一类三维随机动力系统的最大Lyapunov指数

基于一维扩散过程的奇异边界理论,使用摄动方法研究了白噪声参激的一类余维二分岔系统的最大Lyapunov指数渐近表达式和数值解,主要讨论了一维相扩散过程同时存在两类奇异边界以及FPK方程存在平稳解的一般性条件. 通过对参激噪声作用项系数矩阵的分析,给出了不变测度的解析解及其相应的Monte Carlo数值仿真结果,并导出了一维相扩散过程P分岔点的确定方法. 对于一类特殊情形,给出了最大Lyapunov指数的渐近表达式;对于参激噪声作用项系数矩阵的一般情形,则给出了系统最大Lyapunov指数的数值结果.      

二维Logistic映射中的一种新型激变、回滞和分形

研究了二维Logistic映射不动点的性质,给出了在参数空间中二维Logistic映射发生第一次分岔的边界方程。采用相图、分岔图、功率谱、Lyapunov指数计算和分维数计算方法,揭示出具有二次耦合项的二维Logistic映射从规则运动转化到混沌运动所具有的普适特征:①系统是按Pomeau-Manneville途径通向混沌的,且其间歇性与Hopf分岔有关;②系统中存在一种新型循环激变:当参数连续变化时,不稳定周期轨道按固定顺序循环与奇怪吸引子的几个小部分相遇,并导致小部分两两合并,产生出较大的奇怪吸引子;③最大Lyapunov指数的曲线具有“回滞”特征,且回滞现象常伴随循环激变的出现。同时,作者对二维Logistic映射的Mandelbrot-Julia集(简称M-J集)的研究表明:M-J集的结构由控制参数决定,且它们的边界是分形的。     

Synchronization and circuit simulation of a new double-wing chaos

A new three-dimensional double-wing chaotic system with three quadratic terms was proposed. And the parameters which can induce the system are analyzed. The system with five equilibrium points has sophisticated dynamical behaviors and it is further investigated in details, including phase trajectory, Lyapunov exponent spectrum, Poincaré map, spectrogram map and dissipativity analysis. The circuit simulation results of the chaotic attractors are in agreement with numerical simulations. Furthermore, numerical simulations indicate that mismatch synchronization can be achieved and circuit simulations of the system synchronization are also presented.     

The maximal Lyapunov exponent of a co-dimension two-bifurcation system excited by a bounded noise

In the present paper,the maximal Lyapunov exponent is investigated for a co-dimension two bifurcation system that is on a three-dimensional central manifold and subjected to parametric excitation by a bounded noise.By using a perturbation method,the expressions of the invariant measure of a one-dimensional phase diffusion process are obtained for three cases,in which different forms of the matrix B,that is included in the noise excitation term,are assumed and then,as a result,all the three kinds of singular boundaries for one-dimensional phase diffusion process are analyzed.Via Monte-Carlo simulation,we find that the analytical expressions of the invariant measures meet well the numerical ones.And furthermore,the P-bifurcation behaviors are investigated for the one-dimensional phase diffusion process.Finally,for the three cases of singular boundaries for one-dimensional phase diffusion process,analytical expressions of the maximal Lyapunov exponent are presented for the stochastic bifurcation system.     

Nonlinear vibrations and stability of elastic and viscoelastic systems under random stationary loads

The paper deals with numerical analysis of nonlinear vibrations of viscoelastic systems under a stochastic action in the form of a Gaussian stationary process with rational spectral density. The analysis is based on numerical simulation of the original stationary process, numerical solution of the differential equations describing the motion of the system, and computation of the maximum Lyapunov exponent if the stability of this motion is studied. An example of a plate subjected to a random stationary load applied in its plane is used to consider specific issues concerning the application of the proposed method and the peculiarities of the behavior of geometrically nonlinear elastic and viscoelastic stochastic systems. Special attention is paid to the interaction of a deterministic periodic action and a stochastic action from the viewpoint of stability of the system motion. It is shown that in some cases imposing a “colored” noise may stabilize an unstable system subjected to a periodic load.     

Moment Lyapunov exponent of three-dimensional system under bounded noise excitation

In the present paper,the moment Lyapunov exponent of a codimensional two-bifurcation system is evaluted,which is on a three-dimensional central manifold and subjected to a parametric excitation by the ...     

Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems

An n degree-of-freedom (DOF) non-integrable Hamiltonian system subject to light damping and weak stochastic excitation is called quasi-non-integrable Hamiltonian system. In the present paper, the stochastic averaging of quasi-non-integrable Hamiltonian systems is briefly reviewed. A new norm in terms of the square root of Hamiltonian is introduced in the definitions of stochastic stability and Lyapunov exponent and the formulas for the Lyapunov exponent are derived from the averaged Itô equations of the Hamiltonian and of the square root of Hamiltonian. It is inferred that the Lyapunov exponent so obtained is the first approximation of the largest Lyapunov exponent of the original quasi-non-integrable Hamiltonian systems and the necessary and sufficient condition for the asymptotic stability with probability one of the trivial solution of the original systems can be obtained approximately by letting the Lyapunov exponent to be negative. This inference is confirmed by comparing the stability conditions obtained from negative Lyapunov exponent and by examining the sample behaviors of averaged Hamiltonian or the square root of averaged Hamiltonian at trivial boundary for two examples. It is also verified by the largest Lyapunov exponent obtained using small noise expansion for the second example.     

Unified Analysis of Complex Nonlinear Motions via Densities

A unified approach of using densities to analyze bothdeterministic and stochastic complex responses including chaotic andrandom motions of nonlinear engineering systems is illustrated in thisstudy. Motivations to examine deterministic nonlinear dynamical systemsvia densities are first discussed. Essential mathematical background andtechniques pertinent to the analyses of both deterministic chaos andrandom chaotic processes are briefly summarized. Densities of nonlinearresponses are computed by numerically solving the Fokker–Planckequation to examine stochastic properties of random chaotic responses.It is demonstrated that, by introducing random perturbations in anotherwise deterministic excitation, the existence of attractors can beefficiently and clearly depicted by the evolution of a uniqueprobability density over the physical phase space. Two distinctasymptotic behaviors of densities: (i) invariance and (ii) sweeping, ofcomplex motions and their relationship to response stabilities predictedby the Foguel Alternative Theorem are numerically demonstrated.Applications using the probability densities to compute reliabilityindices of an engineering system are demonstrated.     

Bifurcation characteristics of a two-dimensional structurally non-linear airfoil in turbulent flow

The dynamics of a structurally non-linear two-dimensional airfoil in turbulent flow is investigated numerically using a Monte Carlo approach. Both the longitudinal and vertical components of turbulence, corresponding to parametric (multiplicative) and external (additive) excitation, respectively, are modelled. The properties of the airfoil are chosen such that the underlying non-excited, deterministic system exhibits binary flutter; the loss of stability of the equilibrium point due to flutter then leads to a limit cycle oscillation (LCO) via a supercritical Hopf bifurcation. For the random system, the results are examined in terms of the probability structure of the response and the largest Lyapunov exponent. The airfoil response is interpreted from the point of view of the concepts of D- and P-bifurcations, as defined in random bifurcation theory. It is found that the bifurcation is characterized by a change in shape of the response probability structure, while no discontinuity in the variation of the largest Lyapunov exponent with airspeed is observed. In this sense, the trivial bifurcation obtained for the deterministic airfoil, where the D- and P-bifurcations coincide, appears only as a P-bifurcation for the random case. At low levels of turbulence intensity, the Gaussian-like bell-shaped bi-dimensional PDF bifurcates into a crater shape; this is interpreted as a random fixed point bifurcating into a random LCO. At higher levels of turbulence intensity, the post-bifurcation PDF loses its underlying deterministic LCO structure. The crater is transformed into a two-peaked shape, with a saddle at the origin. From a more universal point of view, the robustness of the random bifurcation scenario is critiqued in light of the relative importance of the two components of turbulent excitation.     

含三次耦合项的两自由度Duffing系统的共振及混沌行为

研究了一类含三次耦合项的两自由度Duffing系统的动力学行为。首先应用多尺度方法近似求解系统的一阶稳态响应。通过讨论系统的主共振和1∶1内共振,分析了三次耦合项对系统响应的影响。随后研究系统随外加周期力强度变化的分岔过程,发现除了常见的倍周期分岔通向混沌外,还存在一种直接由周期运动进入混沌的突发路径。结合对系统的最大Lyapunov指数,相轨图及Poincar啨映射的分析验证了上述结论。     

Noise-induced chaos in the elastic forced oscillators with real-power damping force

The chaotic behavior of the elastic forced oscillators with real-power exponents of damping and restoring force terms under bounded noise is investigated. By using random Melnikov method, a mean square criterion is used to detect the necessary conditions for chaotic motion of this stochastic system. The results show that the threshold of bounded noise amplitude for the onset of chaos in the system increases as the intensity of the random frequency increases, and decrease as the real-power exponent of damping term increase. The threshold of bounded noise amplitude for the onset of chaos is determined by the numerical calculation via the largest Lyapunov exponents. The effects of bounded noise and real-power exponent of damping term on bifurcation and Poincaré map are also investigated. Our results may provide a valuable guidance for understanding the effect of bounded noise on a class of generalized double well system.     

受周期和白噪声激励的分段线性系统的吸引域与离出问题研究

离出行为是随机非线性系统的重要现象之一,而离出问题是除随机动力系统理论以外考察随机非线性系统随机稳定性的另一种重要的方法.分段线性系统是一个经典的非线性动力学模型,受随机激励后成为随机系统,但并不是严格的随机动力系统,因而此时随机动力系统理论也不适用.为了研究同时受周期和白噪声激励的分段线性系统,首先使用Poincaré截面模拟其在无噪声时确定性的动力学行为,然后使用Monte Carlo模拟对其在白噪声激励下的离出行为进行了数值仿真分析.其次,为了考察离出问题中的重要参数,系统的平均首次通过时间（mean first-passage time,MFPT）,使用van der Pol变换,随机平均法,奇异摄动法和射线方法进行了量化计算.通过对理论结果与模拟结果的对比分析,得到结论:当系统吸引子对应的吸引域边界出现碎片化时,理论结果与模拟结果的误差极大;而当吸引域边界足够光滑的以后,理论结果与模拟结果才会相当吻合.      

RESEARCH FOR ATTRACTING REGION AND EXIT PROBLEM OF A PIECEWISE LINEAR SYSTEM UNDER PERIODIC AND WHITE NOISE EXCITATIONS

离出行为是随机非线性系统的重要现象之一,而离出问题是除随机动力系统理论以外考察随机非线性系统随机稳定性的另一种重要的方法.分段线性系统是一个经典的非线性动力学模型,受随机激励后成为随机系统,但并不是严格的随机动力系统,因而此时随机动力系统理论也不适用.为了研究同时受周期和白噪声激励的分段线性系统,首先使用Poincaré截面模拟其在无噪声时确定性的动力学行为,然后使用Monte Carlo模拟对其在白噪声激励下的离出行为进行了数值仿真分析.其次,为了考察离出问题中的重要参数,系统的平均首次通过时间（mean first-passage time,MFPT）,使用van der Pol变换,随机平均法,奇异摄动法和射线方法进行了量化计算.通过对理论结果与模拟结果的对比分析,得到结论：当系统吸引子对应的吸引域边界出现碎片化时,理论结果与模拟结果的误差极大;而当吸引域边界足够光滑的以后,理论结果与模拟结果才会相当吻合.     

Principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation

The principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation is studied by using the method of multiple scales and numerical simulations. The first-order approximations of the solution, together with the modulation equations of both amplitude and phase, are derived. The effects of the frequency detuning, the deterministic amplitude, the intensity of the random excitation and the time delay on the dynamical behaviors, such as stability and bifurcation, are studied through the largest Lyapunov exponent. Moreover, the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. It is found that the appropriate choice of the time delay can broaden the stable region of the trivial steady-state solution and enhance the control performance. The theoretical results are well verified through numerical simulations.     

Research on parametric resonance in a stochastic van der Pol oscillator under multiple time delayed feedback control

Analytical derivations and numerical calculations are employed to gain insight into the parametric resonance of a stochastically driven van der Pol oscillator with delayed feedback. This model is the prototype of a self-excited system operating with a combination of narrow-band noise excitation and two time delayed feedback control. A slow dynamical system describing the amplitude and phase of resonance, as well as the lowest-order approximate solution of this oscillator is firstly obtained by the technique of multiple scales. Then the explicit asymptotic formula for the largest Lyapunov exponent is derived. The influences of system parameters, such as magnitude of random excitation, tuning frequency, gains of feedback and time delays, on the almost-sure stability of the steady-state trivial solution are discussed under the direction of the signal of largest Lyanupov exponent. The non-trivial steady-state solution of mean square response of this system is studied by moment method. The results reveal the phenomenon of multiple solutions and time delays induced stabilization or unstabilization, moreover, an appropriate modulation between the two time delays in feedback control may be acted as a simple and efficient switch to adjust control performance from the viewpoint of vibration control. Finally, theoretical analysis turns to a validation through numerical calculations, and good agreements can be found between the numerical results and the analytical ones.     

Duffing-van der Pol系统的随机分岔

应用广义胞映射图论方法(GCMD)研究了在谐和激励与随机噪声共同作用下的Duffing-van der Pol系统的随机分岔现象. 系统参数选择在多个吸引子与混沌鞍共存的范围内. 研究发现, 随着随机激励强度的增大,该系统存在两种分岔现象: 一种为随机吸引子与吸引域边界上的鞍碰撞, 此时随机吸引子突然消失; 另一种为随机吸引子与吸引域内部的鞍碰撞, 此时随机吸引子突然增大. 研究证实, 当随机激励强度达到某一临界值时, 该系统还会发生D-分岔(基于Lyapunov指数符号的改变而定义), 此类分岔点不同于上述基于系统拓扑性质改变所得的分岔点.     

A new chaotic system with fractional order and its projective synchronization

Based on Rikitake system, a new chaotic system is discussed. Some basic dynamical properties, such as equilibrium points, Lyapunov exponents, fractal dimension, Poincaré map, bifurcation diagrams and chaotic dynamical behaviors of the new chaotic system are studied, either numerically or analytically. The obtained results show clearly that the system discussed is a new chaotic system. By utilizing the fractional calculus theory and computer simulations, it is found that chaos exists in the new fractional-order three-dimensional system with order less than 3. The lowest order to yield chaos in this system is 2.733. The results are validated by the existence of one positive Lyapunov exponent and some phase diagrams. Further, based on the stability theory of the fractional-order system, projective synchronization of the new fractional-order chaotic system through designing the suitable nonlinear controller is investigated. The proposed method is rather simple and need not compute the conditional Lyapunov exponents. Numerical results are performed to verify the effectiveness of the presented synchronization scheme.     

磁场中轴向运动条形板强非线性超谐共振及混沌运动

针对磁场环境中周期外载作用下轴向运动导电条形板的非线性振动及混沌运动问题进行研究。应用改进多尺度法对横向磁场中条形板的强非线性振动问题进行求解,得到超谐波共振下系统的分岔响应方程。根据奇异性理论对非线性动力学系统的普适开折进行分析,求得含两个开折参数的转迁集及对应区域的拓扑结构分岔图。通过数值算例,分别得到以磁感应强度、轴向拉力、激励力幅值和激励频率为分岔控制参数的分岔图和最大李雅普诺夫指数图,以及反映不同运动行为区域的动力学响应图形,讨论分岔参数对系统呈现的倍周期和混沌运动的影响。结果表明,可通过相应参数的改变实现对系统复杂动力学行为的控制。     

The moment Lyapunov exponent of a Timoshenko beam under bounded noise excitation

The moment Lyapunov exponents and the Lyapunov exponent of a two-dimensional system under bounded noise excitation are studied in this paper. The method regular perturbation is applied to obtain the small noise expansion of the pth moment Lyapunov exponent and the Lyapunov exponent. The results are applied to the study of the almost-sure and moment stability of the stationary solutions of the elastic beam subjected to the stochastic axial load. The boundaries of the almost-sure and moment stability of the elastic beam as the function of the damping coefficient and characteristics of the stochastic force are obtained.     

A stochastic interrogation method for experimental measurements of global dynamics and basin evolution: Application to a two-well oscillator

An experimental study of local and global bifurcations in a driven two-well magneto-mechanical oscillator is presented. A detailed picture of the local bifurcation structure of the system is obtained using an automated bifurcation data acquisition system. Basins of attractions for the system are obtained using a new experimental technique: an ensemble of initial conditions is generated by switching between stochastic and deterministic excitation. Using this stochastic interrogation method, we observe the evolution of basins of attraction in the nonlinear oscillator as the forcing amplitude is increased, and find evidence for homoclinic bifurcation before the onset of chaos. Since the entire transient is collected for each initial condition, the same data can be used to obtain pictures of the flow of points in phase space. Using Liouville's Theorem, we obtain damping estimates by calculating the contraction of volumes under the action of the Poincaré map, and show that they are in good agreement with the results of more conventional damping estimation methods. Finally, the stochastic interrogation data is used to estimate transition probability matrices for finite partitions of the Poincaré section. Using these matrices, the evolution of probability densities can be studied.