Synchronization Control Of A Class Of Fractional Autonomous Chaotic Systems

Study of hyperchaotic systems has received greated attention in the past several decades. Hyperchaotic systems with more than one positive Lyapunov exponent are more complex and play a more significant role in nonlinear science. Now, fractional-order systems have attracted more and more people's interest, there are two problems considered, the first problem is when an ODE system is chaotic, under what condition the corresponding fractional-order system is also chaotic. More exactly, for what orders, the fractional-order system is chaotic. The second is that given a chaotic fractional-order differential system, how to design a scheme such that synchronization of many such systems is achieved. The relative problems of hyperchaotic system and fractional order chaos synchronization control are studied in this thesis using the methods of theoretical derivation and numerical simulation. The main achievements contained in the research are as follows:Firstly, a new hyperchaotic system is formulated by introducing an additional state into the third-order unified system. Some of its basic dynamical properties, such as Lyapunov exponent, bifurcation diagram and Poincáre section are investigated. It was found that the system is hyperchaotic in several different regions of the parameters. The analysis of equilibrium points and stability are also given.Secondly, two different methods, i.e. nonlinear hyperbolic function feedback control and tracking control methods are used to control hyperchaos in the new hyperchaotic system. Based on the Routh-Hurwitz criteria, the conditions suppressing hyperchaos to unstable equilibrium point are discussed. A tracking control method is proposed. It is also proved that the strategy can make the system approach to any desired smooth orbit at an exponent rate.Thirdly, analyses the generalized synchronization of fractional Chen system. The author presents a systematic design procedure to generalized synchronize fractional Chen systems based on the state observer design and the pole placement technique.Lastly, numerically investigate hyperchaotic behavior generated from Lorenz system with fractional order. The Lyapunov spectrum of the system was also studied as the parameter changed. Based on the stability theory of fractional order systems, a new approach for constructing lag and projective synchronization of fractional order chaotic system is proposed.