一类含时变间隙的强非线性相对转动系统分岔和混沌

建立一类具有时变间隙的两质量相对转动系统的强非线性动力学方程.应用MLP方法求解出变换参数,并运用多尺度法求解该系统发生1/2亚谐共振时的分岔响应方程,采用奇异性理论分析得到系统稳态响应的转迁集,并且得到系统在非自治情形下的分岔特性以及系统的分岔形态.最后通过数值仿真得到系统在间隙和阻尼参数变化下的分岔和混沌行为,发现随着系统参数变化系统将出现周期运动、倍周期运动以及混沌等多种不同的运动形态.     

Modulated amplitude waves and the transition from phase to defect chaos

The mechanism for transitions from phase to defect chaos in the one-dimensional complex Ginzburg-Landau equation (CGLE) is presented. We describe periodic coherent structures of the CGLE, called modulated amplitude waves (MAWs). MAWs of various periods P occur in phase chaotic states. A bifurcation study of the MAWs reveals that for sufficiently large period, pairs of MAWs cease to exist via a saddle-node bifurcation. For periods beyond this bifurcation, incoherent near-MAW structures evolve towards defects. This leads to our main result: the transition from phase to defect chaos takes place when the periods of MAWs in phase chaos are driven beyond their saddle-node bifurcation.     

一个简单延迟非线性系统的动力学行为及混沌同步

分析一个简单二阶延迟系统的Hopf分支和混沌特性, 包括分支点、分支方向和分支周期解的稳定性, 解析求出退延迟情况下, 这个系统的相轨线方程; 通过数值计算并绘制分岔图, 揭示系统存在由倍周期通向混沌的道路; 利用单路线性组合信号, 反馈控制实现系统的部分完全同步; 利用主动-被动与线性反馈的联合, 实现系统的完全同步; 设计和搭建系统的电子实验线路, 并从实验中观测到与理论分析或数值计算相一致的结果. 关键词： 延迟非线性系统 电路实验 Hopf分支 混沌     

Dynamic characteristics in an external-cavity multi-quantum-well laser

This paper outlines our studies of bifurcation, quasi-periodic road to chaos and other dynamic characteristics in an external-cavity multi-quantum-well laser with delay optical feedback. The bistable state of the laser is predicted by finding theoretically that the gain shifts abruptly between two values due to the feedback. We make a linear stability analysis of the dynamic behavior of the laser. We predict the stability scenario by using the characteristic equation while we make an approximate analysis of the stability of the equilibrium point and discuss the quantitative criteria of bifurcation. We deduce a formula for the relaxation oscillation frequency and prove theoretically that this formula function relates to the loss of carriers transferring between well regime and barrier regime, the feedback level, the delayed time and the other intrinsic parameters. We demonstrate the dynamic distribution and double relaxation oscillation frequency abruptly changing in periodic states and find the multi-frequency characteristic in a chaotic state. We illustrate a road to chaos from a stable state to quasi-periodic states by increasing the feedback level. The effects of the transfers of carriers and the escaping of carriers on dynamic behavior are analyzed, showing that they are contrary to each other via the bifurcation diagram. Also,we show another road to chaos after bifurcation through changing the linewidth enhancement factor, the photon loss rate and the transfer rate of carriers.     

Subharmonic bifurcations and chaotic dynamics of an air damping completely inelastic bouncing ball

We investigate the dynamics of a plastic ball on a vibrated platform in air by introducing air damping effect into the completely inelastic bouncing ball model. The air damping gives rise to larger saddle-node bifurcation points and a chaos confirmed by the largest Lyapunov exponent of a one-dimensional discrete mapping. The calculated bifurcation point distribution shows that the periodic motion of the ball is suppressed and a chaos emerges earlier for an increasing air damping. When the reset mechanism and the linear stability which cause periodic motion of the ball both collapse, the investigated system is fully chaotic.     

Generalized Ginzburg-Landau equation for self-pulsing instability in a two-photon laser

A nonlinear analysis is made for a degenerate two-photon ring laser near its critical point corresponding to a self-pulsing instability by using the slaving principle and normal form theory. It turns out that the system undergoes two kinds of transitions, a usual Hopf bifurcation to a stable or unstable limit cycle and a co-dimension two Hopf bifurcation where the limit cycles disappear. An analytical criterion is given to distinguish the super-from the sub-critical bifurcation. We have also solved the equations numerically to confirm and to supplement our analytical results. In the case of super-critical bifurcation, a period-doubling bifurcation sequence to chaos is also observed with the decrease in pumping.     

Two routes to chaos in the fractional Lorenz system with dimension continuously varying

The object of this paper is to reveal the relation between dynamics of the fractional system and its dimension defined as a sum of the orders of all involved derivatives. We take the fractional Lorenz system as example and regard one or three of its orders as bifurcation parameters. In this framework, we compute the corresponding bifurcation diagrams via an optimal Poincaré section technique developed by us and find there exist two routes to chaos when its dimension increases from some values to 3. One is the process of cascaded period-doubling bifurcations and the other is a crisis (boundary crisis) which occurs in the evolution of chaotic transient behavior. We would like to point out that our investigation is the first to find out that a fractional differential equations (FDEs) system can evolve into chaos by the crisis. Furthermore, we observe rich dynamical phenomena in these processes, such as two-stage cascaded period-doubling bifurcations, chaotic transients, and the transition from coexistence of three attractors to mono-existence of a chaotic attractor. These are new and interesting findings for FDEs systems which, to our knowledge, have not been described before.     

Non-Smooth Bifurcation and Chaos in a DC-DC Buck Converter

A direct-current-direct-current （DC-DC） buck converter with integrated load current feedback is studied with three kinds of Poincaré maps. The external corner-collision bifurcation occurs when the crossing number per period varies, and the internal corner-collision bifurcations occur along with period-doubling and period-tripling bifurcations in this model. The multi-band chaos roots in external corner-collision bifurcation and often grows into 1-band chaos. A new kind of chaotic sliding orbits, which is more complex for non-smooth systems, is also found in this model.     

Bifurcation transitions in an optically injected diode laser: theory and experiment

We show how bifurcation theory and experimental measurements can be used hand-in-hand to analyse transitions to complicated dynamics in a semiconductor laser subject to optical injection. By a direct comparison of theoretical and experimental optical spectra we identify and explain the underlying dynamics in phase space. This is demonstrated with four distinct bifurcation transitions, including a transition near a saddle-node Hopf point and an intermittent transition to chaos.     

The chaotic behavior analysis near the points B_3～B_5 of a nonlinear Fabry-Perot cavity

In this paper a further proof is given to the instability of 2~n bifurcation near the point B_3. The chaos in B_3 seems to arrive via the route of intermittent and tangential bifurcation. It is also shown that a metastability chaos occurs in the transition from B_4 to B_5.     

Dynamical behaviors of the shock compacton in the nonlinearly Schrödinger equation with a source term

In this paper, the dynamics from the shock compacton to chaos in the nonlinearly Schrödinger equation with a source term is investigated in detail. The existence of unclosed homoclinic orbits which are not connected with the saddle point indicates that the system has a discontinuous fiber solution which is a shock compacton. We prove that the shock compacton is a weak solution. The Melnikov technique is used to detect the conditions for the occurrence from the shock compacton to chaos and further analysis of the conditions for chaos suppression. The results show that the system turns to chaos easily under external disturbances. The critical parameter values for chaos appearing are obtained analytically and numerically using the Lyapunov exponents and the bifurcation diagrams.     

Fluctuations and simple chaotic dynamics

We describe the effects of fluctuations on the period-doubling bifurcation to chaos. We study the dynamics of maps of the interval in the absence of noise and numerically verify the scaling behavior of the Lyapunov characteristic exponent near the transition to chaos. As previously shown, fluctuations produce a gap in the period-doubling bifurcation sequence. We show that this implies a scaling behavior for the chaotic threshold and determine the associated critical exponent. By considering fluctuations as a disordering field on the deterministic dynamics, we obtain scaling relations between various critical exponents relating the effect of noise on the Lyapunov characteristic exponent. A rule is developed to explain the effects of additive noise at fixed parameter value from the deterministic dynamics at nearby parameter values.     

The instability of chaotic synchronization in coupled Lorenz systems: from the Hopf to the Co-dimension two bifurcation

We investigate the Hopf bifurcation of the synchronous chaos in coupled Lorenz oscillators. We find that the system undergoes a phase transition along the Hopf instability of the synchronous chaos. The phase transition makes the traveling wave component with the phase difference φ(i)-φ(i+1)=2π/N between adjacent sites unstable. The phase transition also plays a role to relate the Hopf bifurcation with the co-dimension two bifurcation of the synchronous chaos.     

Rolling bearing vibrations—The effects of geometrical imperfections and wear

The geometrical shape and surface properties of the components of rolling bearings will always deviate to some extent from their theoretical design. For bearings of standard tolerances these deviations are large enough to cause measurable levels of vibrations when the bearing is in operation. The purpose of this paper is to show in some detail how these surface irregularities are related to the vibration characteristics of the bearing. The study is restricted to radial bearings, having a radial load and a positive clearance. The approximate methods used render the results useful mainly for lightly loaded bearings operating at low and moderate speeds. Attention has been focused on the effects of inner ring waviness and non-uniform diameters of the rolling elements. A mixed theoretical and experimental impedance approach has been used to treat the bearing when fitted in a simple machine structure, thereby showing how resulting vibrations of the bearing pedestal can be calculated, with account taken of the effects of bearing, rotor and foundation properties. During operation bearings undergo progressive surface and subsurface deterioration. These alterations of geometrical and surface properties of bearing components will always be accompanied by some degree of change of the vibrations characteristics of the bearing. Two common modes of surface deterioration—spalling fatigue and abrasive wear—have been studied, the practical objective being to highlight some possible methods of condition monitoring and prediction of impending bearing failure.     

Bifurcation to chaos in clocked digital systems containing autonomous timing circuits

Observations of chaos and period doubling in a repetitively triggered monostable multivibrator circuit are reported, with the time between trigger pulses as the bifurcation parameter. A theory is presented which predicts the first bifurcation point. Measurements of an experimental circuit confirm the predictions of this theory. The consequences for concurrent digital systems are briefly considered.     

频率扫描法研究倍周期分岔与混沌现象

在一类非线性系统中,应用频率控制方法,对倍周期分岔与混沌行为进行了研究。在V₀-ω外控参数平面上,频率扫描显示了分岔与混沌的整体结构:正的和逆的倍周期分岔序列的对称性;分岔收敛于一点的封闭性。本文中所建议的方法,将是一种研究分岔与混沌现象有效而快速的手段。它不仅能定量测量收敛比δ和标度因子α,分段展开还能定性地观察阵发混沌和嵌套在混沌带中的各种窗口等。分岔与混沌是一类非线性系统的频率响应。 关键词：     

Spatial period-doubling in open flow

Coupled logistic lattices with asymmetric coupling in space, with a fixed boundary condition at the left end, are investigated. The system shows a period-doubling bifurcation to chaos as a lattice point goes downflow. In contrast with usual period-doubling in low-dimensional systems, (i) no scaling behavior has been found, (ii) low noise is important for the bifurcation structures. The system corresponds to a model for an open flow, which may be of use for the study of the onset of turbulence in pipe flows.     

Hopf bifurcation and uncontrolled stochastic traffic- induced chaos in an RED-AQM congestion control system

We study the Hopf bifurcation and the chaos phenomena in a random early detection-based active queue management (RED-AQM) congestion control system with a communication delay. We prove that there is a critical value of the communication delay for the stability of the RED-AQM control system. Furthermore, we show that the system will lose its stability and Hopf bifurcations will occur when the delay exceeds the critical value. When the delay is close to its critical value, we demonstrate that typical chaos patterns may be induced by the uncontrolled stochastic traffic in the RED-AQM control system even if the system is still stable, which reveals a new route to the chaos besides the bifurcation in the network congestion control system. Numerical simulations are given to illustrate the theoretical results.     

Nonlinear parametrically excited vibration and active control of gear pair system with time-varying characteristic

In the present work, we investigate the nonlinear parametrically excited vibration and active control of a gear pair system involving backlash, time-varying meshing stiffness and static transmission error. Firstly, a gear pair model is established in a strongly nonlinear form, and its nonlinear vibration characteristics are systematically investigated through different approaches. Several complicated phenomena such as period doubling bifurcation, anti period doubling bifurcation and chaos can be observed under the internal parametric excitation. Then, an active compensation controller is designed to suppress the vibration, including the chaos. Finally, the effectiveness of the proposed controller is verified numerically.     

有内共振时的扬声器分谐波和混沌

研究了当轴对称模态由驱动力共振激发,并且轴对称模态和非轴对称模态存在2:1内共振时的扬声器辐射体薄壳的分谐波和昆沌。采用多尺度法分析了非线性模态方程的稳态解及其稳定性,由此进一步确定了驱动频率和驱动力平面上的分岔集。给出了所考虑情形下扬声器分谐波的阈值电压公式,该阈值电压低于无内共振时的阈值电压。除出现非轴对称模态的1/2分谐波振动外,2个模态的振幅经Hopf分岔后作极限环运动,并经倍周期分岔进入混沌运动。混沌出现是由于2个模态间能量的强烈交换。理论结果和实验结果基本吻合,该一致性表明了所建扬声器非线性薄壳模型的正确性。