The Stochastic Chaos And Asymptotic Stability With Probability 1 Of Quasi Integrable Hamiltonian Systems Under Bounded Noise Excitations And The Response Of The Systems With Delayed Feedback Control

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.