Normally elliptic singular perturbationproblems: local invariant manifolds and applications

Dynamical system;Invariant manifold;Homoclinic orbit

Lu, Nan ; Mathematics

University:Georgia Institute of Technology

Department:Mathematics

关键词: Dynamical system;    Invariant manifold;    Homoclinic orbit;   

Others  :  https://smartech.gatech.edu/bitstream/1853/41090/1/Lu_Nan_201108_phd.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

In this thesis, we study the normally elliptic singular perturbation problems including both finite and infinite dimensional cases, which could also be non-autonomous. In particular, we establish the existence and smoothness of O(1) local invariant manifolds and provide various estimates which are independent of smallsingular parameters. We also use our results on local invariant manifolds to studythe persistence of homoclinic solutions under weakly dissipative and conservative per-turbations. We apply Semi-group Theory and Lyapunov-Perron Integral Equations with somecareful estimates to handle the O(1) driving force in the system so that we can approximate the full system through some simpler limiting system. In the investigation of homoclinics, a diagonalization procedure and some normal form transformation should be first carried out. Such diagonalization procedure is not trivial at all. We discuss this issue in the appendix. We use Melnikov type analysis to study the weaklydissipative case, while the conservative case is based on some energy methods. As a concrete example, we have shown rigrously the persistence of homoclinic solutions of an elastic pendulum model which may be affected by damping, externalforcing and other potential fields.