图灵斑图动力学中的分支问题

发布时间：2018-01-06 01:17

本文关键词：图灵斑图动力学中的分支问题　出处：《北京交通大学》2016年硕士论文　论文类型：学位论文

【摘要】：斑图(pattern)是指在空间上或者时间上,具有某种规律性的非均匀的宏观结构.自然界普遍存在着各种各样的斑图结构,所以我们才能看到这个五彩缤纷的世界.因而了解为什么会有斑图的形成,斑图是怎样形成的等问题,对于揭开自然界形成之谜具有重大的意义.1952年,英国著名的数学家图灵(Alan MathionTuring),在其著名的论文《形态发生的化学基础》中,成功地用一个反应扩散模型说明了某些生物的体表所显现的图纹,如斑马身上的斑图是怎样形成的.随着时代的发展,这一理论已经引起了化学、物理学、数学、生物学等学科的国内外众多研究者的兴趣和重视.如今,这一理论已然成为了反应扩散理论中最基础的理论之一般地在数学上,我们把Turing斑图动力学系统的数学机制的描述为：常微系统的稳定常数平衡态在加入扩散后会发生稳定性反转,从而在其附近就会产生图灵斑图.本文在绪论第一节中具体介绍了斑图动力学理论的背景来源,并且在第二节中给出了Lyapunov急定性理论,然后在第三节、第四节分别给出了一维空间和n维空间自由扩散系统Turing不稳定发生的条件,第五节中给出了进行分叉理论研究的准备知识.当前的研究基本只是对Turing不稳定系统何时出现分叉进行了一些研究,但是具体出现的分叉曲线是什么样子,对分叉解的稳定性如何没有什么深入的探索,所以本文主要针对这两个问题进行了研究.在第二章中,对一维空间自由扩散系统当Turing不稳定发生时,根据Crandall-Rabinowitz分叉理论研究了发生分叉的条件及其能够发生分叉的点,并对分叉点处的局部分叉解进行了研究,然后给出了相应的局部分叉图,在第三节中详细说明了局部分叉解的稳定性.在第三章第一节中,对n维空间自由扩散系统能否发生分叉进行了讨论,第二节我们以二维矩形区域上的自由扩散系统为例,分析了其分叉出现的条件,并对第一特征值处的局部分叉曲线作了详细研究,最后在第四章中对本文的研究进行了总结.
[Abstract]:The pattern (pattern) refers to the space or time, with some regularity of the non uniform macro structure. In nature exist patterns of various structure, so we can see the world. Thus a riot of colours to understand why the formation of pattern, pattern is how it formed. To solve the mystery of the nature of the formation is of great significance in.1952, the famous British mathematician Turing (Alan MathionTuring), the chemical basis of the famous paper > < form, successfully used a reaction diffusion model that some biological surface emerging patterns, such as zebra pattern how is it formed. With the development of the times, this theory has caused chemistry, physics, mathematics, many researchers at home and abroad biology interest and attention. Now, this theory has become the anti The basic theory of diffusion in general in mathematics, we put the mathematical mechanism of Turing Pattern Dynamics System Description: constant equilibrium differential systems will occur in reverse stability with diffusion, resulting in the vicinity of the Turing spot diagram. In the preface of this article will be introduced in the first quarter the pattern dynamics theory of background sources, and in section second gives the Lyapunov stability theory, and then in the third section, fourth section are given one dimension and n-dimensional space free diffusion system Turing instability conditions given in section fifth of the bifurcation preparation knowledge theory research. The current study is the basic the instability of Turing system when the bifurcation are studied. The bifurcation curves appear but specific what is, on the stability of bifurcation solutions without what depth To explore, so this paper researches the two problems. In the second chapter, the one-dimensional space free diffusion system when Turing instability occurs, according to the theory of Crandall-Rabinowitz bifurcation of the bifurcation condition and to the bifurcation point, and the bifurcation point bifurcation solutions is studied. And then the local bifurcation diagram, in section third, a detailed description of the stability of bifurcation solutions. In the first section of the third chapter, the n-dimensional space free diffusion system can bifurcation are discussed, I have second day free diffusion system with a rectangular domain as an example, the bifurcation condition of the the analysis and the first eigenvalue of local bifurcation curves have been studied in detail, finally in the fourth chapter of this study are summarized.

【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175

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