Analysis And Anti-Control Of Chaos

Chaos is a pseudo-random phenomenon generated in the nonlinear deterministic systems. Its most prominent feature is highly sensitive to the initial states, i.e. commonly known as "The Butterfly Effect". This feature is an important symbol which distinguished the chaos from other physical phenomena. It also plays a key role for determining the great application values of chaos in many disciplines.In recent years, chaos has flourish energetically in semiconductor lasers, stock economy, quantum mechanics, neural network, electronic circuits, oscillating chemical, biomedical and so on. Its research contents and methods have also been greatly expanded. In the early work, scholars have mainly focused on the suppression and elimination of chaotic behavior, for making the system status tend to be stable or obtaining the required periodic motion. However, with the deepening of research, some scholars have found that chaos is beneficial to a considerable part of the systems and has potential applications in engineering technology, such as secure communication, weather forecast, etc. Therefore, a new subject, namely generation, maintenance and enhancement of chaotic motion, began to break into the vision of scholars, which is academically to be said "anti-control of chaos" or "chaotification". Anti-control of chaos which breaks the old and inappropriate understanding of chaos is a further development of chaos theory. It really makes chaos began to transition from theory analysis to practical application. Also, abundant advantages of chaos all have a role to play in practice. Though there are some investigations on chaotification recently, the study of chaotification is still in the preliminary stage on the whole, and many problems need to be explored and solved. Based on this point, this dissertation studied the problem of chaotification in depth. Some new chaotic systems are proposed, and the chaotification of linear system and complex network are realized by using impulsive and feedback control. The contents of this dissertation are summarized as follows:Based on the Genesio-Tesi system, eight new chaotic systems with similar form are proposed. These systems have seven terms and one nonlinearity. One of the systems is studied in detail. The stability and Hopf bifurcation of the system are analyzed. The existence of the homoclinic orbit for the system is proved by applying the undetermined coefficient method, and the Si’lnikov criterion guarantees that the system has Smale horseshoe chaos. An analog circuit diagram is also designed by Multisim software to realize the chaotic system.A new three-dimensional system with multiple chaotic attractors is reported. One remarkable feature of the system is that it can generate multiple chaotic and multiple periodic attractors from different initial values for given system parameters. The presence of coexisting chaotic and periodic attractors in the system is investigated. Moreover, it is easily found that the system also can generate four-scroll chaotic attractor in a wide range of system parameters.The switching method and coordinate transformation method are used to generate multi-wing chaotic attractors from a Lorenz-like system. Both methods which can be directly applied to the original system for constructing multi-wing attractors have general applicability. The principle of the switching method is specifically introduced by analysizing the4-wing and6-wing attractors. Meanwhile, the effectiveness of the coordinate transformation method is presented by simulation.The problem of chaotification of linear system is studied. It is verified that the system can transition from non-chaotic state to chaotic state by using the implusive control. The specific implusive gain functions are designed for the one, two, and n-dimensional linear systems. Moreover, the implusive control is applied to complex network, and a sufficient condition for the emergence of chaos in the network is obtained.The chaotification of complex network via state feedback control with mod-operation is investigated. Based on the definition of Li-Yorke chaos, sufficient condition for transition the complex network from non-chaotic to chaotic was obtained by rigorous mathematically proof. The effectiveness and the correctness of the theoretical analysis are illustrated intuitively by simulation experiments.The dynamical behaviors including bifurcation and chaos of a delayed neural network system with three elements are investigated. The existence of the Hopf bifurcation is well studied by analyzing the corresponding characteristic equation. It is shown that the Hopf bifurcation occurs as the time delay reaches a certain value. Also, the chaotic behavior of the system is presented by simulation.Finally, a summary has been presented for our investigations in this dissertation. The directions and contents of the further researches are made for the anti-control of chaos.