KAM Theorems For Non-twist Area-preserving Mappings And Their Applications

In this thesis,we first use the KAM technique to prove two invariant curve theorems of non-twist area-preserve mappings.Then we apply these results to study Lagrange stability of several classes of second order systems.This article contains six chapters.The first chapter introduces the background of areapreserve mappings,the source of the problem of invariant curves and some of the classic KAM results.These results include Moser’s twist theorem(small twist theorem)and certain KAM-type theorems with weaker twist conditions.In addition,the applications of these theorems to Lagrange stability of second order systems are also mentioned.Finally,we introduce an important mechanical model related to area-preserve mappings,that is,the ping-pang model.In Chapter 2,we study a family of non-twist area-preserve mappings on annulus and then prove that for many sufficiently small parameters the mapping admits an invariant closed curve.In Chapter 3,we consider a family of quasi-periodic planar area-preserve mappings,which depends on a small parameter.Without assuming any twist condition,we prove that for many sufficiently small parameters the planar mapping has an invariant curve.In Chapter 4,we apply these invariant curve theorems obtained in Chapter 2 and 3 to study Lagrange stability of three classes of second order systems,including asymptotic linear Duffing equations,asymmetric oscillations and quasi-periodically forced asymmetric oscillations,and then under weaker assumptions of nonlinear terms,we prove Lagrange stability of these systems.In Chapter 5,we study the boundedness and unboundedness orbits of the ping-pang model.This model considers a ball bouncing off an infinitely heavy plate,which moves periodically in the vertical direction.We first construct an unbounded orbit of the ball in the gravity field,which is also a counterexample to Moser’s twist theorem for the smoothness condition.Besides,we prove that in the quadratic potential all orbits of the ball are bounded if the motion of the plane only satisfies smoothness conditions.This boundness result depends on the invariant closed curve obtained in Chapter 2.The Chapter 6 dedicates to summary of study in this thesis and prospect of research in the future.