Bifurcation And Chaos Study Of Nonlinear Vibrations And Fluctuations In The System

Based on a complete summary and examination of the history and the actuality of the bifurcation and chaos research, a systematic investigation into the fundamental theory and application of the bifurcation and chaos phenomenon is made with the ordinary mathematic theory that is familiar to the researchers in the mechanics and vibration field. Some important achievements are obtained in this dissertation.The whole paper consists of seven chapters. Chapter 1 is an introduction of the dissertation. The importance of the research of the bifurcation and chaos is introduced, the advances and actuality of the research are summarized, and the main contents of the dissertation are reported as well. Chapter 2 deals with the fundamental theory and methods of bifurcation. The conditions giving rise to the static bifurcation are derived at first, then, the improved LS reduction method is studied and the centre manifold method is simplified. In chapter 3, a new method that gives the universal unfolding of the bifurcation functions is established. In chapter 4, the bifurcation of the complex nonlinear oscillator of van der Pol- Duffing-Mathieu's is studied and the common bifurcation property of the oscillator in the situations of nonresonance, principal resonance, ultraharmonic resonance and non-subharmonic resonance is revealed. The bifurcation of strongly nonlinear oscillator is also explored in this chapter. The research work on the chaos is carried out in chapter 5. The properties of chaos sensitively dependent on the initial conditions and the perturbations are proved and a method of controlling chaos of Duffing oscillator is investigated as well. In chapter 6, the bifurcation and chaos of continuous nonlinear systems are researched. The method of change partial differential equation into ordinary differential equation is classified, the bifurcation and chaos in nonlinear Schrodinger system is detected and the effectiveness of controlling chaos of buckling beam with parametric perturbation method is studied in details. Chapter 7 is the summary and conclusion of the dissertation.