Analysis Of Chaotic Systems Based On Hybrid Control

Chaos is a special phenomenon in nonlinear systems, which arises from deterministic systems in a random and non-regular fashion. It is ubiquitous and complex. The most remarkable character is that chaotic systems exhibit high sensitivity to initial conditions. In other words, a nonlinear system called chaotic system is unpredictable. Chaos breaks the rigorous division between deterministic systems and random systems, which is un-repeatable, local un-stable and whole stable. The study on chaos enriches our knowledge to the thing's evolvement greatly, which makes us not only achieve correct acquaintance with some complex behaviors of nonlinear systems, but also expect to resolve some problems beyond our knowledge all along. Investigations in the last decades indicate that chaos is more and more close to engineering technology and it is found to be frequently useful in information processing, physics, medicine, biology engineering, chemistry engineering, and so on. People's attentions have been focused on that How to make a non-chaotic system generating chaos, or strengthen a chaotic system.It is not always feasible to validate that a system is chaotic or not with chaos definitions, because it is very difficult to satisfy conditions of the definitions. In this dissertation, in the view of control strategy, hybrid control strategy has been put forward and expression of the largest Lyapunov exponent has been calculated based on that a chaotic system is sensitive to initial conditions, and then some useful results have been achieved.A special piece-wised linear function was adapted to chaotify a class of linear systems. For the switching systems, dynamical behaviors at every equilibrium point have been analyzed, and sufficient conditions for emergence of horseshoe-type chaos were given via Silnikov theorem.For a class of linear systems, impulsive control was used to chaotify the systems. No matter the linear systems are continuous or discrete, based on the impulsive controlled model, boundedness of the controlled systems was analyzed, and an inequality was given that ensured the largest Lyapunov exponent of impulsive systems was positive, then sufficient conditions are derived for the chaotification of impulsive controlled systems.In engineering, time-delay exists among variables in plenty of dynamical systems, in other words, the evolvement of systems not only depends on the current state, but also depends on the previous states. Time-delayed systems can present much more complex dynamical behaviors than common differential equations. For a class of linear discrete time-delayed systems, the dynamical behaviors were analyzed. The original systems were simplified to augmented systems, which were simpler linear impulsive systems with higher dimension and no time-delay. Based on the theoretical results in chapter 4, sufficient conditions for chaotifying linear discrete time-delayed systems were derived via impulsive control.In recent years, complex networks have attracted many interests of researchers in the fields of science and engineering. Networks can be found everywhere in nature and society, such as WWW, Internet, metabolic network, society network and etc. We are living in a world which is full of various networks. Chaotificaiton of complex networks becomes an important research task. In this dissertation, for a class of nonlinear complex networks, the sensitivity to initial conditions was analyzed and conditions of boundedness for the whole networks were derived via Bellman inequality, then sufficient conditions to chaotify the complex networks were deduced.Abroad application of neural networks has resolved a lot of problems that people can't solve with other methods. For a class of Hopfield neural networks with distributed time-delays, the boundedness and the sensitivity to initial conditions were studied respectively, when the kernel is the weak one and the strong one. Sufficient conditions were given for the chaotification of Hopfield neural networks.Finally, a summary has been done for all discussions in the dissertation. The research works in further study are presented.