Property Analysis And Control Of The Fractal Sets Generated From Several Complex Systems

For the development of technology science, the nonlinear phenomena have sprung up in lots of subjects which contribute to the formation of the nonlinear dynamics. Chaos, Fractals, Solitons are regarded as three most typical nonlinear phenomena and they are closely related with each other. Fractal sets can be generat-ed from several ways in which the complex system is the most basic one. The fractal sets generated from complex systems have been applied in the fields of physics, bi-ology, image processing, geology and so forth.There are two kinds of fractal sets that generated from complex systems:Julia set and Mandelbrot set. Based on the computer technique and complex analysis, the two kinds of fractal sets have been investigated in its visualization, topology analysis, properties analysis, effect of noise, control and some other theoretical fea-tures. As mentioned above, the fractal sets could describe some nature forms and processes, and they have good application background in the fields of physics, im-age processing and so on. So it has stimulated scholars’ enthusiasm for seeking the fractal sets generated from some more general complex systems such as:alter-nated complex maps, rational maps, time-delay system, hypercomplex systems and higher-dimensional systems. A striking feature of M-J sets is that the rule of iteration is simple while the fractal structure is complicated. Thus for the mentioned compli-cated complex systems, their M-J sets must have more complicated structures and more interesting properties. In this paper, we investigate the properties of M-J sets from some complicated complex system:the boundedness, symmetry, connected-ness properties. Based on these achieved properties, the effective boundary control and synchronization control of their M-J sets are realized. The main contents of this paper are listed as follows:1. On the properties analysis of the fractal sets generated from alternated com-plex map.Some scholars left some questions to be proved about the boundedness and symmetry properties of the fractal sets generated from the alternative complex sys-tem. This paper give the mathematical proofs about them. Then the connectedness properties of the fractal sets generated from the alternative complex system are ana-lyzed via the graphical method.2. Control and synchronization of Julia sets generated by a class of complex time-delay rational map.For a class of complex time-delay rational map, a control strategy is designed to achieve the control of its Julia set. The escape criterion of the controlled system is calculated. Then via the nonlinear coupling method, we achieve the synchronization control of the Julia sets of two different complex time-delay rational maps. A’Syn-chronization index’algorithm is designed to quantify the relationship of the coupling strength and the synchronization degree.3. Effect of noise, synchronization of the spatial Julia sets generated from Lorenz system.We investigate the fractal behavior of a class of complex discrete Lorenz system which is regarded as the most basic chaotic system. The Julia set of complex Lorenz system is defined as CLS Julia set. The structural damages of CLS Julia set under additive and multiplicative noise are analyzed based on the ’JD’ method. Then we give the proof about the symmetry of the 3-D slice of CLS Julia set and then pro-pose a new method named’Symmetry index’to quantify the symmetry changes of the noise-effected 3-D slices. Besides, the synchronization between the Julia sets of complex Lorenz system and complex Henon map is realized via two kinds of non-linear coupling methods. Finally, the proposed’Synchronization index’ algorithm is modified to investigate the synchronization process of Julia sets between Lorenz system and complex Henon map.4. Control of the spatial Mandelbrot set generated in coupled map lattice.We extend the research of planar Mandelbrot set into spatial case by giving the definition of Mandelbrot set (CML M-set for short) generated in complex coupled map lattice which have two strongly coupled parameters. The symmetry property of 3-D slice of CML M-set is proved. A auxiliary reference control method is designed to achieve the symmetry change, position change and size change of the Mandelbrot set in complex coupled map lattice.In conclusion, this dissertation focus on the properties analysis and control of the fractal sets generated from some kinds of complicated complex systems. The visualization method is modified to achieve a better imaging effect. By using the mathematical induction and inequality techniques, the properties of those M-J sets are analysed. The control of those M-J sets are also investigated. These researches enrich the framework of fractal geometry and could provide some limited theoretical support for the applications of fractal theory.