双稳杜芬振子的随机共振及其动力学机制

把矩方法应用于高斯白噪声和弱周期信号驱动的双稳杜芬振子,发现矩方法的收敛快慢与阻尼系数的大小有关,即在固定非线性参数的前提下,阻尼系数越大,收敛速度越快。在阻尼系数较大的情形,对于不同频率的弱周期输入信号,系统输出功率谱增益因子的演化防噪声强度呈单峰或双峰结构,亦即对于不同的激励频率,系统可表现出单峰或者重峰随机共振结构。为了解释这些共振结构,通过考察由波动谱密度定义的非零频率峰对噪声强度依赖性,发现重峰随机共振的发生在于噪声一方面抑制了井内运动,另一方面诱发了势垒上振动。研究结果为已有结论的修正,在统计力学等方面具有显著意义。     

STOCHASTIC P-BIFURCATION OF TRI-STABLE VAN DER POL-DUFFING OSCILLATOR

分析了乘性和加性噪声作用下三稳态Van der Pol-Duffing振子的随机P分岔. 首先用随机平均法得到系统的随机微分方程,求得系统响应幅值的稳态概率密度函数. 然后应用分岔分析的奇异性理论,求得随机P分岔发生的临界参数条件,得到多种定性不同的稳态概率密度曲线. 讨论了2种激励噪声强度和系统阻尼对响应稳态概率密度曲线峰的个数、各峰值相对大小的影响. 通过Monte-Carlo数值模拟对理论计算结果进行了验证.该方法可用于其他系统的随机P分岔分析.     

随机激励对软弹簧杜芬振子动力学的分散作用

讨论了有界噪声激励对软弹簧杜芬振子的倍周期分岔至混沌运动的影响。利用蒙特卡罗方法,通过对系统受侵蚀安全盆的变化状况进行了观察,并由此对后继动力学分析的初始点进行了选取。系统的相图、倍周期分岔图以及庞加莱映射图等方面的数值结果表明,外加随机激励的作用往往掩盖原确定性系统内在的规则运动,对原确定性系统的运动具有较典型的分散作用,可延缓系统的倍周期分岔,也可使得系统内在随机行为提前发生,即可使得系统更容易出现混沌运动。     

含时滞反馈与涨落质量的记忆阻尼系统的随机共振

针对具有记忆效应的欠阻尼系统,存在时滞反馈与涨落质量,本文主要研究了其输出稳态响应振幅的随机共振效应.首先通过引入新变量和运用小时滞近似展开理论,将具有非马尔科夫特性的原系统转化为等价的两维马尔科夫线性系统,再利用Shapiro-Loginov公式和Laplace变换获得了系统响应的一阶稳态矩和稳态响应振幅的解析表达式.结果表明:当系统参数满足Routh-Hurwitz稳定条件时,稳态响应振幅随质量涨落噪声强度、周期驱动信号频率以及时滞的变化均存在随机共振现象,其中随机多共振现象也被观察到.在适当范围内,通过控制时滞反馈,系统的随机共振效应随着时滞的增大而增强,而较长的记忆时间及增大阻尼参数均对共振行为呈现抑制作用.有效调控时滞反馈与记忆效应的变化关系将有助于增强系统对周期驱动信号的响应强度.最后,通过数值模拟计算验证了理论结果的有效性.     

非惯性参考系中弹性薄板的动力学分析：参数激励与强迫激励联 …

本文研究了非惯性参考系中弹性薄板的大范围运动与大变形运动相互耦合时的非共振分岔，在建立了该动力系统运动控制方程的基础上，利用多尺度法得到了参数激励与强迫激励联合作用下辈在惯性参考系中弹性薄板非共振时的分岔响庆方程及其在反动和几何尺寸两个分岔参数影响下的空间分岔集，讨论了该动力系统的稳定性，并给出了它的非共振分岔响应曲线。     

含时滞反馈与涨落质量的记忆阻尼系统的随机共振

针对具有记忆效应的欠阻尼系统, 存在时滞反馈与涨落质量, 本文主要研究了其输出稳态响应振幅的随机共振效应. 首先通过引入新变量和运用小时滞近似展开理论, 将具有非马尔科夫特性的原系统转化为等价的两维马尔科夫线性系统, 再利用Shapiro-Loginov公式和Laplace变换获得了系统响应的一阶稳态矩和稳态响应振幅的解析表达式. 结果表明: 当系统参数满足Routh-Hurwitz稳定条件时, 稳态响应振幅随质量涨落噪声强度、周期驱动信号频率以及时滞的变化均存在随机共振现象, 其中随机多共振现象也被观察到. 在适当范围内, 通过控制时滞反馈, 系统的随机共振效应随着时滞的增大而增强, 而较长的记忆时间及增大阻尼参数均对共振行为呈现抑制作用.有效调控时滞反馈与记忆效应的变化关系将有助于增强系统对周期驱动信号的响应强度. 最后, 通过数值模拟计算验证了理论结果的有效性.      

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.      

损伤效应对悬索2:1内共振响应影响分析

损伤是结构振动测试和运营维护中不可避免的问题，损伤效应会导致结构振动特性发生改变．本文以受损悬索为例，探究该非线性系统同时发生主共振和2:1内共振时，损伤效应对其面内耦合共振响应影响．首先基于哈密顿变分原理，引入与损伤程度、范围和位置相关的三个无量纲参数，建立受损悬索面内动力学模型，并推导其无穷维非线性运动微分方程．以2:1耦合共振为例，采用Galerkin法和多尺度法得到系统直角坐标形式的调谐方程．数值算例表明：损伤会导致悬索固有频率降低，使得频率间公倍关系发生改变，影响系统耦合共振响应；损伤会引发系统振动特性发生明显定量和定性改变，尤其是共振响应幅值及弹簧特性；损伤对直接激励模态响应幅值的影响比对内共振激发对响应幅值的影响要明显；损伤会导致霍普夫、鞍节点、叉形和倍周期分岔的位置发生偏移，从而影响分岔点附近系统的动力学行为；系统动态解和周期运动与损伤密切相关，损伤会导致系统展现出完全不同类型的吸引子．     

双频激励下Filippov系统的非光滑簇发振荡机理

旨在揭示含双频周期激励的不同尺度Filippov系统的非光滑簇发振荡模式及分岔机制.以Duffing和Van der Pol耦合振子作为动力系统模型,引入周期变化的双频激励项,当两激励频率与固有频率存在量级差时,将两周期激励项表示为可以作为一慢变参数的单一周期激励项的代数表达式,给出了当保持外部激励频率不变,改变参数激励频率的情况下,快子系统随慢变参数变化的平衡曲线及因系统出现的fold分岔或Hopf分岔导致的系统分岔行为的演化机制.结合转换相图和由Hopf分岔产生稳定极限环的演化过程,得到了由慢变参数确定的同宿分岔、多滑分岔的临界情形及因慢变参数改变而出现的混合振荡模式,并详细阐述了系统的簇发振荡机制和非光滑动力学行为特性.通过对比两种不同情形下的平衡曲线及分岔图,指出虽然系统有相似的平衡曲线结构,却因参数激励频率取值的不同,致使平衡曲线发生了更多的曲折,对应的极值点的个数也有所改变,并通过数值模拟,对结果进行了验证.     

双频激励下Filippov系统的非光滑簇发振荡机理

旨在揭示含双频周期激励的不同尺度Filippov系统的非光滑簇发振荡模式及分岔机制. 以Duffing和Van der Pol耦合振子作为动力系统模型,引入周期变化的双频激励项,当两激励频率与固有频率存在量级差时,将两周期激励项表示为可以作为一慢变参数的单一周期激励项的代数表达式,给出了当保持外部激励频率不变,改变参数激励频率的情况下,快子系统随慢变参数变化的平衡曲线及因系统出现的fold分岔或Hopf分岔导致的系统分岔行为的演化机制.结合转换相图和由Hopf分岔产生稳定极限环的演化过程,得到了由慢变参数确定的同宿分岔、多滑分岔的临界情形及因慢变参数改变而出现的混合振荡模式,并详细阐述了系统的簇发振荡机制和非光滑动力学行为特性.通过对比两种不同情形下的平衡曲线及分岔图,指出虽然系统有相似的平衡曲线结构, 却因参数激励频率取值的不同,致使平衡曲线发生了更多的曲折,对应的极值点的个数也有所改变,并通过数值模拟, 对结果进行了验证.      

Analysis of period-doubling bifurcation in double-well stochastic Duffing system via Laguerre polynomial approximation

The Laguerre polynomial approximation method is applied to study the stochastic period-doubling bifurcation of a double-well stochastic Duffing system with a random parameter of exponential probability density function subjected to a harmonic excitation. First, the stochastic Duffing system is reduced into its equivalent deterministic one, solvable by suitable numerical methods. Then nonlinear dynamical behavior about stochastic period-doubling bifurcation can be fully explored. Numerical simulations show that similar to the conventional period-doubling phenomenon in the deterministic Duffing system, stochastic period-doubling bifurcation may also occur in the stochastic Duffing system, but with its own stochastic modifications. Also, unlike the deterministic case, in the stochastic case the intensity of the random parameter should also be taken as a new bifurcation parameter in addition to the conventional bifurcation parameters, i.e. the amplitude and the frequency of harmonic excitation.     

Responses of Duffing oscillator with fractional damping and random phase

This paper aims to investigate dynamic responses of stochastic Duffing oscillator with fractional-order damping term, where random excitation is modeled as a harmonic function with random phase. Combining with Lindstedt–Poincaré (L–P) method and the multiple-scale approach, we propose a new technique to theoretically derive the second-order approximate solution of the stochastic fractional Duffing oscillator. Later, the frequency–amplitude response equation in deterministic case and the first- and second-order steady-state moments for the steady state in stochastic case are presented analytically. We also carry out numerical simulations to verify the effectiveness of the proposed method with good agreement. Stochastic jump and bifurcation can be found in the figures of random responses, and then we apply Monte Carlo simulations directly to obtain the probability density functions and time response diagrams to find the stochastic jump and bifurcation. The results intuitively show that the intensity of the noise can lead to stochastic jump and bifurcation.     

An approach to the problem of closure in non-linear stochastic mechanics

An approach combining the method of moment equations and the statistical linearization technique is proposed for analysis of the response of non-linear mechanical systems to random excitation. The adaptive statistical linearization procedure is developed for obtaining a more accurate mean square of responses. For these, a Duffing oscillator and an oscillator with cubic non-linear damping subject to white noise excitation are considered. It is shown that the adaptive statistical linearization proposed yields good accurate results for both weak and strong non-linear stochastic systems.Presented at the First European Solid Mechanics Conference, September 9–13, 1991. Munich, Germany     

Stochastic resonance phenomenon in an underdamped bistable system driven by weak asymmetric dichotomous noise

Stochastic resonance in an underdamped bistable system subjected to a weak asymmetric dichotomous noise is investigated numerically. Dichotomous noise is a non-Gaussian color noise and more complex than Gaussian white noise, whose waiting time complies with the exponential distribution. Utilizing an efficiently numerical algorithm, we acquire the asymmetric dichotomous noise accurately. Then the system responses and the averaged power spectrum as the signatures of the stochastic resonance are calculated by the fourth-order Runge?CKutta algorithm. The effects of the noise strength, the forcing frequency, and the asymmetry of dichotomous noise on the system responses and the effects of the forcing frequency on the averaged power spectrum are discussed, respectively. It is found that the increasing of the noise strength or the forcing frequency could strengthen the passage between the stable points of the system, and the system responses also display the asymmetry for the asymmetric dichotomous noise, which has not been discovered in other investigated results. Additionally, the averaged power spectrum exhibits the sharp peaks, which indicates the occurrence of stochastic resonance, and we also discover two critical forcing frequencies: one denoting the transformation of the peaks and another for the optimum on stochastic resonance.     

Bifurcation analysis in the system with the existence of two stable limit cycles and a stable steady state

Van der Pol–Duffing oscillator, which can be used a model for many dynamical system, has been widely concerned. However, most of the systems by scholars are either stable steady states or limit cycles. Here, the self-sustained oscillator with the coexistence of steady state and limit cycles, which is famous for describing the flutter of airfoils with large span ratio in low-speed wind tunnels, is treated in this paper. Using the energy balance method, the deterministic bifurcation of the tristable system with time-delay feedback is investigated. The presence of time-delay feedback expands the bifurcation range of the parameters, making the bifurcation phenomenon more abundant. In addition, according to the stationary probability density function obtained by the stochastic averaging method, stochastic bifurcation of the system with time-delay feedback and noise is explored theoretically. The numerical results confirm the correctness of the theoretical analysis. Transition between the unimodal structure, the bimodal structure and the trimodal structure is found. Many rich bifurcations are available by adjusting the time-delay and noise intensity, which may be conductive to achieve the desired phenomenon in the real-world application.

     

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.     

Stochastic P-bifurcation analysis of a fractional smooth and discontinuous oscillator via the generalized cell mapping method

The smooth and discontinuous oscillator with fractional derivative damping under combined harmonic and random excitations is investigated in this paper. The short memory principle is introduced so that the evolution process of the oscillator with fractional derivative damping can be described by the Markov chain. Then the stochastic generalized cell mapping method is used to obtain the steady-state probability density functions of the response. The stochastic response and bifurcation of the oscillator with fractional derivative damping are discussed in detail. We found that both the smoothness parameter, the noise intensity, the amplitude and frequency of the harmonic force can induce the occurrence of stochastic P-bifurcation in the system. Monte Carlo simulation verifies the effectiveness of the method we adopt in the paper.     

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.