The Application Of Catastrophe Theory In The Research Of Complex System Brittleness Theory

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.