Moser’s Theorem For Twist Mappings

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.