The Homoclinic Bifurcation And Chaos Of Piece-Smooth And Discontinuous System Subjected To Impluse Excitation

Impulse excitation is the significant nonsmooth factor occurred in contemporary sciences and engineering applications. Due to the strong nonlinear and singularities caused by the impulse excitation, the dynamic response of the nonlineai dynamic system is very complex. But the application of the conventional dynamic theory is strictly limited by transient impulse. It is urgent to set up a theory of impulsive nonlinear dynamics to develop the methodologies for the impulsive nonlinear dynamical system in science and engineering. In this paper, some models of nonlinear systems subjected to impluse excitation and periodic excitation are constructed. And the well-known Melnikov theory for smooth systems is expanded into the impulsive nonlinear dynamical system. The efficiency of theoretical results is verified by the numerical analysis, which reveals the effects of impact impluse excitation and non-impact impluse excitation on the behavior of nonlinear dynamical systems. The concrete content of this paper is arranged as follows:In Chapter1, relevant research backgrounds and methods are introduced. The classification of non-smooth dynamical systems and the nature of impulse dynamical systems are briefly described. The present situation of research all around the word is analyzed and the main contents, innovations and limitations of the paper are summarized.In Chapter2, based on the traditional pendulum, the bilateral rigid impact model under periodic excitation is established and the existence of non-smooth homoclinic orbits of the impact system is discussed. And then, the corresponding Melnikov function of non-smooth homoclinic orbits is introducted to analyze the homoclinic bifurcation and chaos of the impact system by use of the application and numerical simulation.In Chapter3, the homoclinic bifurcation and chaos of a class of piece-smooth and discontinuous system under a kind of non-impact impluse excitation are studied. First of all, the inverted pendulum model under a kind of non-impact impluse excitation is introduced to illustrate that impulse plays a special constrained role in the system. And a class of impulsive differential equations with discontinuous homoclinic orbits is proposed. And then, the Melnikov function of homoclinic orbit about this class of non-smooth system under non-impact impluse excitation is detected, which leads to the generalization of Melnikov theory in the field of non-smooth with impulse. At last, the theoretical results are verified in two ways. On the one hand, the unity of the Melnikov function is proved theoretically. On the other hand, the efficiency of the criteria for bifurcation and chaos mentioned above is verified by the threshold curves, bifurcation diagram and Poincaré surface of section.In Chapter4, the homoclinic bifurcation and chaos of a class of segmented motion systems caused by both non-impact impluse excitation and impact impluse excitation are studied. Firstly, the simple pendulum model under both non-impact impluse excitation and impact impluse excitation is introduced to illustrate the special constraint effect about impulse in the system. And a uniform impulsive differential equation with discontinuous homoclinic orbits is proposed. Secondly, the Melnikov function of homoclinic orbit about this class of rigid impact system under non-impact impulse perturbations is obtained, which achieves the further innovation of Melnikov theory in the field of non-smooth with impulse. Finally, the unity of the Melnikov function is proved and the efficiency of the criteria for bifurcation and chaos mentioned above is verified by numerical simulation.In Chapter5, we summarize the major work of paper and provide a prospect on the further development of Melnikov theory in the multiple-degrees-of-freedom nonlinear systems subjected to impluse excitation.