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 small
singular parameters. We also use our results on local invariant manifolds to study
the persistence of homoclinic solutions under weakly dissipative and conservative per-
turbations. We apply Semi-group Theory and Lyapunov-Perron Integral Equations with some
careful 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 weakly
dissipative 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, external
forcing and other potential fields.
| Identifer | oai:union.ndltd.org:GATECH/oai:smartech.gatech.edu:1853/41090 |
| Date | 18 May 2011 |
| Creators | Lu, Nan |
| Publisher | Georgia Institute of Technology |
| Source Sets | Georgia Tech Electronic Thesis and Dissertation Archive |
| Detected Language | English |
| Type | Dissertation |
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