Computer-Assisted Verification Of Behavior Complexity Of Dynamical Systems

In order to study the complex dynamical behavior of the system which cannot be studied by analyzing method, this paper develops theories and methods of computer-assisted verification. The computer-assisted verification of dynamical behavior is a new method to combine computer algorithm with profound mathematics theories, which is a very difficult job. The chaoticity verification is a hard problem, since it needs not only deeply dynamical theories, but also a great deal of numerical calculation and computer simulation as well as algorithm analysis.In order to study the chaoticity of dynamical system, the theory of topological horseshoe is introduced; then, series of algorithms for computer-assisted verification based on the theory of topological horseshoe are put forward and applied to the study of chemical systems and neural network systems. Algorithms and corresponding steps are given for computer-assisted verification. Methods for finding horseshoe in single-scroll chaotic attractor, double-scroll chaotic attractor and multi-scroll chaotic attractor are described, respectively. Furthermore, based on the discovery of horseshoe, a quantificational description of complexity of the system can be given.For the sake of investigating the dynamics effectively, this paper studies the algorithm for (un)stable manifold of dynamics and brings forward two new algorithms. New algorithms are economical and have an error deduction.Based on the above theories and algorithms, this paper analyzes several models of Hopfield neural network systems, Celluar neural network systems and chemical dynamical systems. This paper studies these systems by computer simulation, Lyapunov exponent calculation, computer-assisted verification and so on. Rich dynamical behaviors are found, such as chaos, hyperchaos, limit cycle, two-torus and so on. In this paper, two-torus phenomena are discovered in four-dimensional Hopfield neural networks; two-torus phenomena are discovered in four-dimensional Cellular neural networks. Furthermore, computer-assisted verifications of existence of two-tori are given in this paper. At the same time, verifications of chaoticity of several chaotic systems are given by the combination of topological horseshoe theory and computer-assisted verification. Using Poincarésection and Poincarémap, this paper gives an effective verification for the existence of limit cycles and the existence of attracting sets in chaotic system. The analysis and verifications of neural network systems and chemical systems show that the method of computer-assisted verification developed in this paper is effective, which can give computer-assisted verifications for the existence of limit cycles, the existence of two-tori, the chaoticity of dynamical system and the existence of attracting sets. Meanwhile, the method of computer-assisted verification is helpful in giving quantificational descriptions of complexity of the systems.