Noise-Induced Chaos in Duffing Oscillator with Double Wells

Stochastic Melnikov method is employed to predict noise-induced chaotic response in the Duffing oscillator with double wells. The safe basin is simulated to show the noise-induced fractal boundary. Three cases are considered: harmonic excitation, both harmonic and Gaussian white noise excitations, and Gaussian white noise excitation. The leading Lyapunov exponent estimated by Rosenstein's algorithm is shown to quantify the chaotic nature of the sample time series of the system. The results show that the boundary of the safe basin can be fractal even if the system is excited only by external Gaussian white noise.     

耦合Duffing-van der Pol系统的首次穿越问题

利用拟不可积Hamilton系统随机平均法，研究了高斯白噪声激励下耦 合Duffing-van der Pol系统的首次穿越问题. 首先给出了条件可靠性函数满足的后向 Kolmogorov 方程以及首次穿越时间条件矩满足的广义Pontryagin方程. 然后根据 这两类偏微分方程的边界条件和初始条件，详细分析了在外激与参激共 同作用以及纯外激作用等情况下系统的可靠性与首次穿越时间的各阶矩. 最后以图表形式给 出了可靠性函数、首次穿越时间的概率密度以及平均首次穿越时间的数值结果.     

FIRST-PASSAGE FAILURE OF PREISACH HYSTERETIC SYSTEMS

The first-passage failure of a single-degree-of-freedom hysteretic system with non- local memory is investigated. The hysteretic behavior is described through a Preisach model with excitation selected as Gaussian white noise. First, the equivalent nonlinear non-hysteretic sys- tem with amplitude-dependent damping and stiffness coefficients is derived through generalized harmonic balance technique. Then, equivalent damping and stiffness coefficients are expressed as functions of system energy by using the relation of amplitude to system energy. The stochastic aver- aging of energy envelope is adopted to accept the averaged It5 stochastic differential equation with respect to system energy. The establishing and solving of the associated backward Kolmogorov equation yields the reliability function and probability density of first-passage time. The effects of system parameters on first-passage failure are investigated concisely and validated through Monte Carlo simulation.     

Effect of noise on erosion of safe basin in power system

We study the effect of Gaussian white noise on erosion of safe basin in a simple model of power system whose safe basin is integral in the absence of noise. The stochastic Melnikov method is first applied to predict the onset of basin erosion when the noise excitation is present in system. And then the eroded basins are simulated according to the necessary restrictions for the system’s parameters. It is found that for the noisy power system when the noise intensity σ is greater than a threshold, basin erosion occurs and as σ is further increased basin erosion is aggravated. These studies imply that random noise excitation can induce and enhance the basin erosion in the power 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.     

Non-linear stochastic response of a shallow cable

The paper considers the stochastic response of geometrical non-linear shallow cables. Large rain-wind induced cable oscillations with non-linear interactions have been observed in many large cable stayed bridges during the last decades. The response of the cable is investigated for a reduced two-degrees-of-freedom system with one modal coordinate for the in-plane displacement and one for the out-of-plane displacement. At first harmonic varying chord elongation at excitation frequencies close to the corresponding eigenfrequencies of the cable is considered in order to identify stable modes of vibration. Depending on the initial conditions the system may enter one of two states of vibration in the static equilibrium plane with the out-of-plane displacement equal to zero, or a whirling state with the out-of-plane displacement different from zero. Possible solutions are found both analytically and numerically. Next, the chord elongation is modelled as a narrow-banded Gaussian stochastic process, and it is shown that all the indicated harmonic solutions now become instable with probability one. Instead, the cable jumps randomly back and forth between the two in-plane and the whirling mode of vibration. A theory for determining the probability of occupying either of these modes at a certain time is derived based on a homogeneous, continuous time three states Markov chain model. It is shown that the transitional probability rates can be determined by first-passage crossing rates of the envelope process of the chord wise component of the support point motion relative to a safe domain determined from the harmonic analysis of the problem.     

Dynamical responses of airfoil models with harmonic excitation under uncertain disturbance

In this paper, we investigate nonlinear dynamical responses of two-degree-of-freedom airfoil (TDOFA) models driven by harmonic excitation under uncertain disturbance. Firstly, based on the deterministic airfoil models under the harmonic excitation, we introduce stochastic TDOFA models with the uncertain disturbance as Gaussian white noise. Subsequently, we consider the amplitude–frequency characteristic of deterministic airfoil models by the averaging method, and also the stochastic averaging method is applied to obtain the mean-square response of given stochastic TDOFA systems analytically. Then, we carry out numerical simulations to verify the effectiveness of the obtained analytic solution and the influence of harmonic force on the system response is studied. Finally, stochastic jump and bifurcation can be found through the random responses of system, and probability density function and time history diagrams can be obtained via Monte Carlo simulations directly to observe the stochastic jump and bifurcation. The results show that noise can induce the occurrence of stochastic jump and bifurcation, which will have a significant impact on the safety of aircraft.     

Response of quasi-integrable Hamiltonian systems with delayed feedback bang–bang control

The response of quasi-integrable Hamiltonian systems with delayed feedback bang–bang control subject to Gaussian white noise excitation is studied by using the stochastic averaging method. First, a quasi-Hamiltonian system with delayed feedback bang–bang control subjected to Gaussian white noise excitation is formulated and transformed into the Itô stochastic differential equations for quasi-integrable Hamiltonian system with feedback bang–bang control without time delay. Then the averaged Itô stochastic differential equations for the later system are derived by using the stochastic averaging method for quasi-integrable Hamiltonian systems and the stationary solution of the averaged Fokker–Plank–Kolmogorov (FPK) equation associated with the averaged Itô equations is obtained for both nonresonant and resonant cases. Finally, two examples are worked out in detail to illustrate the application and effectiveness of the proposed method and the effect of time delayed feedback bang–bang control on the response of the systems.     

Stochastic averaging of quasi-integrable Hamiltonian systems under combined harmonic and white noise excitations

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems to combined harmonic and white noise excitations. According to the proposed method, an n+α+β-dimensional averaged Fokker-Planck-Kolmogorov (FPK) equation governing the transition probability density of n action variables or independent integrals of motion, α combinations of angle variables and β combinations of angle variables and excitation phase angles can be constructed when the associated Hamiltonian system has α internal resonant relations and the system and harmonic excitations have β external resonant relations. The averaged FPK equation is solved by using the combination of the finite difference method and the successive over relaxation method. Two coupled Duffing-van der Pol oscillators under combined harmonic and white noise excitations is taken as an example to illustrate the application of the proposed procedure and the stochastic jump and its bifurcation as the system parameters change are examined.     

Stochastic bifurcation in duffing system subject to harmonic excitation and in presence of random noise

A global analysis of stochastic bifurcation in a special kind of Duffing system, named as Ueda system, subject to a harmonic excitation and in presence of random noise disturbance is studied in detail by the generalized cell mapping method using digraph. It is found that for this dissipative system there exists a steady state random cell flow restricted within a pipe-like manifold, the section of which forms one or two stable sets on the Poincare cell map. These stable sets are called stochastic attractors (stochastic nodes), each of which owns its attractive basin. Attractive basins are separated by a stochastic boundary, on which a stochastic saddle is located. Hence, in topological sense stochastic bifurcation can be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value. Through numerical simulations the evolution of the Poincare cell maps of the random flow against the variation of noise intensity is explored systematically. Our study reveals that as a powerful tool for global analysis, the generalized cell mapping method using digraph is applicable not only to deterministic bifurcation, but also to stochastic bifurcation as well. By this global analysis the mechanism of development, occurrence, and evolution of stochastic bifurcation can be explored clearly and vividly.     

First-passage failure of strongly non-linear oscillators with time-delayed feedback control under combined harmonic and wide-band noise excitations

A procedure for studying the first-passage failure of strongly non-linear oscillators with time-delayed feedback control under combined harmonic and wide-band noise excitations is proposed. First, the time-delayed feedback control forces are expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method. A backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function, the conditional probability density and moments of first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. An example is worked out in detail to illustrate the proposed procedure. The effects of time delay in feedback control forces on the conditional reliability function, conditional probability density and moments of first-passage time are analyzed. The validity of the proposed method is confirmed by digital simulation.     

First passage of stochastic fractional derivative systems with power-form restoring force

In this paper, the first-passage failure of stochastic dynamical systems with fractional derivative and power-form restoring force subjected to Gaussian white-noise excitation is investigated. With application of the stochastic averaging method of quasi-Hamiltonian system, the system energy process will converge weakly to an Itô differential equation. After that, Backward Kolmogorov (BK) equation associated with conditional reliability function and Generalized Pontryagin (GP) equation associated with statistical moments of first-passage time are constructed and solved. Particularly, the influence on reliability caused by the order of fractional derivative and the power of restoring force are also examined, respectively. Numerical results show that reliability function is decreased with respect to the time. Lower power of restoring force may lead the system to more unstable evolution and lead first passage easy to happen. Besides, more viscous material described by fractional derivative may have higher reliability. Moreover, the analytical results are all in good agreement with Monte-Carlo data.     

First-passage failure of strongly nonlinear oscillators under combined harmonic and real noise excitations

First-passage failure of strongly nonlinear oscillators under combined harmonic and real noise excitations is studied. The motion equation of the system is reduced to a set of averaged Itô stochastic differential equations by stochastic averaging in the case of resonance. Then, the backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function and the conditional probability density and mean first-passage time are obtained by solving the backward Kolmogorov equation and Pontryagin equation with suitable initial and boundary conditions. The procedure is applied to Duffing–van der Pol system in resonant case and the analytical results are verified by Monte Carlo simulation.     

Noise influenced elastic cantilever dynamics with nonlinear tip interaction forces

In this work, the authors study the influence of noise on the dynamics of base-excited elastic cantilever structures at the macroscale and microscale by using experimental, numerical, and analytical means. The macroscale system is a base excited cantilever structure whose tip experiences nonlinear interaction forces. These interaction forces are constructed to be similar in form to tip interaction forces in tapping mode atomic force microscopy (AFM). The macroscale system is used to study nonlinear phenomena and apply the associated findings to the chosen AFM application. In the macroscale experiments, the tip of the cantilever structure experiences long-range attractive and short-range repulsive forces. There is a small magnet attached to the tip, and this magnet is attracted by another one mounted to a high-resolution translatory stage. The magnet fixed to the stage is covered by a compliant material that is periodically impacted by the cantilever’s tip. Building on their earlier work, wherein the authors showed that period-doubling bifurcations associated with near-grazing impacts occur during off-resonance base excitations of macroscale and microscale cantilevers, in the present work, the authors focus on studying the influence of Gaussian white noise when it is included as an addition to a deterministic base excitation input. The repulsive forces are modeled as Derjaguin–Muller–Toporov (DMT) contact forces in both the macroscale and microscale systems, and the attractive forces are modeled as van der Waals attractive forces in the microscale system and magnetic attractive forces in the macroscale system. A reduced-order model, based on a single mode approximation is used to numerically study the response for a combined deterministic and random base excitation. It is experimentally and numerically found that the addition of white Gaussian noise to a harmonic base excitation facilitates contact between the tip and the sample, when there was previously no contact with only the harmonic input, and results in a response that is nominally close to a period-doubled orbit. The qualitative change observed with the addition of noise is associated with near-grazing impacts between the tip and the sample. The numerical and experimental results further motivate the formulation of a general analytical framework, in which the Fokker–Planck equation is derived for the cantilever-impactor system. After making a set of approximations, the moment evolution equations are derived from the Fokker–Planck equation and numerically solved. The resulting findings support the experimental results and demonstrate that noise can be added to the input to facilitate contact between the cantilever’s tip and the surface, when there was previously no contact with only a harmonic input. The effects of Gaussian white noise are numerically studied for a tapping mode AFM application, and it is shown that contact between the tip and the sample can be realized by adding noise of an appropriate level to a harmonic excitation.     

Noise-induced chaotic motions in Hamiltonian systems with slow-varying parameters

This paper studies chaotic motions in quasi-integrable Hamiltonian systems with slow-varying parameters under both harmonic and noise excitations. Based on the dynamic theory and some assumptions of excited noises, an extended form of the stochastic Melnikov method is presented. Using this extended method, the homoclinic bifurcations and chaotic behavior of a nonlinear Hamiltonian system with weak feed-back control under both harmonic and Gaussian white noise excitations are analyzed in detail. It is shown that the addition of stochastic excitations can make the parameter threshold value for the occurrence of chaotic motions vary in a wider region. Therefore, chaotic motions may arise easily in the system. By the Monte-Carlo method, the numerical results for the time-history and the maximum Lyapunov exponents of an example system are finally given to illustrate that the presented method is effective.     

Stochastic Response of Offshore Structures via Statistical Cubicization

This paper deals with the stochastic response of single-degree-of-freedom structures with polynomial restoring force excited by random Morisons forces with mean current. The problem is recast by expressing the excitation by means of a cubic polynomial of the wave elevation, which in turn is assumed as a stationary zero-mean Gaussian process, whose spectral density is given by the output of a cascade of two second order linear filters having a Gaussian white noise as primary excitation. Thus, Itôs stochastic differential calculus becomes applicable, and the solution is pursued with a moment equation approach by using a suitable closure scheme. The results of the applications compare well with digital simulation.     

高斯色噪声和谐波激励共同作用下耦合SD振子的混沌研究

耦合SD振子作为一种典型的负刚度振子, 在工程设计中有广泛应用. 同时高斯色噪声广泛存在于外界环境中, 并可能诱发系统产生复杂的非线性动力学行为, 因此其随机动力学是非线性动力学研究的热点和难点问题. 本文研究了高斯色噪声和谐波激励共同作用下双稳态耦合SD振子的混沌动力学, 由于耦合SD振子的刚度项为超越函数形式, 无法直接给出系统同宿轨道的解析表达式, 给混沌阈值的分析造成了很大的困难. 为此, 本文首先采用分段线性近似拟合该振子的刚度项, 发展了高斯色噪声和谐波激励共同作用下的非光滑系统的随机梅尔尼科夫方法. 其次, 基于随机梅尔尼科夫过程, 利用均方准则和相流函数理论分别得到了弱噪声和强噪声情况下该振子混沌阈值的解析表达式, 讨论了噪声强度对混沌动力学的影响. 研究结果表明, 随着噪声强度的增大混沌区域增大, 即增大噪声强度更容易诱发耦合SD振子产生混沌. 当阻尼一定时, 弱噪声情况下混沌阈值随噪声强度的增加而减小; 但是强噪声情况下噪声强度对混沌阈值的影响正好相反. 最后, 数值结果表明, 利用文中的方法研究高斯色噪声和谐波激励共同作用下耦合SD振子的混沌是有效的.本文的结果为随机非光滑系统的混沌动力学研究提供了一定的理论指导.      

Stochastic averaging method for estimating first-passage statistics of stochastically excited Duffing–Rayleigh–Mathieu system

The first-passage statistics of Duffing-Rayleigh- Mathieu system under wide-band colored noise excitations is studied by using stochastic averaging method. The motion equation of the original system is transformed into two time homogeneous diffusion Markovian processes of amplitude and phase after stochastic averaging. The diffusion process method for first-passage problem is used and the corresponding backward Kolmogorov equation and Pontryagin equation are constructed and solved to yield the conditional reliability function and mean first-passage time with suitable initial and boundary conditions. The analytical results are confirmed by Monte Carlo simulation.     

Gauss白噪声外激下Rayleigh振子的平稳响应与首次穿越

研究了Rayleigh振子在Gauss白噪声外激下的平稳响应和首次穿越。首先利用随机平均法给出了系统随机平均It^o微分方程,对平均方程的稳态概率密度做了数值分析;然后建立了条件可靠性函数的后向Kolmogorov方程及首次穿越时间条件矩的Pontragin方程;最后对三组不同的参数值分析了首次穿越的概率统计特性。