Identification Control On Fractal Of Complex Dynamic System

Fractal is the geometric solid in the real world. And the fractal theory is much attention in the nonlinear science. Its study object is the irregular and unsmooth ge-ometric solid in nature or nonlinear systems. Its applications are involved in almost all of the natural science, and even in the social science.The fractal set of dynamic system is a very active research field in the fractal geometry in the past few years, and one class of them is gotten from the analytic mapping iteration on the complex plane. The complex plane is divided into two parts due to the analytic mapping iteration. One part is called Julia set, and the other part is called Fatou set. Mandelbrot-Julia set which is found to have a fine and complex structure plays an important role in fractal.In theoretical study, many researchers are interested in the dimensions, prop-erties, and the kinetic characteristics of the Julia sets and their generalized forms. Moreover, the properties of the Newton’s transformation Julia sets and stable re-gions of Julia sets are paid much attention. However, the nonlinear attractive domain ranges and parameters of the fractal sets are demanded according to the actual sit-uations. It is very necessary and significant to study the synchronous control and parameter identification of the fractal sets. At present, some significant results of the fractal synchronous control have been reported, such as the nonlinear coupling method, the gradient control, and the optimum control, which realize the control of the stable region and synchronous control for the Julia sets of complex systems.It is pointed that these methods are only applied to the synchronous control and generalized synchronous control of basic Julia sets. And it is only feasible when the parameters of the drive system are obtained. The parameters are usually not derived actually, so the synchronous control in the drive-response system is not solved by the method above. In this paper, the identification control on fractal of the complex dynamic sys-tems which include the generalized Julia sets, basic Julia sets, trigonometric function Julia sets and one class of spatial Julia sets, is innovatively achieved. A new method to realize synchronous control and parameter identification of the drive-response system is obtained. And it is applied to synchronous control in the drive-response system when parameters of drive system are unknown, and parameters of the drive system are identified. The main content is following.1Based on the nonlinear feedback controller method and the stability theory in difference equations, the parameter identification on generalized Julia sets and trigonometric function Julia sets of the complex dynamic systems is studied. The generally applicable adaptive synchronous controller and parameter adaptive ana-lytic expression are designed. It is proved that the controllers make generalized Julia sets and trigonometric function Julia sets achieve the synchronization, and the un-known parameters of the Julia sets can be identified. Particularly, this method is also applied to the basic Julia sets.2Based on variable structure control theory, the zero asymptotic sliding vari-ables are applied to the fractal identification control for the complex dynamic sys-tems. A new method to realize drive-response system synchronous control and parameter identification is derived. And the problems of synchronous control, for generalized Julia sets and trigonometric function Julia sets of the complex dynamic systems, are solved when the drive system parameters are unknown. The unknown parameters of the drive system can be identified in the asymptotic synchronization process. Moreover, this method is also applied to the basic Julia sets.3A new method to realize drive-response system synchronous control for the spatial Julia sets is derived. And the parameter identification of the spatial Julia sets is innovatively solved. The widely used adaptive synchronous controller and the analytical expression of the parameter adaptive law are designed by this method. Meanwhile, the unknown parameters of the drive system can be identified in the asymptotic synchronization process. It is successful to realize the synchronization control in the case of the unknown parameters. Particularly, the method is also ap-plied to the basic Julia sets.These studies are provided certain theoretical bases to the spatial Julia sets and the generalized Julia sets of complex dynamic systems for better applications.