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The Construction and Smoothness of Invariant Manifolds by the Deformation Method

Marsden, Jerrold and Scheurle, Jürgen (1987) The Construction and Smoothness of Invariant Manifolds by the Deformation Method. SIAM Journal on Mathematical Analysis, 18 (5). pp. 1261-1274. ISSN 0036-1410. http://resolver.caltech.edu/CaltechAUTHORS:20100908-094725042

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Abstract

This paper proves optimal results for the invariant manifold theorems near a fixed point for a mapping (or a differential equation) by using the deformation, or Lie transform, method from singularity theory. The method was inspired by the difficulties encountered by the implicit function theorem technique in the case of the center manifold. The idea here is simply to deform the given system into its linearization and to track this deformation using the flow of a time-dependent vector field. Corresponding to the difficulties with the center manifold encountered by other techniques, we run into a "derivative loss" in this case as well, which is overcome by utilizing estimates on the differentiated equation. A survey of the other methods used in the literature is also presented.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1137/0518092DOIUNSPECIFIED
http://www.cds.caltech.edu/~marsden/bib/1987/05-MaSc1987/AuthorUNSPECIFIED
Additional Information:© 1987 Society for Industrial and Applied Mathematics. Received by the editors May 9, 1986; accepted for publication (in revised form) August 14, 1986. The research of this author was partially supported by National Science Foundation contract DMS 84-04506. We thank Ethan Akin, Marty Golubitsky, Jack Hale, Morris Hirsch, Pat McSwiggen and Charles Pugh for their comments.
Funders:
Funding AgencyGrant Number
NSFDMS 84-04506
Subject Keywords:invariant manifold, deformation method, center manifold, Lie transform
Classification Code:AMS(MOS) subject classification: 58F
Record Number:CaltechAUTHORS:20100908-094725042
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100908-094725042
Official Citation:The Construction and Smoothness of Invariant Manifolds by the Deformation Method Jerrold Marsden and Jurgen Scheurle, SIAM J. Math. Anal. 18, 1261 (1987), DOI:10.1137/0518092
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19826
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:09 Sep 2010 15:13
Last Modified:26 Dec 2012 12:24

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