Fractal And Chaos Control In Complex Dynamical System

The iteration of nonlinear complex dynamical system can make very com-plicated phenomenon, and Fractals and Chaos are two typical problems in the nonlinear system while they are closely related. At the same time, due to the the importance of stability in the system analysis, it is essential to consider the alteration of the stable region. As a very important set in Fractals, Julia set is a depiction on the change of the complex system’s stable region, so the control of Julia set has an important theoretical and practical significance. On the other hand, the chaotic systems have an enormous application in the analysis of many problems, which appeal many scholars to study the control of chaos regarding to their complicated dynamical property. In this paper, we mainly focus on the control of switching complex system’s Julia set and complex chaotic systems, and the specific work involves with1, The control of switching complex system’s generalized Julia setWe first give a definition on the generalized Julia set of a switching complex system with delay. With the help of inequality technique, we prove that the gen-eralized Julia set is bounded. Nonlinear feedback control is employed to make the fixed point become local stable, and point in the attraction basin of the fixed point will come into the generalized filled Julia set, thus we achieve the control of the generalized Julia set and alter the upper bound that Julia set exists. Further, we consider the control on generalized Julia set of perturbed system.2, Finite time adaptive control of switching complex chaotic systemFirst the conclusion on finite time control of non-switching system is extended to switching system. We get that the switching system will become stable in fi-nite time if the multi Lyapunov function and average dwelling time satisfy certain constrains. For a class of switching complex chaotic system, we separate the real part of the complex variable from the imaginary part, then we get two systems with real variables. Utilizing conclusion on finite time stability, the appropriate controller and adaptive law are designed to solve the finite time adaptive control of the two real systems, and we achieve the finite time control of the switching complex chaotic system. Finally, a switching complex chaotic system with per- turbation is proposed to verify our method.3, Adaptive control of a class of high order complex chaotic systemThe adaptive control of a class of high order complex chaotic system with unknown parameters is analyzed. For a simplified case, the sliding mode control is applied to solve the adaptive control of this complex chaotic system. For the complicated case, we use back-stepping method and extend the concept of tuning function in the real domain system to complex system, then we get the design-ing approach to the controller and adaptive law. In this chapter, the Lyapunov function that we employed used the complex states direcly, not the method to separate the real and imaginary part of the states. This is also a improvement. At last, examples are given to illustrate our conclusion.4, The analyze of cascade complex chaotic systemFor a cascade complex chaotic system with special formulation, we propose a controller only relating to one subsystem state to stabilize the other subsystem. Moreover, we confirm that this controller can ensure the both subsystem to be sta-ble under some conditions. The adaptive control of the cascade complex chaotic system is also taken into consideration, and we put forward a robust adaptive law. The numerical examples put forward in the last demonstrate the effectiveness of our method.