【作者】 刘畅；

【导师】 李勇；

【作者基本信息】 东北师范大学 ， 应用数学， 2023， 博士

【摘要】 本文利用KAM理论研究扭转映射的Moser定理,从可积扭转映射的定义出发,给出具有相交性质的扭转映射的不变环面在小摄动下的保持性.经典的KAM理论是20世纪60年代由Kolmogorov,Arnold和Moser建立的,考虑的是动力系统在小摄动下的运动或轨道的稳定性问题.对于离散的动力系统,Moser最早给出保面积扭转映射的不变曲线在小摄动下的保持性,即Moser不变曲线定理.它是研究许多物理系统动力学稳定性的重要工具.本文主要考虑作用-角变量维数可以不相同的含参数扭转映射和有限可微扭转映射的Moser定理,以及保频Moser定理.本文由五章组成,主要内容如下:第一章介绍本文的研究背景与动机,主要介绍KAM理论和Moser不变曲线定理的发展进程以及相关研究成果,同时给出本文的主要工作以及全文安排.第二章回顾相关预备知识和结果,主要包括在KAM迭代过程中引入的近恒等变换所必须满足的差分方程的唯一解及其估计,以及Paley-Wiener估计和Cauchy估计等.此外,介绍求解新摄动项时需要用到的隐函数定理和本文中用到的范数以及连续模的定义.最后,对于有限可微的动力系统,介绍现如今人们常用的光滑函数逼近方法及其发展.第三章考虑解析的含参数扭转映射,假设该映射的作用-角变量的维数可以不相同并且映射满足相交性质.利用KAM迭代方法给出对应的Moser定理及其详细证明过程.因为1个极半径变量的保体积扭转映射显然满足相交性质,作为推论,本章将给出对应的含参数保体积扭转映射的Moser定理.Moser不变曲线定理中考虑的映射是充分光滑的,这在实际中却难以应用,因此涌现了大量关于映射的最优正则性的研究.第四章考虑有限可微的扭转映射,仍然假设其作用-角变量的维数可以不相同且该映射满足相交性质,利用光滑函数逼近方法,本章得到有限可微映射对应的Moser定理.同样地,当考虑只有1个作用量的保体积映射时,可以得到保体积的有限可微扭转映射的Moser定理.在迭代过程中保持频率不变一直是KAM理论中的一个基本问题,但也是一个十分困难的问题.第五章中沿用新近人们给出的方法,考虑作用-角变量维数相同的有相交性质的扭转映射,假设拓扑度条件和弱凸条件成立,并且摄动项关于作用-角变量是实解析的,频率关于作用量是模连续的,利用KAM迭代方法能够得到保频Moser定理.据我们所知,这是第一个使得扭转映射的频率在迭代过程中保持不变的结论.当考虑保面积扭转映射时,该结论实际上就是保频Moser不变曲线定理.此外,作为推论,还可以得到保频Herman定理,只要假设频率和摄动项关于参数是连续的.更多还原

【Abstract】 This thesis studies Moser’s theorem for twist mappings using KAM theory and states the persistence of invariant tori of twist mappings with intersection property under small perturbations starting from the integrable twist mappings.The well-known KAM theory was established by Kolmogorov,Arnold,and Moser in the 1960s.It concerns the stability of motions or orbits of dynamical systems under small perturbations.For discrete dynamical systems,Moser first stated the persistence of invariant curves of area-preserving twist mappings under perturbations,i.e.,Moser’s invariant curve theorem.It is an important tool for studying the dynamic stability of many physical systems.This thesis mainly concerns Moser’s theorem for twist mappings with a parameter and for finitely differentiable twist mappings,as well as Moser’s theorem with frequency-preserving.Moreover,the action and angular variables could have different dimensions in the first two cases.Our main research in this thesis is divided into the following five chapters.Chapter 1 introduces the backgrounds and motivations of this thesis including the KAM theory,Moser’s invariant curve theorem,and the related results.Meanwhile,the main results and framework of this thesis are given.Chapter 2 recalls some related preliminaries and results required for this thesis,mainly containing the unique solution of difference equations and estimates,as well as the Paley-Wiener estimates and Cauchy estimates.More precisely,the nearly identical transformations induced by the iteration processes must satisfy the differentiable equations.Additionally,introducing the Implicit Function Theorem which is used to deal with new perturbations in the next KAM step,as well as the norms and the definition of modulus of continuity used in this thesis.For finitely differentiable dynamical systems,the approximation methods used today for smooth functions and their development are presented at the end of Chapter 2.Chapter 3 considers the real analytic twist mappings containing a parameter.Assume that the action and angular variables could have different dimensions,and the mappings have intersection property.The KAM iteration processes are used to present the corresponding Moser’s theorem.This chapter gives the details of the proof as well.Obviously,the volume-preserving mappings with only one polar radius variable satisfy the intersection property.This chapter will show the corresponding Moser’s theorem for volume-preserving mappings with a parameter as a corollary.The systems considered in Moser’s invariant curve theorem are sufficiently smooth,which is difficult to apply in reality.A large number of studies on the optimal regularity of mappings has therefore arisen.Chapter 4 considers the finitely differentiable mappings.Assume that the action and angular variables could have different dimensions,and mappings satisfy the intersection property.The corresponding Moser’s theorem for finitely differentiable mappings is obtained by using the approximation method of smooth functions.Similarly,it is able to obtain Moser’s theorem for volume-preserving finitely differentiable mappings when there is only one action.Frequency-preserving during iteration has always been a fundamental problem in KAM theory,but it is also very difficult.Chapter 5 follows approaches developed recently and considers a system of twist mappings with intersection property of the same dimension of the action and angular variables.Assume that the topological condition and weak convexity condition hold and that the perturbations are real analytic with respect to the action-angular variables and the frequency is continuous about the action.The Moser’s theorem with frequency-preserving is gained using KAM iteration processes.To the best of our knowledge,this is the first conclusion of Moser’s theorem with frequency-preserving for twist mappings.When considering area-preserving twist mappings,this conclusion is in fact Moser’s invariant curve theorem with frequency-preserving.Furthermore,as a corollary,it can also obtain Herman’s theorem with frequency-preserving when the frequency and perturbation are only continuous about the parameters.更多还原