Transition to chaos in a driven pendulum with nonlinear dissipation

The Melnikov-Holmes method is used to study the onset of chaos in a driven pendulum with nonlinear dissipation. Detailed numerical studies reveal many interesting features like a chaotic attractor at low frequencies, band formation near escape from the potential well and a sequence of subharmonic bifurcations inside the band that accumulates at the homoclinic bifurcation point.     

气轨上单摆的混沌实验

设计并自制了一个气轨上单摆的混沌演示装置,建立了系统的动力学方程,直观地显示了混沌的一些基本特征。     

一个电张弛振子中的瞬态激变

报道一例映孔导致激变后发生的奇异排斥子的拓扑突变.这种突变以其分数维的突变为标志 ，并引起激变后长混沌瞬态运动行为的突变，因而应该具有基础理论和实践上的重要意义. 关键词： 激变 奇异排斥子 混沌瞬态     

复摆振动中的混沌现象

利用组合物理实验仪,研究了复摆振动中的混沌现象,分析了产生混沌的物理条件,并给出不同条件下的混沌图形.     

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.     

Distinguishing the transition to chaos in a spherical pendulum

Complex responses are studied for a spherical pendulum whose support is excited with a translational periodic motion. Governing equations are studied analytically to allow prediction of responses under various excitation conditions. Stability for certain cases of damping is predicted by means of existing analysis and compared with experimental data. Numerical time-step integration of the governing equations is developed to predict responses for various types of excitation and damping conditions. Predicted results are compared with corresponding motions measured in an experimental spherical pendulum system. A data acquisition system is included whereby detailed digitized time histories of the pendulum motion can be established and various parameters can be computed to characterize the type of motion present. Two new vector spaces are defined for describing complex responses which occur for certain specified excitation conditions. It is shown in these parameter spaces that the transition from quasiperiodic to chaotic motions can be carefully quantified in systems with very light damping. This discovery provides a convenient means for comparison of complex motions in the numerical and experimental air pendulum systems. The implications of the results are important for dynamic response in various applications, including fluid motions in satellite tanks and other nonlinear time-dependent physical processes which include very light damping. (c) 1995 American Institute of Physics.     

单摆系统通向混沌的道路

对受迫非线性单摆系统进入混沌的道路进行了研究,发现单摆系统的运动是极其复杂的.目前在其他系统发现的进入混沌的通道,在该系统中几乎均可找到.这是一个介绍混沌运动的典型系统.     

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.     

Computational chaos in nonlinear optics

A few models of nonlinear optical systems, known experimentally to possess both stable and unstable dynamical modes, are approximated by different dynamical models and integrated by different numerical methods. It is shown that the onset of instabilities and chaotic behavior in the same physical system may be dependent on the model used and on the numerical method applied. Finite order difference schemes should be applied with caution to infinite dimensional dynamical systems displaying irregular behavior.     

Chaos in nonlinear paradoxical games

Summary Paradoxical games are nonconstant sum conflicts, where individual and collective rationalities are at variance and refer to a dyadic antagonism where the contestants blackmail each other. The state space dynamics of a class of such games has been previously studied in the planar case of two variablesx ₁,x ₂ (representing the propensities of the two parties to cooperate), for which phase space portraits have been obtained for a wide range of control parameters. In this paper, we extend the analysis to 3 dimensions, by allowing two of these parameters (the so-called ?tempting factors?) to oscillate in time. We observe on a Poincaré surface of section that the invariant manifolds of twounstable fixed pointsU ₁ andU ₁ intersect, and form heteroclinic and homoclinic orbits. Thus, sufficiently close toU ₁ andU ₂, one finds ?horseshoe? chaos and extremely sensitive dependence to initial conditions. Moreover, since the equations of motion can be written in Hamiltonian form, all the known phenomena of periodic, quasi-periodic and chaotic orbits can be observed around twostable fixed points, where the two parties become ?deadlocked? in an inconclusive exchange that never ends. To speed up publication, the authors of this paper have agreed to not receive the proofs for correction.     

Symmetry and chaos in the motion of the damped driven pendulum

The damped, driven pendulum equation is studied numerically. A relation is pointed out between the symmetry of the initial period-m dynamical state of am×2ⁿ period-doubling sequence and the form of the chaotic attractor for the final chaotic dynamical state reached after completion of the inverse-doubling sequence. Effects of extrinsic noise are also mentioned.     

Chaos without nonlinear dynamics

A linear, second-order filter driven by randomly polarized pulses is shown to generate a waveform that is chaotic under time reversal. That is, the filter output exhibits determinism and a positive Lyapunov exponent when viewed backward in time. The filter is demonstrated experimentally using a passive electronic circuit, and the resulting waveform exhibits a Lorenz-like butterfly structure. This phenomenon suggests that chaos may be connected to physical theories whose underlying framework is not that of a traditional deterministic nonlinear dynamical system.     

倒摆运动的混沌行为

设计一个受周期外力驱动的倒摆实验装置,并建立该系统的动力学方程,用线性稳定性分析方法讨论了平衡点附近邻域的稳定性,利用数值计算并结合多种分析方法求解非线性方程并判断解的性质．通过改变系统参数,画出时域图、相图及分岔图,计算分析和实验发现,这个简单的力学系统存在十分丰富的动力学行为（分岔、混沌）．理论分析、数值模拟和实验结果一致．     

Optical chaos in nonlinear photonic crystals

We examine the spatial evolution of lightwaves in a nonlinear photonic crystal with a quadratic nonlinearity, when a second harmonic and a sum-frequency generation are simultaneously quasi-phase-matched. We find the conditions for a transition to Hamiltonian chaos for different amplitudes of lightwaves at the crystal boundary.