| The complex dynamical system mainly studies the iteration of analytic func-tions. Its main object is the Julia set with a fractal structure generally, and the map used to produce Julia sets is chaotic. So it is closely linked with chaos and frac-tal. This paper focuses on the qualitative theory of the complex dynamical system and a series of basic researches on its control and synchronization, including con-trol and synchronization of spatial Julia sets and synchronization of chaotic complex systems.1. Control and synchronization of Julia sets in coupled map latticeBased on the stability theory of fixed points for the classical Julia set, the stable condition of the fixed plane for the Julia set in coupled map lattice was given. As the fixed plane was known, the gradient control and auxiliary reference control was respectively used to control the stability of the fixed plane. But in many practical circumstances, the fixed plane was not easily obtained. For such systems, the optimal function control is applied to control the stability. In addition, the synchronization of two different Julia sets in coupled map lattice was also achieved by synchronizing their movement trajectories using the gradient control and optimal function control respectively. At last, the coupling of two different Julia sets in coupled map lattice was also analyzed by coupling their movement trajectories using the linear coupling.2. Control and synchronization of spatial chaotic Julia setsBased on the basic properties of the classical Julia set, the properties of spa-tial chaotic Julia sets and the stable regions of the complex parameter c were given. According to the stable conditions of the fixed plane, the scope of the control pa-rameter was obtained by using the auxiliary reference control so as to control the spatial chaotic Julia set. Moreover, the definition of the generalized synchronization between two different spatial chaotic Julia sets was given. The linear generalized synchronization of spatial Julia sets was achieved by linear feedback control. In addition, the nonlinear generalized synchronization of spatial chaotic Julia sets was also analysed by constructing a multivariate polynomial transformation and using nonlinear feedback control.3. Anti-synchronization of a new complex Lorenz-like system and its dynami-cal propertiesBased on chaotic real systems and their basic properties, a new complex Lorenz-like system was constructed and its dynamical properties was also discussed. The anti-synchronization of the new complex Lorenz-like systems was separately inves-tigated by active control and nonlinear control methods. Although the both methods used to achieve the anti-synchronization of the new complex Lorenz-like system were simple, nonlinear control was preferable for personal purposes and simpler for computations. Numerical simulations verified that both methods are effective.4. Adaptive anti-synchronization of chaotic complex systems with unknown parametersThe adaptive anti-synchronization of a class of chaotic complex systems with fully uncertain parameters, which were described by a united mathematical expres-sion was presented. Based on Lyapunov stability theory, we developed an adaptive control scheme and adaptive laws of parameters to anti-synchronize two unknown chaotic complex systems. The anti-synchronization of two identical complex Lorenz systems and two different complex Chen and Lu systems were taken as two exam-ples to verify the feasibility and effectiveness of the presented scheme.5. Observer-based projective synchronization of chaotic complex systemsBased on the assumed output of the uncertain chaotic complex system, its ob-server was designed. According to the nonlinear state observer and the pole place-ment technique, we got the feedback gain matrix and achieved the projective syn-chronization between uncertain chaotic complex systems and its observer. The pro-posed synchronization scheme was confirmed by numerical simulations of two well known chaotic complex systems.6. Robust adaptive full state hybrid projective synchronization of chaotic com- plex systems with unknown parameters and external disturbancesBased on the definition of full state hybrid projective synchronization (FSHPS) of chaotic real system, we gave the definition of FSHPS of chaotic complex system. By introducing a dynamic compensator and using nonlinear control and adaptive control, we proposed the robust adaptive FSHPS scheme, which can achieve adap-tive FSHPS of two different chaotic complex systems asymptotically with a small error bound. The adaptive laws of the unknown parameters were given, and the suf-ficient conditions of realizing FSHPS were derived as well. Finally, the proposed control scheme was successfully applied to two identical chaotic complex systems and two different chaotic complex systems.In conclusion, this dissertation focuses on the controls of spatial fractal sets and synchronization of chaotic complex systems. The control was introduced successful-ly into the spatial Fractal, which had important practical significance to further study the spatial fractal and explain the more complex phenomena. All kinds of synchro-nization of chaotic complex systems was achieved firstly, which provided theoretical basis for further enhancing the security of communications.
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