Research Of Some Problems In Control And Synchronization Of Chaotic Systems

The relative problems of chaos control and chaos synchronization 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, classical feedback method is used to control chaos in Liu dynamical system. Based on the Routh-Hurwitz criteria, the conditions of the asymptotic stability of the steady states of the controlled Liu system are discussed, and they are also proved theoretically. Numerical simulations show that the method can both suppress chaos to unstable equilibrium points and unstable periodic orbits (limit cycles) successfully.Secondly, assuming the Rossler system as a reference, this paper studies two cases of chaotic synchronization of a pair of (master and slave) systems: one with fully uncertain parameters for both, the other where the master system has fixed given parameters while the slave system is endowed with uncertain parameters. The respective adaptive controller based on parameter identification is then designed, according to Lyapunov stability theorem. Then it is proved that the two controllers are capable of making the two (identical) Rossler systems asymptotically synchronized. Numerical simulation results further testify the efficiency of controllers.Thirdly, this paper studies projective synchronization of the Rossler's hyperchaotic system and designs a controller based on techniques from the state observer design and the pole placement technique. It is also proved that the strategy can make the error system stable. This controller has two advantages: first, it is capable to achieve a full-state synchronization; secondly, the scalar factor can be adjusted arbitrarily in due course of control without degrading the controllability. At last, feasibility of the technique is illustrated.Finally, based on techniques from the state observer design and the pole placement technique, we present a systematic design procedure to synchronize a class of chaotic systems by a scaling factor (projective synchronization). Compared with the method proposed by Wen and Xu, this method eliminates the nonlinear item from the output of the drive system. Finally, feasibility of the technique is illustrated for the unified chaotic system.