Dynamical Behaviour Analysis And Existence Verification For Chaos Of Complex Chaotic Systems

Over the past half century, the world has witnessed the rapid development of Chaos theory. Chaos has penetrated into almost every science and engineering field, and its application has shown the tremendous vitality in many aspects of human life. Recently, researchers have turned their close attention from the simple analysis of chaos to deeply analyze the essence of chaos theoretically, so as to control and make use of chaos in engineering fields, such as the chaotic secure communication and the streaming media encryption. In particular, hyperchaos is more complex than chaos, and has stronger randomness and unpredictability. Therefore, hyperchaos has great potential applications in chaos-needed fields. In this dissertation, the existence of chaos and dynamical behavior analysis of some chaotic systems, and also the generation and theoretical analysis of two hyperchaotic systems are investigated systematically. The main work can be summarized as follows:(1) Based on the state feedback control method, two strong four-dimensional autonomous continuous hyperchaotic systems are proposed. By analyzing the Lyapunov exponent spectra, bifurcation diagram, phase portraits, Poincare sections, it can be obtained that the new systems have very rich dynamical behaviors, and they can evolve into period, quasi-period, chaos and hyperchaos. The two new hyperchaotic systems both have wider hyperchaotic parameter region, two relatively larger positive Lyapunov exponents. Furthermore, the pitchfork bifurcation and Hopf bifurcation in one hyperchaotic system are analyzed in detail. Moreover, a new intermittent route from period to hyperchaos is observed in another hyperchaotic system. The two new hyperchaotic systems have complex and rich nonlinear dynamics, and thus have great theoretical significance and application value.(2) The topological horseshoe dynamics are investigated in three different three-dimensional continuous systems, namely the famous Chen system, a complex chaotic system presented by Qi et al. and an epidemic disease model. By utilizing the recent famous topological horseshoe theory, the existence of topological horseshoe is verified in each system, and the topological entropy is also estimated during the process of proof. Thus, the existence of chaos is proved in a computer assisted manner. The chaotic system proposed by Qi et al. has not only bigger Lyapunov exponent but also bigger lower bound estimation of topological entropy. The result shows that the Qi chaotic attractor has very complex dynamical behaviors.(3) Chaotic dynamics in two different economic systems are investigated in detail. One is a microeconomic model, that is, a Cournot duopoly model proposed by Kopel in 1996. The existence of horseshoe chaos is first verified in this economic model, and then many kinds of nonlinear dynamics, such as transient chaos, nonsmooth business cycle, co-existence of two chaotic attractors, and two different types of economic intermittency, are observed and analyzed. Besides, the influence of chaos on the profits of duopolists is analyzed. It is numerically demonstrated that chaotic market is not totally harmful, that is to say. either of the duopolists could be beneficial from a chaotic market. A state feedback controller is designed to control chaos to the Nash equilibrium, as a result of which the two duopolists share the market with good equity. A macroeconomic model, namely a business cycle model is also learned, the topological horseshoe dynamics are analyzed, and thus the existence of chaos in this model is proved in a computer assisted manner.(4) The fractional-order Chen system and two modified fractional-order Chua circuits are analyzed. The computer-assisted proof for the existence of chaos in fractional-order Chen system is provided. The analog circuit implementation of fractional-order Chen system is also given. When the sine function is applied to replace the nonlinear function of Chua's circuit, and the integer order is replaced by a fractional order, the modified fractional order Chua system can generate n-scroll chaotic attractor. Another modified fractional order Chua system with the sawtooth and staircase function being the nonlinearity, can generate n×m-scroll chaotic attractor.