Control And Applications Of Fractals

Fractal is an active branch in the study of modern mathematics and nonlinear science, specially to the geometrical description about chaotic motion.Since the essence of the world is nonlinear and the chaotic phenomena exist everywhere,the application of fractal geometry is very extensive and it is a hot topic in current research.Julia set and Mandelbrot set are two important sets in fractal theory.Through the method of computer graphics combing with the complex variable function theory and computer drawing,the refined and complicated structures of Julia set and Mandelbrot set are found,which make them have extensive applications in the fields of physics, biology and so on,such as the classical Langevin problem,fractal mechanics,fractal proteins.It is worth noting that the Julia set of the nonlinear system is an important nonlinear character of the system.According to the practical requirement,it is needed to control the scope of the attractive domain and sometimes it requires the systems to show different or similar behaviors.Therefore,it is of great importance to effectively control the Julia set of the nonlinear system.However,up to now,it is regrettable that the study about Julia set is only limited to its drawing and the control of Julia set is a new research field.In this paper,the method of control and synchronization are firstly introduced to fractal and the new definitions of synchronization and generalized synchronization of Julia sets of different systems are given.Also a series of results about spatial Julia set are obtained.Concretely,1.the(product)auxiliary reference feedback control,gradient control are taken to achieve the control of Julia set and Mandelbrot set of different functions,such as polynomial functions,exponential functions and so on;2.the definitions of synchronization and generalized synchronization of Julia sets of different systems are firstly given and nonlinear coupling,gradient synchronization, optimal function methods are taken to achieve the synchronization and generalized synchronization of Julia sets of different functions,such as polynomial functions, trigonometric functions and so on;3.the spatial Julia set is depicted by use of the model of the 2-D complex Henon system and feedback control,gradient control,optimal functions methods are taken to achieve the control and synchronization of its Julia sets.Since Julia set is the boundary of the attractive domain of the attractive fixed point, the control of the attractive domain of the fixed point and the synchronization of the attractive domains about different systems are also achieved.And so the stability of the system is further studied and depicted,and the content of fractal theory is also enriched.