Dynamical Analysis Of The Softening Duffing Oscillator Under The Bounded Noise Excitation

This thesis discussed the effects of the bounded noise excitation on the period-doubling bifurcation and chaotic responses of the softening Duffing oscillator, and the corresponding statistical characteristics were also analyzed. By the dynamical theory, the stochastic oscillation theory and the Monte-Carlo method, the author studied the complicated dynamical behavior of the softening Duffing oscillator with the bounded noise excitation. The noise-induced chaotic responses were studied based on the numerical results for the Poincare map and the maximum Lyapunov exponent. Due to the difficulty from the strongly nonlinear and stochastic nature of the system, the method from Chen and Cheung was employed to obtain the stochastic averaging equation and the FPK equation, from which the stationary probability density function was simulated. It was shown that, the regular motions in the original deterministic system would be masked by the external bounded noise excitation, and the stochastic excitation played a dispersive role to the dynamical behavior and could delay the bifurcation of the system. In addition, the bounded noise excitation could bring forward the internal chaos, e.g., chaotic responses would arise more readily. The thesis was arranged as follows:The first chapter briefly surveyed the studies on chaos, some discussions on the noise-induced chaos by others were also introduced. In Chapter â…¡, the softening Duffing oscillator under the bounded noise excitation was presented, the erosion of the safe basin of the system was discussed in detail. Based on the simulation results from Chapter â…¡, the initial point for simulating the dynamical behavior was chosen, and the bifurcation scenario, the phase portrait and the time histories, etc., were presented. The noise-induced chaos was testified by the Poincare map and the maximum Lyapunov exponent. These results were given in Chapter â…¢. In the fourth chapter, theoretical analysis was performed by using the stochastic oscillation theory, the stochastic averaging equation and the FPK equation were deduced and some simulation results about the stationary probability density function of such system were given. Chapter â…¤ presented the summarization and discussion.