Study On Control And Synchronization Of Complex Dynamical Networks And Chaotic Systems

Firstly in this dissertation, a brief introduction to the historical backgrounds of chaos and complex networks, research progress and typical methods for chaotic control and synchronization are given. Then this dissertation is devoted to the study on control and synchronization for some network models and chaotic systems. The research included tracking control of unidirectional coupling network, synchronization in complex networks with adaptive coupling, control chaos to low-period motion based on adaptive pulse perturbation, impulsive synchronization of uncoupled chaotic systems driven by an external signal, chaos synchronization induced by sliding-mode control for chaotic systems with parameter mismatches, generalized synchronization via impulsive control and H?lder continuity for generalized synchronization manifolds. The detail research results focuse mainly on the followings:(1) Based on Lyapunov stability theory, a tracking controller is designed for the tracking control of Lorenz systems in unidirectional coupling network. A controller is put to only one node of this network, then it can be realized that some output signals of this network approach to any desired orbit.(2) A new concept about the asymptotic stability of balance point, component leading asymptotic stability, is presented. A new kind model of complex networks with adaptive coupling is constructed. Generally it is very difficult to realize synchronization for some complex networks. In order to synchronize, the coupling strength coefficient of networks has to be very large, especially when the number of coupled nodes is larger. However we proved by using the well-known LaSalle invariance principle, that the state of such a complex network can synchronize as long as the coupling strength coefficient is positive. Moreover it is noted that there is a couple only between the first state equations for each node of the network.(3) A new controller is designed, which input variable comes from some state variable of a pulse perturbed chaotic system, to direct many chaotic systems towards low-period motions via adaptive pulse perturbing. Many chaotic systems can be controlled to be different periodic orbits simultaneously. When the same input state variable of the controller is used, many chaotic systems can be controlled to be different periodic synchronizations.(4) In oreder to synchronize two uncoupled systems, a new method is designed that is to drive impulsively the parameters of two uncoupled R?ssler systems via using external chaotic signal. By using the existing results of impulsive control theory, it is proved theoretically that chaotic and periodic synchronizations can be implemented. The key point is that there is no couple between two chaotic systems, in practice, sometimes it is difficult even impossible to couple two chaotic systems. So it is more useful and easier to realize in practice, and hope have better application. (5) Using the sliding mode control technique, a new adaptively control law is established which realizes the synchronization of two unified chaotic systems with parameter mismatches. Both theoretical analysis and illustrative examples have been presented to verify the validity of the developed control scheme.(6) A sufficient condition for the occurrence of impulsively generalized synchronization of two completely different dynamical systems is given. Moreover a method of constructing a response system to achieve impulsively generalized synchronization with drive system is presented. By using Lyapunov theory, the stability of this impulsively generalized synchronization is demonstrated.(7) It is also proved theoretically that generalized synchronization can occur in two linear coupled dynamical systems, and the generalized synchronization manifolds are H?lder continuous under certain conditions. By using Temam's amended interial manifolds theory. Under the presumption that two linear coupled systems have attractor and the basin of attraction, the existence of generalized synchronization manifold can be attained by Schauder's Fixed-Point Theorem via defining a map over a functional category. This kind of manifold has positive invariant property. It is also proved that there is an exponential attractor with a fraction number dimension. The intersection set of a H?lder continuous inertial manifold and an exponential attractor has exponential attraction.For all the above theoretical results, we proposed numerical simulations which verity the corresponding theoretical results.