Analysis And Controlling In Nonlinear Chaotic System

Chaology is a rising nonlinear science. Its research upsurge started from 1970s', butits origin could be traced back to 1830s'. In recent years, chaos theory is developed rapidlyunder the incessant efforts of scholars over the world, and the research areas of chaologyare also expanded thoroughly. With the intercrossing and penetration of different knowl-edge, chaos theory has been applied in many other fields effectively. The in?uences of chaosresearch towards modern sciences are not bounded in natural science, but cover almost allfields of science. Well, from the point of cybernetic, it yields a natural yet nontrivial ques-tion whether one can control chaos, including chaos control (chaos inhibition) anti-chaosControl(chaos producing and enhancing), and chaos synchronization.Meanwhile, in recent years, the fractional-order systems become a hot research topic.The concepts of fractional derivatives and integral were proposed three hundred years ago.Fractional calculus is the theory on the research and application of derivatives and integralof arbitrary order. It is a natural extension of the classical mathematics. In the last three orfour decades, many researchers have made a great effort to apply this knowledge in practiceand in different research fields. Now, the application domains of fractional calculus have in-creased significantly. Real systems in general are fractional-order systems, although in sometypes of systems the order is very close to an integer order. The control systems used so farwere all considered as integer-order systems, regardless of the reality, the reason is the highercomplexity and the absence of adequate mathematical tool. Since major advances have beenmade in fractional calculus in the last few years, the knowledge of fractional calculus be-gins to be applied in control theory and control engineering. It provides new landscape forcontrol theory and control engineering based on integer-order differential equations. Amongthose, the research of the fractional-order systems and chaos systems become increasinglyclose, because of their own nature, such as the fractal dimension, self-similarity and so on.However, as the theory of chaotic systems and fractional-order systems being in developing,many issues still need further study and discussion. Based on above, the research and contributions in this dissertation are divided into twoaspects: one is about controlling chaos in the integer-order nonlinear chaos systems; theother is study and discussion the analysis and controlling the fractional-order nonlinear chaossystems. Specifically,- By using Melnikov method of perturbation criteria, the conditions of periodicparamtric perturbations to control chaos in Lorenz system are discussed.- The single input state feedback approach for chaotifying a stable system is presented..Based on the Marotto theorem, it is proven theoretically that the closed-loop systemis chaotic in the sense of Li and Yorke.- The PI~αregulator of system states to control a kind of fractional-order chaos systemsis first introduced in this dissertation. At same time, a stability theorem of nonlinearfractional-order differential equations is proven theoretically. Then, according to that,a new criterion is derived for designing the controller gains for stabilization this kindof fractional-order chaos systems.- Based on the integer-order system, the harmonic balance principle is used to analyzethe parameter domain of nonlinear fractional-order system.- Twoapproachestosynchronizeakindofincommensuratefractional-orderchaoticsys-tem are addressed in this thesis. According to the different characteristic, the differentprocesses of theoretical proof are presented respectively.