随机干扰与随机参数激励联合作用下的Hopf分叉

本文研究van der Pol-Duffing型的非线性振子在随机干扰和随机参数联合作用下的Hopf分叉现象。本文所得结果证实了当系统处在于Hopf分叉点附近时,对系统的参数的变化具有敏感性。在研究过程中,我们利用Markov扩散过程逼近系统的随机响应,得到了沿稳定矩的概率1稳定和矩稳定的条件。对于非线性振子,我们得到了振幅过程的稳态概论密度函数。研究发现,确定性系统的Hopf分叉点在随机参数作用下具有漂移现象,这种漂移是由系统的性质所决定的,当分叉点为超临界的,分叉点向前漂移;而当分叉点为亚临界时,这种漂移是向后的。当系统处在外部随机干扰作用下时,系统出现非零响应。另外我们发现,稳态矩的分叉与其阶数无关。     

随机ARNOLD系统的稳定性与分叉

本文详细讨论了当ｎ＝２时Ａｒｎｏｌｄ系统在小强度的随机参数激励扰动下，系统的运动稳定性及分叉。为了研究系统响应的统计特性，本文使用了Ｍａｒｋｏｖ近似技巧。在线性系统的情形，给出了系统矩稳定及样本稳定的充分必要条件。在非线性情形，本文的结果表明随机扰动可使系统的分叉点发生漂移     

一般力学中动力系统的非线性和混沌的最新进展与展望

本文从力学的角度出发综述了摈睛为国内外在压力系统的非线线和混沌方面的主要成果和发展前景。首先，本文阐述了非线性动力系统发展的必要性，其欠，我们详细的综述了非线性振动和局部分叉，全局分叉和混沌，非线笥随机系统的振动和分叉，和某些应用基础理论问题等四个方面近期国内外的主要研究成果。     

C^r随机中心流形定理

随机中心流形定理在非线性随机分叉理论的研究中具有关键作用。本文对Ｃ＾ｒ非线性随机系统给出Ｃ＾ｒ中心流形的存在性定理，并讨论了中心流形的稳定性及例子。     

非线性随机振动理论的近期进展

本文评述非线性随机振动理论近１０年来的进展,内容包括精确平稳解,等效非线性系统法,非线性系统的窄带随机激励,随机分叉,以及其他非线性随机响应预测方法。      

裂纹转子分叉、混沌行为研究中的映射延拓综合法

介绍一种能全面计算周期激励下非线性系统周期响应，拟周期响应和混沌响应的新算法－映射延拓综合法，它可方便地确定拟周期响应和混沌响应对应的系统参数区间。应用此算法对具有非线性刚度的裂纹转子系统裂纹扩展故障特征问题进行了研究，得到了以裂纹深度为分叉参数的系统稳态响应分叉解图。     

非线性Hill系统周期响应和分叉理论

本文首先给出并证明了解一类弱非线性问题的广义Ｇｒｅｅｅｎ法，利用这一方法求得非线性Ｈｉｌｌ振动系统在非共振和共振二种民政部下的周期响就以及描述周期响应特征的二次近似分叉方程应用具有Ｚ２对称的奇异性理论，建立了模参数与各物理参数之间的对应关系，通过对Ｚ２余维数≥３周期分叉解的普适性分类，全面分析了共振情况下物理参数对周期分叉解特征的影响。从而使二次近似分叉方程是否能够在拓扑意义下完全描述原系统的周期     

一类强非线性振动系统的分叉

对于参数激励和强迫激励共同作用的一类强非线性系统，本文先用改进的Ｌ－Ｐ方法求出了变换参数，使该系统的解能展为小参数的幂级数．然后利用多尺度法求出了该系统的分叉响应方程．研究了这类强非线性系统的余维１分叉问题，画出了转迁集和分叉图     

非稳态非线性油膜力作用下柔性轴裂纹转子的动力特性分析

分析了在动载轴承非稳态非线性油膜力作用下，具有横向裂纹柔性轴Jeffcott转子在非线性涡动影响下的动力特性。通过数值计算表明，在油膜失稳转速前，随着裂纹轴刚度变化比的增大，系统在低转速区域内具有丰富的非线性动力行为，出现倍周期分叉及混沌现象，涡动振幅随转速升高而减小，直到非稳态非线性油膜失稳，在无裂纹转子油膜临界失稳点处发现了类Hopf分叉现象，系统运动由平衡变为拟周期运动；裂纹转子在油膜临界失稳时的系统运动亦为拟周期运动，裂纹转子轴刚度变化对油膜失稳点及油膜失稳之后转子的运动影响不大，转子系统作拟周期运动。     

水下航行体纵倾航行稳定性研究

水下航行体运动方程含有诸多的非线性项,用传统的分析方法全面处理非线性问题有一定的难度.运用非线性科学中的分叉理论,系统地分析在纵倾控制系统作用下,水下航行体在退化平衡点处的航行稳定性.利用等价变换可将高维系统约化到低维的包含了原系统全部动力学特性的中心流形上来研究,得到跨临界分叉范式;分叉图表明姿态失稳及不规则弹道的机理;用系统状态方程的数值计算结果验证了系统的分叉现象.为水下航行体纵倾控制系统的参数设计提供了理论依据.     

随机激励的耗散的Hamilton系统理论的研究进展

近几年中,利用Ｈａｍｉｌｔｏｎ系统的可积性与共振性概念及Ｐｏｉｓｓｏｎ括号性质等,提出了高斯白噪声激励下多自由度非线性随机系统的精确平稳解的泛函构造与求解方法,并在此基础上提出了等效非线性系统法,提出了拟Ｈａｍｉｌｔｏｎ系统的随机平均法,并在该法基础上研究了拟Ｈａｍｉｌｔｏｎ系统随机稳定性、随机分岔、可靠性及最优非线性随机控制,从而基本上形成了一个非线性随机动力学与控制的Ｈａｍｉｌｔｏｎ理论框架．本文简要介绍了这方面的进展．     

Exact solutions for stationary responses of several classes of nonlinear systems to parametric and/or external white noise excitations

The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external white noise excitations are constructed by using Fokker-Planck-Kolmogorov equation approach. The conditions for the existence and uniqueness and the behavior of the solutions are discussed. All the systems under consideration are characterized by the dependence of nonconservative forces on the first integrals of the corresponding conservative systems and are called generalized-energy-dependent (G.E.D.) systems. It is shown taht for each of the four classes of G.E.D. nonlinear stochastic systems there is a family of non-G.E.D. systems which are equivalent to the G.E.D. system in the sense of having identical stationary solution. The way to find the equivalent stochastic systems for a given G.E.D. system is indicated and as an example, the equivalent stochastic systems for the second order G.E.D. nonlinear stochastic system are given. It is pointed out and illustrated with example that the exact stationary solutions for many non-G.E.D. nonlinear stochastic systems may be found by searching the equivalent G.E.D. systems.Project Supported by The National Natural Science Foundation of China. Accepted by XVIIth International Congress of Theoretical and Applied Mechanics.     

On homoclinic bifurcation system in the presence of stochastic perturbation

By virture of the singular point theory for one-dimension diffusion process and the stochastic averaging approach of energy envelop, the bifurcation behavior of a homoclinic bifurcation system, which is in the presence of parametric white noise and is concealed behind a codimension two bifurcation point, is investigated in this paper. Supported by the National Science Foundation of China under Grant No. 19602016.     

Parametric random excitation of nonlinear coupled oscillators

The stochastic bifurcation and response statistics of nonlinear modal interaction under parametric random excitation are studied analytically, numerically and experimentally. Two basic definitions of stochastic bifurcation are first introduced. These are bifurcation in distribution and bifurcation in moments. bifurcation in moments is examined for the case of a coupled oscillator subjected to parametric filtered white noise. The center frequency of the excitation is selected to be close to either twice the first mode or second mode natural frequencies or the sum of the two. The stochastic bifurcation in moments is predicted using the Fokker-Planck equation together with gaussian and non-Gaussian closures and numerically using the Monte Carlo simulation. When one mode is parametrically excited it transfers energy to the other mode due to nonlinear modal interaction. The Gaussian closure solution gives close results to those predicted numerically only in regions well remote from bifurcation points. However, bifurcation points predicted by the non-Gaussian closure are in good agreement with those estimated by numerical simulation. Depending on the excitation level, the probability density of the excited mode is strongly non-Gaussian and exhibits multi-maxima as predicted by Monte Carlo simulation. Experimental tests are carried out at relatively low excitation levels. In the neighborhood of stochastic bifurcation in mean square the measured results exhibit different regimes of response characteristics including zero motion and occasional small random motion regimes. These two regimes are characterized by the phenomenon of on-off intermittency. Both regimes overlap and thus it is difficult to locate experimentally the bifurcation point.     

Stochastic Hopf bifurcation in quasi-integrable-Hamiltonian systems

A new procedure is developed to study the stochastic Hopf bifurcation in quasiintegrable-Hamiltonian systems under the Gaussian white noise excitation. Firstly, the singular boundaries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system‘s energy levels with respect to the stochastic averaging method. Secondly, the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones. Lastly, a quasi-integrable-Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure.Moreover, simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure. It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system‘s parameters. Therefore, one can see that the numerical results are consistent with the theoretical predictions.     

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 and stability of strongly non-linear oscillators under wide-band random excitation

A new stochastic averaging procedure for single-degree-of-freedom strongly non-linear oscillators with lightly linear and (or) non-linear dampings subject to weakly external and (or) parametric excitations of wide-band random processes is developed by using the so-called generalized harmonic functions. The procedure is applied to predict the response of Duffing–van der Pol oscillator under both external and parametric excitations of wide-band stationary random processes. The analytical stationary probability density is verified by digital simulation and the factors affecting the accuracy of the procedure are analyzed. The proposed procedure is also applied to study the asymptotic stability in probability and stochastic Hopf bifurcation of Duffing–van der Pol oscillator under parametric excitations of wide-band stationary random processes in both stiffness and damping terms. The stability conditions and bifurcation parameter are simply determined by examining the asymptotic behaviors of averaged square-root of total energy and averaged total energy, respectively, at its boundaries. It is shown that the stability analysis using linearized equation is correct only if the linear stiffness term does not vanish.     

随机扰动的同宿分岔系统研究

利用一维扩展过程的奇点理论并结合能量包络的随机平均法，考查“隐藏在余维２分岔点之后”的同宿分岔系统受参激白噪声影响的分岔行为。