【作者】 郭健；

【导师】 金鸿章；

【作者基本信息】 哈尔滨工程大学 ， 控制理论与控制工程， 2004， 博士

【摘要】 随着科学技术的发展，系统的规模越来越大。人们为了追求舒适的生活，也会使得系统的功能复杂、规模巨大。也正因为系统的功能复杂、规模巨大使得对复杂系统的控制与描述变得越来越困难。而且，对整个复杂系统的研究也面临了许多难以想象的麻烦。 突变理论的创立是非线性科学的重要成就之一。突变理论(CatastropheTheorv)是法国数学家Renè Thom于20世纪60年代提出的一种拓扑学理论，能直接处理不连续性的问题，特别适用于内部作用机理未知系统的研究。因此，突变理论在解决类似的问题时具有一定的优势。 以突变理论、静态分叉理论、奇异性理论等为基础，对复杂系统的脆性问题进行了深入的研究。描述了随着系统的发展，规模的扩大，层次的增加，使得复杂系统的脆性的研究具有必需性。 首先，对本论文的研究对象——复杂系统进行了较深入的研究。在结合古今中外学者对复杂系统研究的基础上，给出了复杂系统的语言描述。基于复杂系统的语言描述，给出了复杂系统的定性描述。为了对复杂系统有一个更深入的认识，给出了复杂系统的判断方法，子系统重要度的概念。 其次，在进行脆性问题分析的过程中，描述了复杂系统脆性问题存在的客观性。将材料力学中的脆性名词，引申到复杂系统中来，给出了新的语言描述定义，并且也给出了其的数学描述。为了更好地理解复杂系统中的脆性问题，定义了一些与脆性相关的概念，崩溃、脆性过程、脆性关联性等概念。基于脆性的数学描述，给出了脆性的基本特性：隐藏性，伴随性、作用结果的危害性等。基于信息熵原理，预测了干扰对脆性激发的能力。 再次，基于突变评价级数法，建立了脆性评价指标。由于客观实际中，系统的势函数并不容易建立，本文根据两个单个系统及三个系统之间的演化关系，通过方程转换变成与势函数相似的形式，再进行脆性分析。又给出了对两个典型的非线性系统的脆性分析，分析的关键所在是：如何确立系统的分叉点(分叉点集)。又给出了两种基于数据探索分析法确定变点的方法。 最后，根据数理统计中的假设检验思想，进行显著性检验。采用双总体哈尔滨工程大学博士学位论文的滑动I检验法，对SARS危机事件进行突变检测分析，证明了在此过程中确实存在突变。又根据多项式拟合，分析了北京每天新增的确诊人数和医务人员人数之间的脆性关联性，并进行了扼要的脆性分析。关键词:复杂系统;脆性;突变理论;SARS更多还原

【Abstract】 With the development of science technology, the scales of system become larger and larger. People make the function of system be complexity and the scale of system become large in order to pursuit comfortable living. The controlling and description of complex system are more and more difficulty because of the complexity function and large scale of system. Moreover the research to the whole complex system confronts numerous unthinkable troubles.The set-up of catastrophe theory is one of the most important achievements of nonlinear science. French mathematician Rene Thom presented catastrophe theory, which is one kind topology theory, in twenty century. Catastrophe theory can directly deal with discontinuity problems, especially applicable to research such system that its inside behavior mechanism is unknown. Therefore catastrophe theory is of definite vantage in the solution of analogous problems.Based on the catastrophe theory, static bifurcation theory and singularity theory, to research the brittleness problems of complex system. Describing with the development of system, enlarge of scales, increase of hierarchy, it is necessary to research the brittleness of complex system.At first, we research the complex system, which is the research object of this article. On the basis of combination the research to complex system of experts, who are belong to ancient or modern, or who are foreigners or Chinese, the qualitative description of complex system is given. In order to further realization of complex system, the judgment method of complex system is given and the conception of important degree of subsystem is also given.In the next place, in the analytical process of brittleness problem, we describe the existential objectivity of brittleness problem of complex system. The brittleness terminology of material mechanics is quoted to complex system, the new explanation of it is given in this article. On the basis of linguistic description of brittleness, the mathematic description is also given. In order to further understand the brittleness problem of complex system, we definite some concepts relative to brittleness, collapse, brittleness procedure and brittleness relevance. On the basis of mathematic description of brittleness, the basic properties ofbrittleness, recessive, concomitant, severity of results, are also given. Otherwise, based on the entropy of information principles, to forecast the ability of interference to motivate brittleness.Thirdly, based on the catastrophe series appraisal method, we establish the evaluation index of brittleness. In fact, the potential function of system is not easily built up. So in this article, on the basis of evolution relation between two systems or among three systems, we can translate it to the formality of potential’ function through equation translation. Next, the analysis of brittleness is given. The brittleness analyses of two representative nonlinear systems are also given. The key of the analysis is to define the bifurcate point or bifurcate point sets of systems. On the basis of datum exploration analysis, two kinds of methods to define catastrophic point are also given.At last, based on the thought of hypothesis testing of mathematical statistics, we take the signification test. Couple ensemble slide / detection is taken to analyze SARS catastrophes. We can prove that there is catastrophe in SARS. Based on the match of polynomial, to analyze the brittleness relevance between new supernumerary number of accurate diagnosis and new supernumerary number of physicals every day in Beijing and the simple brittleness analysis between them is given.更多还原