【作者】 常沛军；

【导师】 王兴元；

【作者基本信息】 大连理工大学 ， 计算机应用技术， 2005， 硕士

【摘要】 分形是非线性科学中富有挑战性和广阔应用前景的学科。分形理论中Mandelbrot集和Julia集都是非常复杂的对象。本文主要研究了广义Mandelbrot集和Julia集的分形特征,内容如下:提出了用于探讨广义M-J集对应关系的周期轨道搜索比较技术,结合Lyapunov指数和周期点查找技术,本文分析了广义M-J集的分形特征。利用上述技术,本文构造了一系列复映射z→zα+c(α∈R)的广义M-J集,研究了广义M-J集的结构拓扑不变性和裂变演化规律;建立了复映射z→zα+c(α∈R)的广义M-J集之间内在机制的等价定理、拓扑不变性和裂变演化规律;探索了广义M-J集的分叉嵌套序列、吸引周期花瓣的分布规律、轨道的混沌特征;把计算机试验与数值计算相结合,从广义M集对应的不同周期的广义J集周期轨道入手,对广义J集周期轨道的特征进行分析,建立定性、定量化的标准与统计特性的指数,来描述、刻画广义M-J集对应关系;并且在此基础上阐述了此类广义M-J集的物理意义。这一研究成果即将发表在《自然科学进展》上。研究了广义高斯和的分形序列及其M-J集。本文从理论上分析了分形序列的生成规则,给出了二次高斯和所生成的分形序列的标度及维数;利用逃逸时间算法,构造了广义高斯和的M-J集,分析了M-J集的周期性和结构特征,并给出了相应的理论证明。把噪声对动力系统的影响引入到广义M-J集的研究中,分析了加项扰动的广义M-J集分形特性,探讨了加项扰动的广义M-J集的结构变化及周期性规律;并在加项扰动的广义M集上取点构造了对应加项扰动的广义J集,探讨了扰动广义M-J集的对应关系;从理论上研究了扰动参数对广义M集的影响。更多还原

【Abstract】 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.更多还原