Lagrange Stability For Two Kinds Of Second Order Ordinary Differential Equations

KAM theory is one of the most important achievements about the solutions’ stability of differential equations in twentieth century. It derives from the study of N-body problem. Now, it has been the important tool to study solutions’ stability for differential equations. As a part of KAM theory, Moser’s twist theorem is usually applied to prove the Lagrange stability of planar systems. Recently, after R.Ortega, Liu et al. extended Moser’s twist theorem, the extended theorem can solve more problems on solutions’stability for differential equations.Duffing equation is a simple but important mathematical equation. It describes lots of physical phenomena, such as resonance, harmonic vibration, sub-harmonic vibration, quasi-periodic vibration, almost periodic vibration, singular attractors and chaos phenomena, etc. As we known, the reversible system has significant applications and tremendous theoretical value in fluids, mechanics and optics. Therefore, in the differential equations’study, to learn the nature of solutions for Duffing equation and reversible system is very popular.This thesis, which is divided into three parts, is concerned with the Lagrange stability of one type of Duffing equation and one kind of reversible system, which is solved by the variants of Moser’s twist theorem.In the preface, the author simply introduces the history and development of the solutions’stability of Duffing equations and reversible systems, and then brings up the problems we investigated on.The first chapter is a brief introduction of classical Moser’s twist theorem and its extensions.The second chapter studies Lagrange stability for a type of Duffing equations. The results of Wang are extended. Namely, the Lagrange stability of Duffing equation with jumping term and time periodic potential can be proved by Ortega’s twist theorem.The third chapter is concerned with boundedness for a kind of reversible systems. The result of Liu [4] is extended. Namely, the existence of bounded solution for reversible system at resonance is proved by the invariant of twist theorem[5].