Chaos Control And Semiclassical Quantization Of Nonintegrable Hamiltonian Systems

The studying of chaotic dynamics of nonintegrable Hamiltonian systems is a very important theoretic and applicable topic, and it's one of the frontier problems of nonlinear fields. By studying Henon-Heiles system which is a typical nonintegrable with two degrees of freedom, this paper analyses and discusses theorizing and numbering value theoretically and numerically by chaotic and periodic trajectories. Based on these discussing, we further put forward the chaotic movement of nonintegrable Hamiltonian systems, and provide the theoretical basement for studying the dynamical behavior of molecular systems. The main contents and innovation points are organized as follows:1. In nonintegrable Hamiltonian systems the periodic orbits play the important role which in the study of the dynamics of the systems, control the system's chaos and quantize the nonintegrable system. This paper presents the Multipoint Shooting Method for locating the unstable periodic orbits, and we find some periodic orbits in Henon map, Henon-Heiles system, DKP system. We plot the Poincare surface of section and trajectories, and discuss the Multi-shooting Method's accuracy and higher efficiency contrasting to Newton Method. It shows that the method utilizes all the forecast points of period n, which can eliminate the long-time exponential instability of unstable orbits by splitting an orbit into a number of short segments, each with a controllable expansion rate.2. This paper studies the phase space trajectories in Henon-Heiles system by utilizing the pictures of Poincare surface of section. It analyses the character of dynamics, shows the phenomena in process of producing chaos, and summarizes the rule. At the same time, this paper studies the system's three kinds of period 1 trajectories. By discussing the structure stability of the system and its local topological structure in the space, we get a series of critical values when the periodic points lose its stability.3. When the energy is a constant, we utilize LDG method in the period 1 orbits and period 3 orbits which we find to control, and we can make the system switch the target orbits on purpose. Furthermore, we discuss the virtue and disadvantage about the method.4. Action integrals of the periodic orbits of nonintegrable two-mode system of Henon-Heiles are analyzed to show ample linear relations. By calculating the system's action integrals of some periodic orbits and at some energy, we quantize and analyses the nonintegrable system, and show the intrinsic correspondence of classical and quantal properties.