紧区间上保向微分同胚的光滑嵌入流

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.

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。

嵌入流及其弱问题为连接离散动力系统和连续动力系统的桥梁，是动力系统理论的基本问题之一。迭代可产生离散动力系统，其复杂性并非源于被迭代映射的非线性性，而是源于它的非单调性。对具有有限个非单调点的逐段单调函数，研究非单调高度这一刻画其复杂度的量，通过周期点给出函数高度为无穷的充分必要条件并进一步给出高度无穷函数在逐段单调函数类里的稠密性，说明高度是一个敏感的量且不具有稳定性；对具有无穷多个非单调点的平台函数，引入平整度的概念来刻画此类函数的复杂性并对其展开研究，将所有单平台函数按复杂度进行了分类，并对其中平台区间随迭代保持不变的函数给出其所有迭代根的构造方法。对于双曲情形下的保向微分同胚，通过Julia方程的正则解给出更多可全局光滑嵌入流的函数，克服之前需要给定函数在不动点附近线性的限制。微分方程的解可视为一个时间1映射所嵌入的光滑流。研究含有“状态依赖”时滞的微分方程的解的存在性问题，最大的困难在于解在其存在区间上的返回性。在不要求“不动点型”初值条件的情况下给出解的存在性。对于研究估计中所涉及的不等式，我们研究了一个带有限求和项的幂次积分不等式的的连续性、有界性、渐近性和连续依赖性，它从形式上包含之前所有有限求和积分不等式，估计式的存在区间也延拓到了无穷区间。以上结果部分发表于Discrete Contin. Dyn. Syst.，J. Math. Anal. Appl.和Results Math.等国际重要学术期刊，另有1篇已返回审稿意见并要求修改，1篇处于审稿中。