Chaotic Dynamics Study Of Several Classes Of Systems

Chaos is one of popular research topics in recent years, and the rigorous verification ofchaotic behavior is one of difficult work. Traditional numerical study methods, such ascalculating Lyapunov exponents, bifurcation diagram and Poincarésection, possibly leadto wrong verification result because of calculation errors, for example, the maximalLyapunov exponents of system is positive, but the system is not chaotic, so it is necessaryto give rigorous verification of chaoticity of system. In this dissertation, by utilizing thenewly developed topological horseshoe theory and numerical techniques, we present thegeneral method of finding one dimensional stretching topological horseshoe, and verifythe chaoticity of several classes of dynamical system rigorously. The main work can beoutlined as follows:Basing on the theories of symbol dynamics and topological horseshoe, and bymeans of numerical technique, a simple Hopfield neural network model is studied, thegeneral idea and techniques of finding topological horseshoe are given in detail.According to the method of finding one dimensional stretching topological horseshoegiven in this dissertation, through multiple numerical simulations, a proper Poincarésection is selected and the corresponding Poincarémap is established, and on thecross-section a quadrangle is determined to find one dimensional stretching topologicalhorseshoe, the chaoticity of a modified Van der Pol-Duffing circuit system, nonlinearBloch system, and a simple nonlinear state feedback controlled system is rigorouslyverified. The phase portrait of the nonlinear state feedback controlled system is atwo-scroll attractors, the verification method of chaoticity is the application of method offinding one dimensional stretching topological horseshoe.The dynamical behavior of a simple Hopfield neural network model is studied. Firstly,by adjusting the value of one element of the connection matrix, the periodic motion andchaos behavior of the system is analyzed by means of numerical computation; at last, thechaoticity of the system is rigorously verified. The chaotic dynamics of a three species food chain model is investigated. Thedynamical behavior of the system is analyzed briefly by means of numerical method,and the chaotic behavior is rigorously verified by utilizing topological horseshoe theoryand numerical technique.The chaotic behavior of a voltage-mode controlled buck converter model is studied,which is typically a switched piecewise linear system. The chaotic attractor of the modelis 2-dimensional, by selecting a Poincarésection and defining the correspondingPoincarémap skillfully, the one dimensional stretching topological horseshoe embeddedin the planar hybrid system is verified rigorously.