Control And Synchronization Of Chaotic Dynamical Systems

This paper investigates time-delay feedback control of a class of chaotic dynamicalsystems and synchronization of the coupled complex networks. In the study of time-delayfeedback control of chaotic systems, this paper analyzes and proves exponential asymptoticstability of zero solution of the controlled systems, that is, the systems converge to zeroexponentially, which is stronger and more accurate than the general nature asymptotic stabilityof zero solution of the controlled system. In the study of synchronization of coupled complexnetworks, based on present results, this paper weakens some conditions appropriately andobtains conclusions which are more general. What’s more, both the impulsive effects andcoupling time-delays are considered, and the method has certain innovation. The obtainedresults can be applied to many practical complex networks and have certain value.First, this paper presents a time-delay feedback control method for a class ofnon-autonomous chaotic dynamical systems. By constructing a special Lyapunov function andusing the stability theory on Delayed Differential Equations, the global exponential stability ofthe controlled systems is analyzed and proved. Moreover, the theoretical results are applied tothe typical Lorenz system and Josephson system, and the numerical simulations verify thevalidity of the results.Then, this paper popularizes the form of the above chaotic systems appropriately, andconstitutes a class of general complex networks through linear coupling. Furthermore, thispaper investigates the global complete synchronization of this class of complex networks withimpulsive control, and some sufficient conditions are derived to guarantee synchronization ofthe impulsive coupled complex networks. Next, the obtained theoretical results are applied tosome typical complex networks, such as scale-free (SF) networks composing of identical Chennodes and small-world (SW) networks composing of identical Duffing nodes. Moreover,numerical simulations are given to visualize theoretical results and also show the advantagesand effectiveness of the results.Finally, for the above complex networks, this paper continues to study in depth theirglobal exponential synchronization with both impulsive effects and coupling time-delays.Using Lyapunov functional method and Linear Matrix Inequality (LMI) technique, somecriteria are obtained for both delayed-independent global exponential synchronization and delayed-dependent global exponential synchronization. These criteria are simple yet lessconservative. Furthermore, the theoretical results are applied to some practical small-worldnetworks and scale-free networks. Numerical simulations visualize theoretical results and alsoverify the validity of the obtained results.