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.     

Hopf分岔的代数判据及其在车辆动力学中的应用

利用Hurwitz行列式,给出平衡点失稳而发生Hopf分岔的代数判定准则和计算方法,这一方法将Hopf分岔点的求解转化为一个非线性方程的求解问题,从而克服了以前方法在计算Hopf分岔点时,对于参数的每一次变化通过求特征根并判定特征根的实部是否为零的庞大工作量。应用这一方法,我们进行了非线性车辆系统蛇行运动稳定性的研究,得到了轮对系统发生蛇行运动的临界速度的解析表达式。     

轮对非线性动力学系统蛇行运动的解析解

在对铁路车辆系统的极限环幅值和非线性临界速度进行分析时通常采用数值方法, 不便于研究其随系统参数的变化规律. 轮对系统保留了影响车辆系统动力学性能的几个关键要素: 如轮轨几何非线性约束、轮轨接触蠕滑关系和悬挂系统等, 可以反映铁路车辆系统蛇行运动的本质特性. 轮对系统自由度少、参数少, 可以采用解析方法进行分析. 本文选取合适的特征量把轮对非线性动力学方程无量纲化, 得到了带有小参数的两自由度微分方程; 采用多尺度方法对该方程进行了解析求解; 给出了轮对系统极限环幅值的解析表达式并对其稳定性进行了判定; 给出了轮对系统的分岔速度解析表达式, 并进而获得系统的非线性临界速度的解析表达式. 在对得到的解析解用数值结果进行验证后, 用得到的解析解进行了系统参数影响分析. 传统的分岔图计算方法(如降速法、路径跟踪法等)需对微分方程进行大量数值积分计算方可求解系统的非线性临界速度值, 而通过本文获得的解析表达式可直接给出系统的非线性临界速度值和极限环幅值, 便于研究轮对系统动力学特性随参数的变化规律,进行快速方案比对和筛选, 为转向架结构优化设计提供参考.      

Effects of small world connection on the dynamics of a delayed ring network

This paper presents a detailed analysis on the dynamics of a ring network with small world connection. On the basis of Lyapunov stability approach, the asymptotic stability of the trivial equilibrium is first investigated and the delay-dependent criteria ensuring global stability are obtained. The existence of Hopf bifurcation and the stability of periodic solutions bifurcating from the trivial equilibrium are then analyzed. Further studies are paid to the effects of small world connection on the stability interval and the stability of periodic solution. In particular, some complex dynamical phenomena due to short-cut strength are observed numerically, such as: period-doubling bifurcation and torus breaking to chaos, the coexistence of multiple periodic solutions, multiple quasi-periodic solutions, and multiple chaotic attractors. The studies show that small world connection may be used as a simple but efficient “switch” to control the dynamics of a system.     

A new 5D hyperchaotic system based on modified generalized Lorenz system

This paper reports a new five-dimensional (5D) hyperchaotic system with three positive Lyapunov exponents, which is generated by adding a linear controller to the second equation of a 4D system that is obtained by coupling of a 1D linear system and a 3D modified generalized Lorenz system. This hyperchaotic system has very simple algebraic structure but can exhibit complex dynamical behaviors. Of particular interest are the observations that the hyperchaotic system has a hyperchaotic attractor with three positive Lyapunov exponents under a unique equilibrium, three or infinite equilibria, and there are three types of coexisting attractors of this new 5D hyperchaotic system. Numerical analysis of phase trajectories, Lyapunov exponents, bifurcation, Poincaré projections and power spectrum verifies the existence of the hyperchaotic and chaotic attractors. Moreover, stability of hyperbolic or non-hyperbolic equilibria and two complete mathematical characterization for 5D Hopf bifurcation are rigorously studied. Finally, some electronic circuits are designed to implement the 5D hyperchaotic system.     

Stability, bifurcation and chaos of a delayed oscillator with negative damping and delayed feedback control

This paper presents a detailed analysis on the dynamics of a delayed oscillator with negative damping and delayed feedback control. Firstly, a linear stability analysis for the trivial equilibrium is given. Then, the direction of Hopf bifurcation and stability of periodic solutions bifurcating from trivial equilibrium are determined by using the normal form theory and center manifold theorem. It shows that with properly chosen delay and gain in the delayed feedback path, this controlled delayed system may have stable equilibrium, or periodic solutions, or quasi-periodic solutions, or coexisting stable solutions. In addition, the controlled system may exhibit period-doubling bifurcation which eventually leads to chaos. Finally, some new interesting phenomena, such as the coexistence of periodic orbits and chaotic attractors, have been observed. The results indicate that delayed feedback control can make systems with state delay produce more complicated dynamics.     

Neimark–Sacker bifurcation analysis and complex nonlinear dynamics in a heterogeneous quadropoly game with an isoelastic demand function

In this paper, a nonlinear quadropoly game based on Cournot model with fully heterogeneous players is established. This game extends the model introduced by Tramontana and Elsadany (Nonlinear Dyn 68:187–193, 2012) who considered a heterogeneous triopoly game with an isoelastic demand function. Here, four different types of players and potentially different marginal costs are considered. Moreover, the assumption of an isoelastic demand function increases the nonlinearity of the final four-dimensional map. The stability of the resulting discrete-time dynamical system is analyzed. The existence of Neimark–Sacker bifurcation near the Nash equilibrium point of the game is shown. Also, based on the Kuznetsov’s normal form technique for discrete-time system, the stability of the Neimark–Sacker bifurcation is also discussed which indicates that the bifurcation is supercritical. Moreover, it is shown that the Nash equilibrium point of the game undergoes period-doubling (flip) bifurcation. Furthermore, the double route to chaotic dynamics in this model, via flip bifurcations and via Neimark–Sacker bifurcation of the Nash equilibrium point, is illustrated. Coexistence of multi-chaotic attractors of the model is illustrated. Simulation tools like bifurcation diagrams, stability regions of parameters, Lyapunov exponent spectrum, phase plots and basins of attraction are used to verify the complex dynamics of the game.     

A feedback linearization design for the control of vehicle’s lateral dynamics

In this paper, the feedback linearization scheme is applied to the control of vehicle’s lateral dynamics. Based on the assumption of constant driving speed, a second-order nonlinear lateral dynamical model is adopted for controller design. It was observed in (Liaw, D.C., Chung, W.-C. in 2006 IEEE International Conference on Systems, Man, and Cybernetics, 2006) that the saddle-node bifurcation would appear in vehicle dynamics with respect to the variation of the front wheel steering angle, which might result in spin and/or system instability. The vehicle dynamics at the saddle node bifurcation point is derived and then decomposed as an affine nominal model plus the remaining term of the overall system dynamics. Feedback linearization scheme is employed to construct the stabilizing control laws for the nominal model. The stability of the overall vehicle dynamics at the saddle-node bifurcation is then guaranteed by applying Lyapunov stability criteria. Since the remaining term of the vehicle dynamics contains the steering control input, which might change system equilibrium except the designed one. Parametric analysis of system equilibrium for an example vehicle model is also obtained to classify the regime of control gains for potential behavior of vehicle’s dynamical behavior.     

Hopf bifurcation analysis of railway bogie

The railway bogie, the most important running component, has direct association with the dynamic performance of the whole vehicle system. The bifurcation type of the bogie that is affected by vehicle parameters will decide the behavior of the vehicle hunting stability. This paper mainly analyzes the effect of the yaw damper and wheel tread shape on the stability and bifurcation type of the railway bogie. The center manifold theorem is adopted to reduce the dimension of the bogie dynamical model, and the symbolic expression for determining the bifurcation type at the critical speed is obtained by the method of normal form. As a result, the influence of yaw damper on the bifurcation type of the bogie is given qualitatively in contrast to typical wheel profiles with high and low wheel tread effective conicities. Besides, the discriminant of bifurcation type for the wheel tread parameter variation is given which depicts the variation tendency of dynamics characteristics. Finally, numerical analysis is given to exhibit corresponding bifurcation diagrams.     

两系非线性悬挂车辆的运行稳定性与分叉

本文选取两系具有滞后非线性悬挂的车辆为目标，建立其数学模型和运动微分方程，用常微分方程稳定性理论对车辆蛇行运动进行理论分析，并应用分叉理论研究了整车在蛇行失稳后的动力学行为，得出蛇行运动的分叉解及稳定判据，得到防止车辆蛇行运动的充分条件，并研究了系统参数对临界速度的影响、分叉解振幅及稳定性的影响，为车辆设计和参数选取提供依据。     

Stability and Hopf bifurcation of the maglev system with delayed position and speed feedback control

In this paper, the dynamic behavior of suspension system of maglev train with time-delayed position and velocity feedback signal is considered with rigid guideway. The stability conditions of the system are obtained with characteristic root method. The Hopf bifurcation direction and stability of the system at the critical point are also investigated. Based on center manifold reduction and Poincaré normal form theory, the general formula for the direction, the estimation formula of period and stability of Hopf bifurcating periodic solution are also given. It is shown that time delays can change the current complicated dynamic behavior. And the condition that the bifurcation may occur is given to restrain the dynamic response and vibration between vehicle and guideway of the system with time-delayed position and velocity signal.     

Periodic and chaotic dynamics of a sliding driven oscillator with dry friction

We have performed a numerical study of the dynamics of a harmonically forced sliding oscillator with two degrees of freedom and dry friction. The study of the four-dimensional dynamical system corresponding to the two non-linear motion equations can be reduced, in this case, to the study of a three-dimensional Poincaré map. The behaviour of the system has been investigated calculating bifurcation diagrams, time series, periodic and chaotic attractors and basins of attraction. Furthermore, a systematic study of the stability of periodic solutions and their bifurcations has been carried out applying the Floquet theory. The results show rich dynamics being very sensitive to the changes in forcing amplitudes (control parameter), where periodic and chaotic states alternatively appear. It is shown how the system exhibits different types of bifurcational phenomena (saddle-node, symmetry-breaking, period-doubling cascades and intermittent transitions to chaos) into relatively narrow intervals of the control parameter. Moreover, a collection of chaotic attractors was computed to show the evolution of the chaotic regime. Finally, basins of attraction were calculated. In all the cases studied, the basins exhibit fractal structure boundaries and, when more of two attractors are coexisting, we have found Wada basin boundaries.     

Perturbation of a Globally Stable Steady State and Uniform Persistence

In this paper we consider a parametrized family of semi-flows with continuous or discrete time. In the spirit of the global stability result proved by Smith and Waltman (Proc AMS 127:447–453, 1999) we use the upper semi-continuity of a parametrized family of global attractors. Here we investigate the case where the linearized equation of the unperturbed system has a simple dominant eigenvalue 0 in the case of a continuous time system (or 1 in the case of a discrete time system). New difficulties arise since such a system may exhibit a bifurcation. The goal of the paper is to describe the global dynamics of the perturbed system.      

Stability and Hamiltonian Hopf bifurcation for a nonlinear symmetric bladed rotor

The modal interaction which leads to Hamiltonian Hopf bifurcation is studied for a nonlinear rotating bladed-disk system. The model, which is discussed in the paper, is a Jeffcott rotor carrying a number of planar blades which bend in the plane of the motion. The rigid rotating disk is supported on nonlinear bearings. It is supposed that this dynamical system is a Hamiltonian system which is perturbed by small dissipative and nonlinear forces. Krein’s theorem is employed for obtaining a stability criterion. The nonlinear eigenvalue equations on the stability boundary are turned into ordinary differential equations (ODEs) by differentiating them over the rotating speed. By solving these ODEs, the eigenmodes and the eigenvalues on the stability boundary are obtained. The bifurcation analysis is performed by applying multiple scales method around the boundary. The rotor nonlinear behavior and damping effects are studied for different conditions on the rotating speed and nonlinearity type by the bifurcation equation. It is shown that the damping distribution between the blades and bearings may shift the unstable mode. Depending on the nonlinearity type, subcritical and supercritical Hopf bifurcation are possible.     

一类碰撞振动系统在内伊马克沙克-音叉分岔点附近的局部两参数动力学

考虑一类具有对称性的三自由度碰撞振动系统.系统的庞加莱映射在一定条件下存在对称不动点,对应于系统的对称周期运动.根据对称性导出庞加莱映射P是另外一个隐式虚拟映射Q的二次迭代.推导了庞加莱映射对称不动点的解析表达式.根据映射不动点的稳定性及分岔理论,映射P的对称不动点发生内伊马克沙克-音叉(Neimark--Saker-pitchfork)分岔对应于映射Q发生内伊马克沙克-倍化(Neimark--Sakerflip)分岔.利用隐式虚拟映射Q,通过对范式作两参数开折分析,研究了映射P的对称不动点在内伊马克沙克-音叉分岔点附近的局部动力学行为.碰撞振动系统在这个余维二分岔点附近的局部动力学行为可能表现为投影后的庞加莱截面上的单一对称不动点、一对共轭不动点、单一对称拟周期吸引子以及一对共轭拟周期吸引子.数值模拟得到了内伊马克沙克-音叉分岔点附近的各种可能情况.内伊马克沙克-分岔和音叉分岔互相作用可能产生新的结果:对称不动点虽然首先分岔为两个共轭不动点,但是这两个共轭不动点是不稳定的,最终收敛到同一个对称拟周期吸引子.     

The effects of fractional order on a 3-D quadratic autonomous system with four-wing attractor

In this paper, a fractional 3-dimensional (3-D) 4-wing quadratic autonomous system (Qi system) is analyzed. Time domain approximation method (Grunwald–Letnikov method) and frequency domain approximation method are used together to analyze the behavior of this fractional order chaotic system. It is found that the decreasing of the system order has great effect on the dynamics of this nonlinear system. The fractional Qi system can exhibit chaos when the total order less than 3, although the regular one always shows periodic orbits in the same range of parameters. In some fractional order, the 4 wings are decayed to a scroll using the frequency domain approximation method which is different from the result using time domain approximation method. A surprising finding is that the phase diagrams display a character of local self-similar in the 4-wing attractors of this fractional Qi system using the frequency approximation method even though the number and the characteristics of equilibria are not changed. The frequency spectrums show that there is some shrinking tendency of the bandwidth with the falling of the system states order. However, the change of fractional order has little effect on the bandwidth of frequency spectrum using the time domain approximation method. According to the bifurcation analysis, the fractional order Qi system attractors start from sink, then period bifurcation to some simple periodic orbits, and chaotic attractors, finally escape from chaotic attractor to periodic orbits with the increasing of fractional order α in the interval [0.8,1]. The simulation results revealed that the time domain approximation method is more accurate and reliable than the frequency domain approximation method.     

The effects of time-dependent dissipation on the basins of attraction for the pendulum with oscillating support

We consider a pendulum with vertically oscillating support and time-dependent damping coefficient which varies until reaching a finite final value. Although it is the final value which determines which attractors eventually exist, the sizes of the corresponding basins of attraction are found to depend strongly on the full evolution of the dissipation. In particular, we investigate numerically how dissipation monotonically varying in time changes the sizes of the basins of attraction. It turns out that, in order to predict the behaviour of the system, it is essential to understand how the sizes of the basins of attraction for constant dissipation depend on the damping coefficient. For values of the parameters where the systems can be considered as a perturbation of the simple pendulum, which is integrable, we characterise analytically the conditions under which the attractors exist and study numerically how the sizes of their basins of attraction depend on the damping coefficient. Away from the perturbation regime, a numerical study of the attractors and the corresponding basins of attraction for different constant values of the damping coefficient produces a much more involved scenario: changing the magnitude of the dissipation causes some attractors to disappear either leaving no trace or producing new attractors by bifurcation, such as period doubling and saddle-node bifurcation. Finally, we pass to the case of an initially non-constant damping coefficient, both increasing and decreasing to some finite final value, and we numerically observe the resulting effects on the sizes of the basins of attraction: when the damping coefficient varies slowly from a finite initial value to a different final value, without changing the set of attractors, the slower the variation the closer the sizes of the basins of attraction are to those they have for constant damping coefficient fixed at the initial value. Furthermore, if during the variation of the damping coefficient attractors appear or disappear, remarkable additional phenomena may occur. For instance, it can happen that, in the limit of very large variation time, a fixed point asymptotically attracts the entire phase space, up to a zero-measure set, even though no attractor with such a property exists for any value of the damping coefficient between the extreme values.     

Local bifurcation analysis and ultimate bound of a novel 4D hyper-chaotic system

This paper presents a new four-dimensional smooth quadratic autonomous hyper-chaotic system which can generate novel two double-wing periodic, quasi-periodic and hyper-chaotic attractors. The Lyapunov exponent spectrum, bifurcation diagram and phase portrait are provided. It is shown that this system has a wide hyper-chaotic parameter. The pitchfork bifurcation and Hopf bifurcation are discussed using the center manifold theory. The ellipsoidal ultimate bound of the typical hyper-chaotic attractor is observed. Numerical simulations are given to demonstrate the evolution of the two bifurcations and show the ultimate boundary region.     

一类非线性磁流变系统局部分岔特性研究

讨论了一类基于磁流变阻尼器非线性系统的局部分岔与控制问题，建立了该系统的动力 学模型，运用中心流形定理和范式理论，得到该系统双零特征值问题的规范形及其普适开折， 进而探讨了此系统的分岔行为和稳定性；给出了分岔曲线、转迁集；并给出了此类非线性系 统的数值仿真结果.     

A special type of codimension two bifurcation and unusual dynamics in a phase-modulated system with switched strategy

We investigate a new type of codimension two bifurcation and related dynamics in a phase-modulated system with switched strategy. Two curves intersect at a point and are called the crisis–Hopf bifurcation. At the critical crisis–Hopf vertex, a boundary crisis and Hopf bifurcation coincide. Metamorphosis of coexisting attractors can be observed in the vicinity of the vertex. Another novelty is that we discover some sets of measure zero with riddled holes in the neighborhood of the bifurcation point. These sets display some qualitative and quantitative features of riddled basins but they are essentially different from the riddled basin. It may provide a more comprehensive picture for unusual dynamical features.