Bounded Noise Harmony And Incentives Under The Combined Effects Of Nonlinear Chaos

In this dissertation, the chaos and response of a nonlinear dynamical system to combined deterministic and narrow-band random excitations is investigated by using analytical and numerical methods. Some new conclusions are obtained. Based on these conclusions, chaos control by harmonic excitation with proper random phase is presented. The main contents of the dissertation are as follows:Chapter one summarizes current situations of chaos in random dynamical systems and states that stochastic excitations can generate chaos or suppress chaos although their function to the nonlinear system is to be discovered.Chapter two introduces mathematical definition, the basic characteristics, and analytical and numerical methods of chaos in both deterministic and stochastic systems.Chapter three studies the chaos and response of a nonlinear dynamical system to combined deterministic and narrow-band random excitations. The multiple-scale method is used to reduce the system and the result of numerical methods shows good consistence between the reduced system and the original system. For the reduced system, the mean square criterion of stochastic Melnikov process is derived to give the critical values of the probable onset of chaos and the conclusion is that the critical value turns from increase to decrease as the amplitude of Weiner process increases in the interested parameter range. Also, the numerical methods and the analytical method lead to consistent conclusions.Based on above conclusions, chapter five presents a new kind of chaos control by harmonic excitation with proper random phase, i.e. suppressing or generating chaos by properly adjusting the amplitude of random phase. Two illustrative examples, a Duffing oscillator subject to a harmonic parametric control and a driven Murali-Lakshmanan-Chua (MLC) circuit imposed with a weak harmonic control, are presented here to show that the random phase plays a decisive role for control function. The method for computing the top Lyapunov exponent is based onKhasminskii's formulation for linearized systems. Then, the obtained results are further verified by the Poincare map analysis on dynamical behavior of the system, such as stability, bifurcation and chaos. Both two methods lead to fully consistent results.Chapter five concludes the work and points out some aspects to be further studied on nonlinear dynamical system to combined deterministic and narrow-band random excitations.