Synchronization And Control Of Fractional Order Chaotic Systems

Nonlinear science is a foundational discipline which concerns the common properties of nonlinear phenomena. Particularly, chaos theory is one of the important subdisciplines of nonlinear science. The relative problems of fractional order chaos synchronization and control are studied in this thesis using the methods of theoretical derivation and numerical simulation, and then the periodic regions' centers of the general M-sets are computed. The main achievements contained in the research are as follows:Firstly, studies the problem of chaotic synchronization of the fractional order modified coupled dynamos system, designs the activate controller, and then proves that the self-synchronization of the fractional-order modified coupled dynamos system and the fractional-order modified coupled dynamos system's different structure synchronization with the fractional-order Lorenz system can both be arrived theoretically.Secondly, analyzes some Routh-Hurwitz stability conditions generalized to the fractional order case, and discusses the stability region of the fractional order system. The paper analyzes the chaotic behavior of the fractional order modified coupled dynamos system concretely, and provides the conditions suppressing chaos to unstable equilibrium points, then uses the feedback control method to control chaos in the fractional order modified coupled dynamos system.Thirdly, analyzes the dynamical behavior of fractional order unified system, based on the stability criterion of linear systems, a new approach for constructing projective synchronization of fractional order unified system is proposed. Numerical simulations of fractional order Chen system, fractional order Lii system and fractional order Lorenz-like system are achieved via the linear separation method.Fourthly, analyzes the dynamical behavior of fractional order Lu system, and finds out that this system enters into chaos from period-doubling bifurcation and reverse bifurcation. A new approach for constructing synchronization of fractional order Lii system is proposed. Based on the Lyapunov direct method and Routh-Hurwitz criteria, the conditions of arriving synchronization are discussed and are also proved theoretically. This paper achieves the coexistence of anti-phase and complete synchronization.Lastly, analyzes two methods for accurately computing the periodic regions' centers, the first fits for the general M-sets with integer index number, the sond suits the general M-sets with negative integer index number. The paper primarily discusses the general M-sets with negative integer index, analyzes the relationship between the number of periodic regions' centers on the principal symmetric axis and in the principal symmetric interior. It also applies the Newton's method to the transformed polynomial equation associated with the periodic regions' centers. This paper lists some centers' coordinates of general M-sets' k-periodic regions (k = 3,4,5,6) for the index numberα= -25,-24,...,-1, all these coordinates havehighly numerical accuracy.In the studies above, one paper have been accepted by Physics Letters A, one accepted by International Journal of Modern Physics B, and one accepted by Acta Physica Sinica.