紧区间上保向微分同胚的光滑嵌入流

项目名称： 紧区间上保向微分同胚的光滑嵌入流

项目编号： No.11501394

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 曾莹莹

作者单位： 四川师范大学

项目金额： 18万元

中文摘要： 嵌入流问题是动力系统的重要问题之一。特别是对紧区间上保向微分同胚的光滑嵌入流，因其必于两端点处保持不动，因此被视为一全局嵌入且对非局部分岔的研究意义重大。其中，由局部嵌入到全局嵌入过程中所可能出现的无穷振荡给光滑性的全局保持带来极大困难。本项目将对可全局光滑嵌入流的保向微分同胚进行探讨。针对不动点均双曲的情形，尽管可光滑嵌入的函数通有不存在，但仍有研究其存在性的必要。我们希望利用Julia方程的正则解来克服线性化仅适用于不动点局部的缺陷，对函数分别在内点及不动点附近进行改造，构造出可全局C1嵌入流的函数。针对非双曲情形Julia方程已不再适用。我们旨在利用Abel、Böttcher与线性化方程之间的变量替换关系，给出克服仅能局部嵌入的新方法，进而讨论该情形下可全局嵌入函数的构造。基于函数构造过程中桥函数的一定任意性,进一步考虑被改造区间的大小和位置对全局嵌入流的影响，以期解决拟迭代根问题。

中文关键词： 嵌入流；光滑性；全局；共轭；泛函方程

英文摘要： The embedding flow problem is an important problem in the dynamical systems. Specially, for the orientation-preserving differeomorphism on the interval, its smooth embedding problem is actually a global problem and is meaningful in the study of non-local bifurcation theory. This is because the end points of the interval are both fixed points. In the extending process from local case to global case, some great difficulties are caused by a violent oscillation and then lead to a difficulty of keeping smoothness. In this project we will study the orientation-preserving diffeomorphism which can be embedded in a smooth flow. In the case that all the fixed points are hyperbolic，although the functions, available for globally embedding problem, are found “rarely”, it is still necessary to give the existence. We expect to use the regular solutions of the Julia equation to overcome the shortcoming that linearization is only available for a neighborhood of a fixed pint. Then we can construct a function satisfying global embedment by give a pulse in a neighborhood of an inner point or a fixed point. However，Julia equation is not working for the non-hyperbolic case. We want to give a new method by using the relation among Abel,Böttcher and linearization. And then discuss the problem of constructing a globally embeddable function. On the basis of some arbitrariness for the conjugate function, we can further consider the influence caused by the size and position of the reconstructed interval, and give a consideration to the problem of qusi-iterative roots.

英文关键词： embedding flow;smoothness;global;conjugacy;functional equations