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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. doi:10.1137/0518092.

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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.

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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.
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NSFDMS 84-04506
Subject Keywords:invariant manifold, deformation method, center manifold, Lie transform
Issue or Number:5
Classification Code:AMS(MOS) subject classification: 58F
Record Number:CaltechAUTHORS:20100908-094725042
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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
Deposited By: Ruth Sustaita
Deposited On:09 Sep 2010 15:13
Last Modified:08 Nov 2021 23:55

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