神经网络混沌运动的控制--CM方法

基于压缩映射的混沌控制方法——CM方法被应用到小的离散神经网络,通过一个外部输入的小干扰,稳定混沌轨道嵌入在混沌吸引子内的某一不稳周期轨上。利用闭回路对技术估计欲稳定周期轨的近似位置。给出二维和三维神经网络的典型例子,通过数值模拟显示CM方法控制离散神经网络混沌行为的简单和有效性。     

欧拉动弯曲问题的混沌控制

用周期激振力法、参数共振微扰法和延时反馈控制方法对欧拉动弯曲问题的混沌行为进行了控制，通过计算受控系统的Ｌｙａｐｕｎｏｖ指数来确定控制参数，得到受控后系统的高周期、低周期和准周期等稳定周期振动结果，。     

离散达芬映射中由边界激变所诱发的复杂的张弛振荡

多时间尺度问题具有广泛的工程与科学研究背景,慢变参数则是多时间尺度问题的典型标志之一.然而现有文献所报道的慢变参数问题,其展现出的振荡形式及内部分岔结构,大多较为单一,此外少有文献涉及到混沌激变的现象.本文以含慢变周期激励的达芬映射为例,探讨了一类具有复杂分岔结构的张弛振荡.快子系统的分岔表现为S形不动点曲线,其上、下稳定支可经由倍周期分岔通向混沌.而在一定的参数条件下,存在着导致混沌吸引子突然消失的一对临界参数值.当分岔参数达到此临界值时,混沌吸引子可能与不稳定不动点相接触,也可能与之相距一定距离.对快子系统吸引域分布的模拟,表明存在着导致边界激变(boundary crisis)的临界值,在这些值附近,经由延迟倍周期分岔演化而来的混沌吸引子可与2~n(n=0,1,2,···)周期轨道乃至混沌吸引子共存.当慢变量周期地穿过临界点后,双稳态的消失导致原本处于混沌轨道的轨线对称地向此前共存的吸引子转迁,从而使系统出现了不同吸引子之间的滞后行为,由此产生了由边界激变所诱发的多种对称式张弛振荡.本文的结果丰富了对离散系统的多时间尺度动力学机理的认识.     

COMPLEX RELAXATION OSCILLATION TRIGGERED BY BOUNDARY CRISIS IN THE DISCRETE DUFFING MAP

多时间尺度问题具有广泛的工程与科学研究背景，慢变参数则是多时间尺度问题的典型标志之一.然而现有文献所报道的慢变参数问题，其展现出的振荡形式及内部分岔结构，大多较为单一，此外少有文献涉及到混沌激变的现象.本文以含慢变周期激励的达芬映射为例，探讨了一类具有复杂分岔结构的张弛振荡.快子系统的分岔表现为S形不动点曲线，其上、下稳定支可经由倍周期分岔通向混沌.而在一定的参数条件下，存在着导致混沌吸引子突然消失的一对临界参数值.当分岔参数达到此临界值时，混沌吸引子可能与不稳定不动点相接触，也可能与之相距一定距离.对快子系统吸引域分布的模拟，表明存在着导致边界激变（boundary crisis）的临界值，在这些值附近，经由延迟倍周期分岔演化而来的混沌吸引子可与2ⁿ（n=0，1，2，…）周期轨道乃至混沌吸引子共存.当慢变量周期地穿过临界点后，双稳态的消失导致原本处于混沌轨道的轨线对称地向此前共存的吸引子转迁，从而使系统出现了不同吸引子之间的滞后行为，由此产生了由边界激变所诱发的多种对称式张弛振荡.本文的结果丰富了对离散系统的多时间尺度动力学机理的认识.     

齿轮传动系统共存吸引子的不连续分岔

大量的多吸引子共存是引起齿轮传动系统具有丰富动力学行为的一个重要因素.多吸引子共存时,运动工况的变化以及不可避免的扰动都可能导致齿轮传动系统在不同运动行为之间跳跃变换,对整个机器产生不良的影响.目前,一些隐藏的吸引子没有被发现,共存吸引子的分岔演化规律没有被完全揭示.考虑单自由度直齿圆柱齿轮传动系统,构建由局部映射复合的Poincaré映射,给出Jacobi矩阵特征值计算的半解析法.应用数值仿真、延拓打靶法和Floquet特征乘子求解共存吸引子的稳定性与分岔,应用胞映射法计算共存吸引子的吸引域,讨论啮合频率、阻尼比和时变激励幅值对系统动力学的影响,揭示齿轮传动系统倍周期型擦边分岔、亚临界倍周期分岔诱导的鞍结分岔和边界激变等不连续分岔行为.倍周期分岔诱导的鞍结分岔引起相邻周期吸引子相互转迁的跳跃与迟滞,使倍周期分岔呈现亚临界特性.鞍结分岔是共存周期吸引子出现或消失的主要原因.边界激变引起混沌吸引子及其吸引域突然消失,对应周期吸引子的分岔终止.     

参激屈曲梁的倍周期分岔和混沌运动的实验研究

本文对一端固定一端滑动承受轴向简谐载荷的屈曲梁的非线性响应进行了实验。研究了其基本参数共振和主参数共振两种情况，揭示了系统的倍周期分岔和混沌运动等复杂动力学行为。在某些情况下，混沌吸引子和周期吸引子共存，另一些则存在间歇混沌。给出了响应的时间历程、相图、频率谱和Ｐｏｉｎｃａｒｅ映射     

参数激励耦合系统的复杂动力学行为分析

分析了耦合van der Pol振子参数共振条件下的复杂动力学行为．基于平均方程，得到了参数平面上的转迁集，这些转迁集将参数平面划分为不同的区域，在各个不同的区域对应于系统不同的解．随着参数的变化，从平衡点分岔出两类不同的周期解，根据不同的分岔特性，这两类周期解失稳后，将产生概周期解或3—D环面解，它们都会随参数的变化进一步导致混吨．发现在系统的混沌区域中，其混吨吸引子随参数的变化会突然发生变化，分解为两个对称的混吨吸引子．值得注意的是，系统首先是由于2—D环面解破裂产生混吨，该混吨吸引子破裂后演变为新的混吨吸引子，却由倒倍周期分岔走向3—D环面解，也即存在两条通向混沌的道路：倍周期分岔和环面破裂，而这两种道路产生的混吨吸引子在一定参数条件下会相互转换．     

碰撞振动系统中周期轨擦边诱导的混沌激变

基于图胞映射理论, 提出了一种擦边流形的数值逼近方法, 研究了典型Du ng 碰撞振动系统中擦边诱导激变的全局动力学. 研究表明, 周期轨的擦边导致的奇异性使得系统同时产生1 个周期鞍和1 个混沌鞍. 当该周期鞍的稳定流形与不稳定流形发生相切时, 边界激变发生使得该混沌鞍演化为混沌吸引子. 噪声可以诱导周期吸引子发生擦边, 这种擦边导致了1 种内部激变的发生, 表现为该周期吸引子与其吸引盆内部的混沌鞍发生碰撞后演变为1 个混沌吸引子.     

OGY法实现混沌控制的参数辩识研究

给出了在系统方程未知情况下，采用OGY控制策略，通过混沌的部分测试信息辩识控制参数的方法，并对Duffing方程及强迫Brusselator振子方程的参数辩识及混沌控制进行了数值计算；讨论了使用这一方法实现控制时的一些关键问题，给出了实现控制的一些重要参数；发现了对Duffing方程使用同一参数辩识值，可将混沌控制到不同的一倍周期轨道及高倍周期轨道上的新现象。     

有限时间李雅普诺夫指数与哈密顿系统的混沌控制

通过计算相空间各混沌轨道的有限时间李雅普诺夫指数,得到有限时间收敛的区域。利用混沌轨道的有限时间收敛性,将常数周期脉冲方法,应用于控制哈密顿系统的混沌运动,给出了确定脉冲强度及周期不动点的方法,讨论了受控周期轨道的抗噪声能力。     

引导混沌运动到周期运动的自适应控制策略

提出一种自适应控制策略，对控制参数作线性反馈将非线性动力系统由混沌运动引导到指定的周期运动．所解决的关键问题是将反馈控制强度的确定转化为扩维相空间中的极点配置问题．给出了将该策略用于控制Ｌｏｇｉｓｔｉｃ映射和受迫Ｄｕｆｉｎｇ振子的仿真．     

Analysis and control of a hyperchaotic system with only one nonlinear term

In order to promote the development of chaos in nonlinear systems, and explore more convenient controllers for the engineering application, a four-dimensional nonlinear dynamic system with only one nonlinear term was constructed and its complex dynamic characteristics were analyzed, including the phase trajectory map, Lyapunov exponents, and so on. Furthermore, the recursive backstepping method was proposed to design a different controller; the hyperchaotic system was controlled to an equilibrium point and a periodic orbit. Theoretical analysis is in agreement with simulation results. The results show that the recursive backstepping control method can wipe off chaos, and make the hyperchaotic system achieve stable states. The control process is a smooth transition, and the transition time is short.     

Autoparametric amplification of two nonlinear coupled mass–spring systems

The gearboxes of machines generally operate under a time-varying state rather than under steady-state conditions. However, it is difficult to investigate the nonlinear dynamics of a time-varying gear system. A gear system model of a railway vehicle was proposed in consideration of its time-varying mesh stiffness, nonlinear backlash, transmission error, time-varying external excitation, and rail irregularity. To obtain the nonlinear behaviors of a time-varying stochastic gear system, a quasi-static analysis was performed to observe its doubling-periodic bifurcation, chaotic motion, and transition from a lower to a higher power periodic motion. Based on the energy comparison results, the time-varying stochastic gear system was degraded to a time-varying system to simplify the calculation. Furthermore, the nonlinear response of the time-varying system was computed using the Runge–Kutta method and was compared with the results of a quasi-static analysis that employed a short-time Fourier transform method. The results of the quasi-static analysis were consistent with the results of the time–frequency analysis for the time-varying gear system except for the result at 3180 r/min, which represented a short period wherein the process transitioned to chaos. Hence, the comparison demonstrates the applicability of the quasi-static analysis for the nonlinear behavior analysis of a time-varying stochastic system.     

Global bifurcation and chaos analysis in nonlinear vibration of spur gear systems

The global homoclinic bifurcation and transition to chaotic behavior of a nonlinear gear system are studied by means of Melnikov analytical analysis. It is also an effective approach to analyze homoclinic bifurcation and detect chaotic behavior. A generalized nonlinear time varying (NLTV) dynamic model of a spur gear pair is formulated, where the backlash, time varying stiffness, external excitation, and static transmission error are included. From Melnikov method, the threshold values of the control parameter for the occurrence of homoclinic bifurcation and onset of chaos are predicted. Additionally, the numerical bifurcation analysis and numerical simulation of the system including bifurcation diagrams, phase plane portraits, time histories, power spectras, and Poincare sections are used to confirm the analytical predictions and show the transition to chaos.     

Bifurcation and chaos analysis of multistage planetary gear train

A nonlinear time-varying dynamic model for a multistage planetary gear train, considering time-varying meshing stiffness, nonlinear error excitation, and piece-wise backlash nonlinearities, is formulated. Varying dynamic motions are obtained by solving the dimensionless equations of motion in general coordinates by using the varying-step Gill numerical integration method. The influences of damping coefficient, excitation frequency, and backlash on bifurcation and chaos properties of the system are analyzed through dynamic bifurcation diagram, time history, phase trajectory, Poincaré map, and power spectrum. It shows that the multi-stage planetary gear train system has various inner nonlinear dynamic behaviors because of the coupling of gear backlash and time-varying meshing stiffness. As the damping coefficient increases, the dynamic behavior of the system transits to an increasingly stable periodic motion, which demonstrates that a higher damping coefficient can suppress a nonperiodic motion and thereby improve its dynamic response. The motion state of the system changes into chaos in different ways of period doubling bifurcation, and Hopf bifurcation.     

Nonlinear response and nonsmooth bifurcations of an unbalanced machine-tool spindle-bearing system

In this effort, a six-degree-of-freedom (DOF) model is presented for the study of a machine-tool spindle-bearing system. The dynamics of machine-tool spindle system supported by ball bearings can be described by a set of second order nonlinear differential equations with piecewise stiffness and damping due to the bearing clearance. To investigate the effect of bearing clearance, bifurcations and routes to chaos of this nonsmooth system, numerical simulation is carried out. Numerical results show when the inner race touches the bearing ball with a low speed, grazing bifurcation occurs. The solutions of this system evolve from quasi-periodic to chaotic orbit, from period doubled orbit to periodic orbit, and from periodic orbit to quasi-periodic orbit through grazing bifurcations. In addition, the tori doubling process to chaos which usually occurs in the impact system is also observed in this spindle-bearing system.     

求解非线性动力系统周期解推广的打靶法

提出一种确定非线性系统周期轨道及周期的改进打靶算法。首先通过改变系统的时间尺度，将非线性系统周期轨道的周期显式地出现在非线性系统的系统方程中，然后对传统打靶法进行改造，将周期也作为一个参数一起参入打靶法的迭代过程，从而能迅速确定出系统的周期轨道及其周期。该方法对初始迭代参数没有苛刻要求，可以用于分析强非线性系统，而且对参数激励系统同样有效，对高维系统也能迅速、准确地求得周期解。文中应用该方法对三维Rǒssler系统和八维非线性柔性转子-轴承系统的周期轨道和周期进行了求解，通过与四阶Runge-Kutta数值积分结果比较，验证了方法的有效性。     

Bifurcations and chaos in a periodic time-varying temperature-excited bimetallic shallow shell of revolution

The chaotic vibrations of a bimetallic shallow shell of revolution under time-varying temperature excitation are investigated in the present study. The governing equations are established in forms similar to those of classical single-layered shell theory by re-determination of reference surface. The nonlinear differential equation in time-mode is derived by variational method following an assumed spatial-mode. The Melnikov function is established theoretically to estimate regions of the chaos, and the Poincaré map, phase portrait, Lyapunov exponent, and Lyapunov dimension are used to determine if a chaotic motion really appears. Further investigations are developed by means of detailed numerical simulation, and both the bifurcation diagrams and corresponding maximum Lyapunov exponent are illustrated. The influence of static and time-dependent temperature parameters, height parameter of the shell, and damping parameter on the dynamic characteristics is examined. Interesting phenomena such as the onset of chaos, transient chaotic motion, chaos with interior crisis and period window, period-doubling scenario and reversed period-doubling bifurcation leading to chaos, jump phenomena, and chaos suddenly converting to period orbit have been observed from these figures.     

非线性转子-轴承系统的周期解及近似解析表达式

通过对普通打靶方法进行改造提出一种确定非线性系统周期轨道及周期的新型打靶算法。首先通过改变系统的时间尺度，将非线性系统周期轨道的周期显式地出现在非线性系统的系统方程中，然后对传统打靶法进行改造，将周期也作为一个参数一起参与打靶法的迭代过程，迭代过程包含对周期轨道和周期的求解，迭代过程中的增量通过优化方法选择，从而能迅速确定出系统的周期轨道及其周期。应用所求的结果结合谐波平衡方法求得了非线性系统的周期轨道的近似解析表达式，理论上通过增加谐波的阶数任何精度的周期解都可以得到。最后将该方法应用于非线性转子轴承系统，求出了在某些参数下转子的周期解及其近似解析表达式，通过与四阶Runge-Kutta数值积分结果比较，验证了方法的有效性，计算结果对于转子系统运动的定量控制有重要理论指导意义。     

两空间耦合下齿轮传动系统多稳态特性研究

通过将系统参数定义为参数变量, 构成参数空间,研究齿轮传动系统在参数空间和状态空间耦合下的非线性全局动力学特性,以及多参数、多初值和多稳态行为之间的关联特性.首先设计了一个两空间耦合下非线性系统多稳态行为的计算和辨识方法.其次,基于该方法并结合相图、Poincaré映射图、分岔图、最大Lyapunov指数、吸引域等,研究齿轮传动系统在不同参数平面上多稳态行为的存在区域和分布特性,以及多稳态行为在状态平面上的分布特性,揭示了参数平面和状态平面上系统可能隐藏的多稳态行为和分岔,并分析了多稳态行为的形成机理. 结果发现,两空间耦合下系统在参数平面上存在大量多稳态行为并呈"带状"分布, 状态平面上多稳态行为出现两种不同的侵蚀现象, 即内部侵蚀和边界侵蚀.分岔点或分岔曲线对初值的敏感性导致多稳态行为的出现.当齿侧间隙和误差波动在较小的范围内变化时,系统全局动力学特性受间隙和误差扰动的影响较小,受啮合频率的影响较大.两空间耦合下系统全局动力学特性变得丰富和复杂.