Estimation Of Correlation Dimensions And Lyapunov Exponent Of Typical Nonlinear Systems Under The Bounded Noise Excitation

This thesis discussed the effects of the bounded noise excitation on the chaotic responses of the Holmes, softening Duffing type oscillator and Lorenz oscillator. By the dynamical theory and the stochastic oscillation theory, the author studied the complicated dynamical behavior of these three kinds of nonlinear systems under determinate excitation and stochastic noise excitation from their phase portraits, time histories, correlation dimensions and Lyapunov exponents.The Monte-Carlo method and the phase space reconstruction skill are firstly employed to obtain the time series of these three kinds system under the effects of periodic excitation and bounded noise excitation. The G-P algorithm and small data sets method are then used to calculate the correlation dimension and maximum Lyapunov exponent of system's response respectively. The noise-induced chaotic responses were studied based on the numerical results of the correlation dimension and the maximum Lyapunov exponent.This thesis was arranged as follows: The first chapter briefly introduced the theory achievements and developments of nonlinear stochastic dynamics. Some methods used to analyze nonlinear stochastic dynamics, such as the FPK method, the method of stochastic equivalent linearization, stochastic perturbation method were briefly introduced. Besides, the recent results on the Hamiltonian framework developed by Professor Zhu Weiqiu are mentioned to emphasize the need for the current work. In chapter 2, we present the conception of bifurcation and chaos. Several qualitative measures to find chaos, such as Poincare method, the power spectrum method, the correlation dimension and Lyapunov exponent, as well as the focus of the current work are emphasized. In chapter 3, the G-P algorithm was employed to analyze in detail on the complex dynamical behaviors of these three kind systems under different excitation by their corresponding correlation dimensions. In chapter 4, two traditional methods (Wolf method and Small data sets method) were listed for calculate the maximum Lyapunov exponent. For the typical systems studied in the current work, the small data sets method was used to discuss the stochastic dynamical behaviors under different excitations. Chapter 5 presented the summarization and discussion.