Studies On The Fractals And Bifurcations Of A Holmes-Duffing System Under Random Excitations

This thesis discusses the bifurcations and fractals of Holmes-Duffing oscillator under some kinds of excitations. In chapter 1, the history, the development trends and prospects of the research on chaos as well as some principal research methods and important concepts are introduced. In chapter 2, the bifurcations of the system are studied, and the factors are classified by interior factors and exterior factors based on their different effects on the bifurcations.In chapter 3, the Melnikov function of the system is calculated, and the effects of different excitations and factors on the system are discussed. It is found that the Melnikov function cannot be applied to the conservative system excited by single periodic force. The threshold value for chaos changes monotonously following, the change of the amplitude of excitations and the damping coefficient, but not for the excitation frequency. Chaos tends to arise more readily if a periodic excitation is added to a stochastically excited system. This conclusion is also applicable for parametric systems.The erosion of safe basins under some excitations is discussed in chapter 4. The studies are focused on the shape of basin boundary and the erosion speed. The safe basins of the system with or without damping are studied separately, from which a new phenomenon is found, i. e., the safe basin of a parametric system is symmetric about a point.The relation between the stochastic Melnikov function and the phase space flux is analyzed in chapter 5. The threshold value for chaos is calculated by two methods and the results are also testified by the simulations on safe basins. The results show the consistency of these three methods in computing the threshold value for chaos in the system, and the substantial differences are also discussed in the end part of this chapter.