Bounded Noise Perturbations Of Nonlinear Dynamical System Under Study

In this dissertation, the response and stability in several typical strongly stochastic nonlinear systems perturbed by random narrow-band noise and the chaotic motion of a nonlinear system with triple well under harmonic and bounded noise excitations are investigated. The main contents of the dissertation are as follows:Chapter one reviews briefly the current developments and problems of weakly/strongly nonlinear stochastic dynamical system, and introduces the main contents of this dissertation, together with the preparations including the method of MLP, the method of multiple scales, the Routh-Hurwitz criterion and the models of bounded narrow-band noise.Chapter two investigates the 1/2 subharmonic resonant response of a strongly nonlinear Van der Pol-Duffing oscillator subject to parametric random narrow-band excitation. The technique of MLP method is used to introduce a new expansion parameter, and then the multiple scales method is applied to determine the modulation equations for amplitude and phase of the response. The maximum Lyapunov exponent is obtained analytically and the dynamics near the resonant domain is analyzed. Numerical simulation is carried out to verify the analytical results and the excellent agreement between theoretical results and numerical ones can be found immediately. Thus the present method combining the MLP method with the multiple scales method is applicable to solve strongly nonlinear problems to parametric random narrow-band excitation.Chapter three studies the prinple resonant response of a strongly nonlinear Duffing-Rayleigh oscillator to additive random narrow-band excitation. Firstly, a new small expansion parameter is introduced by the parameter transformation technique. Then the multiple scales method is applied to determine the modulation equations for amplitude and phase of the response.The steady state mean square response is obtained by the moment method and perturbation method and its local stability is checked by Routh-Hurwitz criterion. Analytical results are verified by numerical simulations, which indicates that the present method combining the parameter transformation technique with the multiple scales method is adapt to solve strongly nonlinear problems to additive random narrow-band excitation.Chapter four investigates the chaotic behaviors of a Duffing oscillator with triple well possessing both homoclinic and heteroclinic orbits subject to harmonic and bounded noise excitations. From Melnikov theory, the semi-analytical criteria for the occurrence of transverse intersection on the surface of homoclinic and heteroclinic orbits are derived, which are complemented by the numerical simulations from which we show the bifurcation surfaces and the fractality of the basins of attraction. The results reveal that for larger noise intensity the threshold amplitude of bounded noise for onset of chaos will move upwards as the noise intensity increases, which is further verified by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the less possible chaotic domain in parameter space. The effect of bounded noise on Poincare maps of the system responses is also discussed which indicates the chaotic attractor is diffused as the noise intensity increases, and the larger the noise intensity results in the more diffused attractor.Chapter five concludes the work and innovation of this dissertation, and points out some aspects to be further studied on stochastic nonlinear system.