拟可积哈密顿系统中噪声诱发的混沌运动

研究拟可积哈密顿系统在谐和与噪声激励联合作用下的混沌运动。通过对噪声性质的假定，并利用动力系统理论，给出了高维梅尔尼科夫方法应用于随机拟可积哈密顿系统的推广形式。根据这种推广的方法，研究了谐和与高斯白噪声激励联合使用下两自由度拟可积哈密顿系统 同宿分岔，得出了系统发生混沌运动的参数阈值，并由此讨论了噪声对系统的混沌运动的影响。蒙特-卡罗方法模拟、李雅普诺夫指数等数值结果表明，这种推广的方法是有效的。

NOISE-INDUCED CHAOTIC MOTIONS INQUASI-INTEGRABLE-HAMILTONIAN SYSTEMS

Engineering structures are often subjected to stochastic loadings, suchas those occurring due to wind, earthquakes and ocean waves, etc. Suchsystems used to be idealized to the deterministic ones with harmonicexcitations in order to study the chaotic dynaniics by the well-known Melnikovmethod. However, one knows that the validity of such idealized modelsdepends on whether an appropriate choice was made of the amplitude andfrequency of the harmonic function used as an idealization of the actualstochastic excitation. So there exist many drawbacks in such method.Recently, a more natural and effective approach is put forward to studythe chaotic behavior of the stochastic systems, i.e., to apply theMelnikov theory directly to such systems where the noisy excitationsare assumed to be bounded and uniformly continuous. There have been anumber of publications analyzing the effects of noises on the chaoticbehavior of dynamical systems, but till now, these studies are stilllimited to the systems with single degree of freedom. 　　In the presentpaper, the chaotic motions in quasi-integrable-Hamiltonian systems withmulti-degree of freedom under both harmonic and noisy excitations areinvestigated. When the excitations are assumed to besufficiently small and the noisy excitation is ensemble uniformlycontinuous, an extended form of high-dimensional Melnikov method isgiven to analyze the stochasticquasi-integrable-Hamiltonian systems by usingthe dynamical theory. By computing the mean value and the variance ofthe Melnikov integrals which can bedivided into the deterministic part and the stochastic part, thehomoclinic bifurcations of a two DOF quasi-integrable-Hamiltonian systemare studied in detail by this extended method, the parametric threshold forchaotic motions is obtained and the effects of noise on the system'smotions are also discussed. It is shown thatthe addition of extemal excitation of Gauss white noise can make theparametric threshold for chaotic motionsvary in a larger region, so that chaotic motions may occur more easily. 　　Moreover, simulations by Monte-Carlo method and computations of theLyapunov exponents, whichcharacterize the average exponential rate of divergence or convergenceof nearby orbits in phase space, areperformed. It is shown that the system's periodic and quasi-periodicmotions may become chaotic ones andtheir maximal Lyapunov exponents become positive when external Gaussianwhite noise is added to the system.Therefore these numerical results are in excellent agreement with thetheoretical ones and one then canconclude that the extended method given in the present paper isreasonable.