【作者】 骆超；

【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】 本文主要涉及了非线性理论的中有关混沌和分形学若干问题的研究,其中包括一维、二维和高维不同非线性映射中混沌现象及其普适特征分析,广义M-J集嵌套拓扑分布定理的讨论以及混沌学与生物医学工程相结合的多学科交叉研究。 本篇论文在实验分析的基础上,详细讨论了当非线性映射中控制参数改变时,系统的演变规律。计算出Lorenz系统、二维logistic映射等非线性系统从规则运动转化到混沌运动所具有的普适特征,并且详细讨论了二维logistic映射中所含有的自相似现象。 推广了Michelitsch和Rssler所提出的由一个简单非解析映射所构造Julia集和Mandelbrot集的方法,构造出一系列实数阶的广义M-J集。利用复变函数理论和计算机制图相结合的实验数学的方法,对两者的结构和演化进行了研究,结果表明在所分析的映射中:①广义J集的几何结构依赖于参数α、R和c;广义M集则依赖于参数α和R②广义J集和整数阶广义M集具有对称性和分形特征;③小数阶广义M-J集出现了错动和断裂,且其演化过程依赖于相角主值范围的选取。 作者对缺氧窒息而引起的中枢神经损伤实验中仔猪的脑电信号进行了混沌特性分析,根据该实验得出如下结果:①仔猪的EEG(脑电图,Electroencephalogram)信号中存在混沌性②信号中的混沌性随仔猪的生理状态而改变,并实验表明在正常的生理状态下EEG的混沌性较强,而在损伤状态下趋于有序。 作者对由Liley等人提出的脑电动力学模型进行了分析与计算。分析结果如下:①该模型是按Pomeau-Manneville途径通向混沌的,且该途径与Hopf分岔、倍周期分岔和逆分岔有关;②支持了EEG中存在混沌运动的观点。 以上研究内容的相关论文已被《力学学报》、《数学物理学报》等刊物录用或发表。更多还原

【Abstract】 This paper concerns studies of chaos and fractal of nonlinear theory, including analysis on chaotic and general features of different dimensional nonlinear mappings; discussion of generalized M-J sets and analysis on EEG signals by using chaos theory.(1) By using phase space reconstruct technique from a time series and the quantitative criterion and rule of system chaos, different nonlinear mappings are studied.At the base of calculation and anaylize by using phase graphics, bifurcation graphics, power spectra, the computation of the fractal dimension and the Lyapunov exponent, the general features of chaos and "approach to chaos" are discussed.(2) The method constructing the J-M sets from a simple nonanalytic mapping developed by Michelitsch and Rossler was expanded. According to the complex mapping expanded by the author, a series of the generalized J-M sets for real index number were constructed. Using the experimental mathematics method combining the theory of analytic function of one complex variable with computer aided drawing, the fractal features and evolutions of the generalized J-M sets are studied. The results show-. (1) the geometry structure of the generalized Julia sets depends on the parameters of α, R and c;and the Mandelbrot sets depends on a and R (2) the generalized J-sets and the generalized Mandelbrot sets for integer index number have symmetry and fractal feature; (3) the generalized J-M sets for decimal index number have discontinuity and collapse, and their evolutions depend on the choice of the principal range of the phase angle.(3) the author analyses EEG (Electroencephalogram) signals of piglets in the HAI (Hypoxic-Asphyxic Injury) experiments, the following conclusions are shown: (1)The analyses reflect the whole dynamic characteristics of the brains, and they may become a new method of researching EEG quantitatively to early diagnose of brain disease. (2)Under normal physiological conditions, the EEG signals are chaotic, while under injury conditions the signals approach regularity. Analyses and computations are conducted on EEG dynamics model, the following conclusions are shown: (1) Chaotic patterns of the dynamics model may emerge out of Pomeau-Manneville route, and relevant to double-periodic bifurcation, Hopf bifurcation, and reverse bifurcation; (2) To further support the view that chaos exist in EEG signals.更多还原