Control And Synchronization Of Fractal In Several Complex Alternated Systems

The evolution of a natural process can be modeled using discrete dynamical systems, that is to say, maps which apply from one point to another in the certain variables space. In nature, there are many different interactions and therefore sys-tems do not evolve according to an unique dynamics, such as the phenomenon of seasonal alternation, stock movements in economics, the tumor model in biology and so on. Therefore, the evolutions of natural processes should be explained by the alternated iteration of different dynamics. This dissertation is concerned with the control and synchronization of Julia sets of several classes of complex alternated systems. The main contents of this dissertation are as follow:1. Control and synchronization of plane-alternated Julia setsThe plane-alternated system is discussed. According to the basic properties of plane Julia sets and fixed-point theory, the stability of fixed points of plane-alternated systems is discussed. When fixed points of the system can be calculated, the control of plane-alternated Julia sets can be achieved using auxiliary feedback control and gradient control. Due to the complexity of nonlinear systems, fixed points are of-ten difficult to obtain, in this case, the optimal control method can be used to achieve control objectives. According to definitions of the synchronization between differen-t plane-alternated Julia sets, the synchronization between different plane-alternated Julia sets can be accomplished by synchronizing their trajectories. The synchroniza-tion between different plane-alternated Julia sets is achieved using gradient control and optimal control. And the coupling synchronization between different plane-alternated Julia sets is achieved using linear coupling.2. Control and synchronization of generalized-alternated Julia setsGeneralized-alternated systems are discussed. The definitions of generalized-alternated Julia sets and several basic properties are given first. The stability of fixed points of generalized-alternated systems is discussed, and the control of generalized-alternated Julia sets is achieved using auxiliary feedback control, gradient control and optimal control. In addition, according to definitions of the synchronization between different generalized-alternated Julia sets, the synchronization between dif-ferent generalized-alternated Julia sets can be accomplished by synchronizing their trajectories. The synchronization between different generalized-alternated Julia sets is achieved using gradient control and optimal control. And the coupling synchro-nization between different generalized-alternated Julia sets is achieved using linear coupling.3. Control and synchronization of spatial-alternated Julia setsSpatial-alternated systems are discussed. The definitions of spatial-alternated Julia sets and several basic properties are given first. The control of spatial-alternated Julia sets is achieved using auxiliary feedback control. The definition of generalized synchronization is given. The linear generalized synchronization between different spatial-alternated Julia sets is achieved using linear coupling. Moreover, the non-linear generalized synchronization between different spatial-alternated Julia sets is achieved using nonlinear coupling.4. Control and synchronization of Julia sets in 1-D Logistic map1-D Logistic map is discussed, and Julia sets generated by its iterations are studied. The control of Julia sets in 1-D Logistic map is achieved using gradient control, optimal control and auxiliary feedback control. The synchronization of Julia sets between different 1-D Logistic map is given. The synchronization of Julia sets between different 1-D Logistic map can be accomplished by synchronizing their trajectories.And the synchronization of Julia sets between different 1-D Logistic map is achieved using gradient control and optimal control.In summary, this dissertation focus on the control and synchronization of sev-eral classes of complex alternated systems. Julia sets of these different alternated systems can be controlled using gradient control, optimal control and auxiliary feed-back control, and the synchronization, coupling synchronization and generalized synchronization between different alternated Julia sets is achieved. This disserta-tion is helpful in enlarging the study of fractal theory, and it provides a theoretical basis to better understand non-linear phenomena in nature.