【作者】 杨小英；

【导师】 吴庭万； 余新科；

【作者基本信息】 华南理工大学 ， 光学， 2010， 硕士

【摘要】 混沌预测即对给定的系统通过一定的“观测”手段判断其是否具有混沌的特征,在混沌研究领域中占有重要地位。其研究方法大体上可分为三类:一是通过实验观测确定系统是否具有混沌运动的一般特征;二是通过计算机仿真,得到系统在相空间的运动轨迹(相图),计算连续功率谱,计算Lyapunov指数,计算费根鲍母常数,运算poincaré映射图等;三是解析预测,通过数学的逻辑推演,给出系统混沌存在的一般条件。本文对混沌的发展历程及发展趋势、混沌定义、混沌的基本特性和通向混沌的道路做了全面的概述;对混沌的预测方法进行了详细的论述,对各种混沌解析预测方法进行了深入的比较分析,并给出了三阶自治非线性系统混沌存在的一个新的判定定理,该定理扩充了人们对混沌机理的认识,是对解析判定一般方法理论的发展,并为实际应用提供方便的工具;对相图法、功率谱法、庞加莱截面法以及Lyapunov指数法等数值方法的计算进行了较为深入的研究并详细说明了系统处于混沌状态时它们各自的特征。在上述基础上,本文对典型的光学非线性系统的数学模型——Lorenz模型进行了数值方面的混沌预测,在理论分析的基础上仿真分析了其走向混沌的道路;着重研究了系统在(σ,b,r)参数空间中(b , r)截面的混沌现象,利用MATLAB软件仿真出了在系统参数σ=10、1 < b <9、0 <r<+∞变化范围内的混沌域,该混沌岛图在科研及工程技术领域中均有一定的参考价值。混沌解析预测方面,本文选取了几类不同的实际系统如强非线性保守系统、耗散系统等对三阶自治非线性系统混沌存在判定定理进行了数值方面的验证,并判断了其适用范围;结果表明,该定理与世界公认的Melnikov方法和Shilnikov方法相比,具有更广泛的适用范围且具有更强的实用性。更多还原

【Abstract】 For a given system, Chaos prediction is to identify whether it has the characteristics of chaos by a certain "observations", it plays an important role in chaos research. The research method can be roughly divided into three categories: First, determine whether the system is with the general characteristics of chaotic motion through the experimental observations; Second, through computer simulation, we obtain the phase diagram and poincarémaps of the system, calculate the Lyapunov index and the Feigenbaum constants. these are belong to the numerical method; Third, analytic prediction which is presented with general conditions of chaos exists by logical deduction of mathematics.In this paper, the development process and trend of chaos,the definition of chaos, chaotic properties, the process of chaotic production are done a a comprehensive overview; the methods of chaos prediction are systematically discussed; We compared all sorts of chaotic analytic prediction method in detail, introduced a new existence theorem for Chaos in the Third-order nonlinear system of autonomy which expands people’s understanding of the mechanism of chaos, develops the general theory of the analytical method and is a convenient tool for the practical applications. The numerical methods such as the calculation of Lyapunov index and power spectrum on are studied in-depth.Based on the above, this paper is to do the numerical aspects of chaotic prediction to the typical mathematical model of the optical nonlinear systems------Lorenz system. Based on theoretical analysis,Simulate its way to chaos; Focuse on the system in the (σ,b,r) parameter space the chaotic phenomena of the (b ,r)cross section,the chaotic domain with the changes of the parameters (σ=10、1 < b <9、0 <r<+∞) is obtained using MATLAB software, which has some reference value in the field of scientific research and engineering techniques. Prediction of chaotic analysis, the new existence theorem for Chaos in the Third-order nonlinear system of autonomy is verified from the numerical aspects and determined the scope of its applicationby by selecting different types of physical systems such as strongly nonlinear conservative system,dissipative system and so on;the results show that the theorem is with a broader scope and more practical than the Shilnikov and Melnikov method which are all the universally recognized methods.更多还原