Research On Fractal Structures Of Generalized M-J Sets

Fractal is of great challenge and bright application future in nonlinear science. In fractal theory, Mandelbrot set and Julia set are all extraordinary complicated objects. In this thesis the fractal characteristics of generalized Mandelbrot set and Julia set are discussed. And the content is as following.The thesis presents the periodicity orbit search and comparison technique which can be used to discuss the relationship of the generalized M-J sets. The fractal characteristics of generalized Mandelbrot and Julia sets have been analyzed by using Lyapunov exponent and periodic scanning techniques and the method mentioned above. A series of the generalized M-J sets have been generated from the complex mapping z→zα + c(α∈R) by using these techniques. The thesis researches on the structure topological inflexibility and the discontinuity evolution law of the generalized M-J sets, and explores structure and distributing of periodicity "petal" and topological law of periodicity orbits of the generalized M sets, and finds that the generalized M set contains abundant information of structure of the generalized J sets by founding the whole portray of the generalized J sets based on the generalized M set qualitatively. Besides, the movement law of the Brownian particles has been expounded well using the fractal structure characteristics of the generalized M-J sets. The result has been accepted by the journal Progress in Natural Science and will be published soon.The fractal sequence and Mandelbrot-Julia sets of generalized Gauss sums are studied. This thesis analyzes the generated rules of fractal sequence, and gives the scales and dimensions of the fractal sequence constructed by quadratic Gauss sums. The Mandelbrot-Julia sets of the generalized Gauss sums have been constructed by escape time algorithm. The periodicity and structure characteristics of the Mandelbrot-Julia sets are discussed and the corresponding theory prove is given.The characteristics of the additive perturbed generalized M-J sets have been analyzed through introducing the noise in dynamical system to the generalize Mandelbrot map. The variety structures and the periodicity of the additive perturbed generalized M-J sets are discussed. The relationship between additive perturbed M sets and J sets is explored through generating the generalized J sets by choosing points from the generalized M sets. And the perturbed parameters' influences on generalized M sets are investigated.