Research On The Method Of Fractal Generation Based On Complicated Systems And Its Application In The Visualization Of Nonlinear Dynamics

The dissertation makes a systematic study of the method of fractal generation based on complicated systems aimed at exploring solutions to the existing problems in the field. Great progress has been made in the research of the multifarious technologies of fractal generating, which include the fractal creation on the basis of non-analytical complex dynamic systems, the deformation of fractals generated from complex dynamic systems under perturbations, 3-D fractals' generation either from ternary number system or 3-D polynomial maps. Furthermore, the above method of fractal creation has been successively applied to the visualization of the dynamic behavior of chaotic systems and the syntheses of planar four-bar mechanisms.The main work of the dissertation is as follows:Chapter 1 first gives a review of the development course of the fractal theory as well as its influence on relative scopes, and then summarizes the present situation of fractal theory and its application, while the open questions in the research of fractal generation either from complex or multidimensional dynamic systems are also analyzed. Finally, the significance, background and contents of this dissertation are expounded particularly.In chapter 2, the general Mandelbrot sets created from the non-analyticalcomplex dynamic systems are investigated. The parameter equations of the boundaries of the fixed point regions when α is positive are strictly given, while the M-sets' properties with different α are theoretically analyzed and proved. A symmetrical period-checking algorithm is put forward, which colors M-sets according to the period of each point in the complex plane and takes full advantage of the M-sets' properties to accelerate drawing process.Chapter 3 elaborates the theory and method of deforming the fractals generated from complex dynamic systems. An elementary model of complex dynamic systems under perturbations is presented, which enables us to expediently control the integral structure and local details of resulted fractals by altering control parameters such as multiplicative perturbations, linear perturbations and additive perturbations. A 2-D extension factor is constructed, which can be exerted on every point in the fractal sets to transform their positions. The transfiguration algorithm for fractals from complex dynamic systems under perturbations is proposed and experimental results demonstrate that it is simple, intuitive, foreseeable and easy to implement.Chapter 4 puts forward a novel approach to generate 3-D fractal sets based on...