一种通过δ函数求解分片线性映射的不变密度的新方法

发布时间：2018-01-17 00:26

本文关键词：一种通过δ函数求解分片线性映射的不变密度的新方法　出处：《中国科学院研究生院(武汉物理与数学研究所)》2016年硕士论文　论文类型：学位论文

【摘要】：不变密度是动力系统研究一个重要的课题。本文研究区间I=[0,1]上的分片线性映射的不变密度的表达式求解。本文是通过求解F-P算子方程：Pτf=f,以得到τ的不变密度。在求解过程中,本文采用广义微商和类似于偏微分方程中求解线性方程基本解的思想,引入δ函数；在不变密度具有特定形式的假定下,之后将F-P方程化为级数形式,并通过分片线性映射的端点轨道性质求得不变密度形式。本文在第三章中对四个两片线性映射的不变密度进行了讨论,分别是τ(0)=0的Lorenz映射、单峰型映射、一般的Lorenz映射和压缩单峰映射。其中前三者得到了不变密度的形式解,但是压缩单峰映射的F-P方程无法求解,与其不变密度的不存在性吻合,说明了该方法的稳定性。
[Abstract]:Invariant density is an important subject in the study of dynamic systems. Interval I =. [The invariant density of piecewise linear mappings is solved by solving the F-P operator equation: P 蟿 f f, so as to obtain the invariant density of 蟿. In this paper, the concept of generalized derivative and the idea of solving the basic solutions of linear equations similar to partial differential equations are introduced, and 未 functions are introduced. Under the assumption that the invariant density has a particular form, the F-P equation is then transformed into a series form. The invariant density form of piecewise linear mappings is obtained by the properties of extreme orbits. In chapter 3, the invariant densities of four two linear mappings are discussed. They are the Lorenz map of 蟿 0, the unimodal mapping, the general Lorenz map and the contraction unimodal map, where the first three obtain the formal solution of the invariant density. However, the F-P equation of contractive unimodal mapping can not be solved, which is consistent with the nonexistence of its invariant density, which shows the stability of the method.
【学位授予单位】：中国科学院研究生院(武汉物理与数学研究所)
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175

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