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Advances in Mathematical Physics
Volume 2012 (2012), Article ID 505281, 17 pages
http://dx.doi.org/10.1155/2012/505281
Review Article

Reduction of Dynamics with Lie Group Analysis

Department of Complex Systems Science, Graduate School of Information Science, Nagoya University, Nagoya 464-8601, Japan

Received 1 October 2011; Accepted 6 January 2012

Academic Editor: Mariano Torrisi

Copyright © 2012 M. Iwasa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is mainly a review concerning singular perturbation methods by means of Lie group analysis which has been presented by the author. We make use of a particular type of approximate Lie symmetries in those methods in order to construct reduced systems which describe the long-time behavior of the original dynamical system. Those methods can be used in analyzing not only ordinary differential equations but also difference equations. Although this method has been mainly used in order to derive asymptotic behavior, when we can find exact Lie symmetries, we succeed in construction of exact solutions.