Feedback Control In Dynamical System Of Chaos And Complex Network And Its Applications

The phenomenon of chaos are widespread in the real world. Apparently the chaos phenomena is irregular. But regular innate character which hides in the chaos phenomena is gradually revealed. Because of the complexity of the chaos phenomenon, people always expect to find some ways to control the chaos system so that the unstable periodic track of chaotic system is controlled to the expected track of stability. By the understanding of the nature of chaos and the value of a wide range of applications in some fields, the research of control and synchronization of chaotic systems and complexnetworks develops rapidly. Among them, feedback control is one of the most interesting. In fact, a wide variety of approaches have been proposed for feedback control of chaotic systems and complex networks which include linear and nonlinear feedback control, time-delay feedback control, among many others. However, to our best knowledge, the aforementioned methods and many other existing control schemes mainly address control with no significant time of chaotic systems and complex networks without delays. This feature limits obviously complexity of a large class of chaotic systems and complex networks. It is believed that as many chaotic systems and complex networks developed in physics, economics and biology are formulated in term of coupled nonlinear oscillators. Time delay plays an important role in control of such chaotic oscillators. In addition, literature dealing with chaotic control of coupled systems with time delays appears to be scarce due to the difficulties in timely and effective control for the delayed dynamical systems. Moreover, most of the developed methods are valid only for chaotic systems whose feedback control is autonomous. But in practical situation, the control results of some systems cannot be exactly known, the effect of these uncertainties will destroy control results and even break it. Therefore, it is essential to investigate significant time control of delayed chaotic systems. Moreover, different from previous investigations, In this paper, we shall focus on the influences of time-delay on control.In this article, add the appropriate controller to the proposed chaotic system and complex network. It is proved the stability of the controlled systems, through LMI and the Lyapunov-Krasovskii function method, based on the stability of the principle of functional differential equations. The effectiveness of controlled systems is proved, such as the chaotic systems controlled by delay controller, the time-delay chaotic systems controlled by nondelay controller, chaotic systems with time-delay controlled by delay controller and the complex delay networks controlled by delay feedback controller.To this end, the theoretical results will be applied to chaotic systems, chaotic delayed system and complex networks. Numerical simulations are also presented to demonstrate the effectiveness of the approaches.