【作者】 刘中华；

【导师】 朱位秋；

【作者基本信息】 浙江大学 ， 固体力学， 2005， 博士

【摘要】 本论文研究了拟可积哈密顿系统在有界噪声激励下的随机混沌和概率为1渐近稳定性及在高斯白噪声激励与时滞反馈控制下的随机响应。在随机混沌研究中,应用随机Melnikov过程的均方准则研究了单摆在有界噪声激励下发生混沌或随机混沌时有界噪声激励的临界幅值;对于耦合的单摆-谐振子系统,先用Melnikov函数研究了在哈密顿扰动下发生混沌的必要条件;然后用推广的随机Melnikov过程方法研究了在非哈密顿扰动下发生随机混沌的必要条件;用最大Lyapunov指数及Poincaré截面方法结果与上述理论结果作了对比。在概率为1渐近稳定性研究中,先用随机平均法得到关于系统慢变过程的It(?)随机微分方程,再引入新的范数,得到了最大Lyapunov指数的近似表达式,研究系统的概率为1渐近稳定性。在高斯白噪声激励下时滞反馈控制的拟可积哈密顿系统的随机响应研究中,分别对具有时滞bang-bang反馈控制和时滞无界反馈控制两种情形,将时滞的状态变量在平均意义上用无时滞的状态变量近似,由此得到了无时滞受控的拟可积哈密顿系统,再运用随机平均法,得到了系统的响应,据此研究时滞反馈控制的控制效果。结果表明,时滞对控制效果有明显的破坏作用,所有理论结果都与数字模拟结果做了对比。更多还原

【Abstract】 The stochastic chaos and the stochastic asymptotic stability with probability 1 of quasi-integrable Hamiltonian systems under bounded noise excitations and the response of quasi-integrable Hamiltonian systems with delayed feedback control under Gaussian white noise excitations are studied extensively. In the study of the stochastic chaos in simple pendulum, the random Melnikov process is derived and the mean-square criterion is used to determine the threshold amplitude of the bounded noise excitation for the onset of the chaos or random chaos in the system. For the coupled simple pendulum and harmonic oscillator, the Melnikov function is used to determine the condition for the onset of chaos in the case of Hamiltonian perturbations. In the case of non-Hamiltonian perturbation, the generalized random Melnikov process and mean-square criterion are used to determine the threshold amplitude of the bounded noise for the onset of random chaos. The largest Lyapunov exponent and Poincare map are then used to verify all the results obtained by using the Melnikov method. For the study of the asymptotic stability with probability 1 the stochastic averaging method is used to derive the Ito differential equations for the slow varying processes. By introducing a new norm, the approximate formula for the largest Lyapunov exponent is derived. The necessary and sufficient condition for the asymptotic stability with probability 1 is then obtained by using the largest Lyapunov exponent. For the study of the response of the quasi-integrable Hamiltonian systems with delayed feedback control under Gaussian white noise excitations, the time-delayed state variables are approximated with the state variables without time delay. The stochastic averaging method for quasi-integrable Hamiltonian systems is used to predict the response of the quasi-integrable Hamiltonian system without time delay. The effects of time delayed feedback control on the response are studied. All theoretical results are compared with those obtained by using digital simulations.更多还原