Research Of The Interior Structure Of The Generalized M-J Set And The Related Problems

Noneliner science is a new crossed discipline, which concerns the common properties of nonlinear phenonmena. It involves the Soliton, Chaos and Fractal. The theory that corresponds to these concepts makes up of theoretical basis of non-linear together. In this thesis, the basic fractal theory of complex mapping is introduced, the interior structure of the generalized Mandelbrot-Julia set is emphased on, the general rules of Quasi-M set of two-dimensional logistic mapping are discussed.The basic fractal theory of complex mapping is the theory base for us to research on the fractal problems, we give a brief introduce to Julia set and Mandelbrot set in complex mapping, have a discuss on the basic relations between the two sets, and the escaped-time algorithm often used in constructing fractal graphics.The structure and construction of generalized Mandelbrot-Julia set is a meaningful direction. It has been researched in-depth since Mandelbrot constructed and studied Julia set on Z-plane and Mandelbrot set on C-plane of complex mapping f(z) = z~2 + c with the aid of computer. Based on escaped-time algorithm and the algorithm that has been put forward by Engel about constructing the interior structure of Julia set, a series of the interior structure of Mandelbrot-Julia set are constructed. Using the experimental mathematics method combining the theory of analytic function of one complex variable with computer aided drawing, this paper researches on the structure topological inflexibility and the discontinuity evolution law of generalized Mandelbrot-Julia set. The result has been accepted by the Journal of Engineering Graphics and will be published soon.In the following, with the escaped-time algorithm and the cycle-finding algorithm, We show how the parameter plane for two-dimensional mappings can be colored and the general rules of two-dimensional logistic mapping.