh振子的Melnikov混沌

研究了有界噪声与谐和激励作用下的Duffing-Rayleigh振子的动力学行为.首先运用随机Melnikov过程方法得到系统出现混沌的条件,结果表明随着非线性阻尼参数的增加系统会从混沌运动到周期运动,随着Wiener过程强度参数的增加,系统由混沌进入周期的临界幅值会先递增后不变.最后,用两类数值方法即最大Lyapunov指数与Poincare截面验证了上述结果. 关键词： 有界噪声 随机Melnikov过程 混沌运动 周期运动     

量子无规运动与核耗散

从动力学对称性观点出发考察了量子规则运动与无规运动 .用能级动力学研究了从量子规则运动向量子无规运动的过渡 ,给出了导致能级混沌的条件 ,揭示了造成能级混沌的机制 .用混沌态矢的特征解释了原子核的各态历经集体态的衰变特性 .研究了重离子碰撞中核耗散与动力学对称性破坏之间的关系. Quantum regular and irregular motions are investigated from the viewpoint of dynamical symmetry. The transition from quantum regular motion to chaotic motion is studied by level dynamics and computer experiments. The conditions for onset of quantum chaos are presented.The mechanism for causing chaotic level spectrum is unveiled. The decay behavior of the nuclear ergodic collective states is explained in terms of the peculiar property of chaotic states. The connection between nuclear...     

Paul阱中共线三离子体系的经典动力学

研究了在Paul阱囚禁场赝势作用下共线构形的三离子体系经典动力学特性.尽管这是一个非线性体系，但不存在混沌，即体系在任何能量下运动都是规则的，而相空间则由两个轨迹为对称和反对称周期(或准周期)轨道的KAM不变环面构成.体系的两条最简单的周期轨道S和A的周期随能量E的下降而增大，并在E趋于体系的最小值E_min=3.0时分别为反对称和对称谐振动. 关键词：     

The nonlinear dynamics of the damped and driven Toda chain: I. Energy bifurcation diagrams

The dynamics of a periodically driven damped Toda chain with periodic boundary conditions is investigated. With the help of energy cross sections it can be shown that besides periodic motions of different periodicity quasiperiodic and aperiodic (chaotic) motions exist. The method of energy bifurcation diagrams is introduced which allows a survey of the resonance properties of the chain.