- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Advances in Mathematical Physics
Volume 2012 (2012), Article ID 505281, 17 pages
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.
- P. J. Olver, Application of Lie Group to Differential Equations, Springer, Berlin, Germany, 1986.
- G. W. Bluman and S. Kumei, Symmetries and Differential Equations, vol. 81 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
- J. Bricmont and A. Kupiainen, “Renormalization group and the Ginzburg-Landau equation,” Communications in Mathematical Physics, vol. 150, no. 1, pp. 193–208, 1992.
- D. V. Shirkov and V. F. Kovalev, “The Bogoliubov renormalization group and solution symmetry in mathematical physics,” Physics Reports, vol. 352, no. 4–6, pp. 219–249, 2001.
- G. Gaeta, “Asymptotic symmetries and asymptotically symmetric solutions of partial differential equations,” Journal of Physics, vol. 27, no. 2, pp. 437–451, 1994.
- G. Gaeta and R. Mancinelli, “Asymptotic scaling symmetries for nonlinear PDEs,” International Journal of Geometric Methods in Modern Physics, vol. 2, no. 6, pp. 1081–1114, 2005.
- M. Iwasa and K. Nozaki, “A method to construct asymptotic solutions invariant under the renormalization group,” Progress of Theoretical Physics, vol. 116, no. 4, pp. 605–613, 2006.
- M. Iwasa and K. Nozaki, “Renormalization group in difference systems,” Journal of Physics, vol. 41, no. 8, Article ID 085204, 7 pages, 2008.
- H. Chiba and M. Iwasa, “Lie equations for asymptotic solutions of perturbation problems of ordinary differential equations,” Journal of Mathematical Physics, vol. 50, no. 4, Article ID 042703, 18 pages, 2009.
- M. Iwasa, “Solution of reduced equations derived with singular perturbation methods,” Physical Review E, vol. 78, no. 6, Article ID 066213, 6 pages, 2008.
- H. Chiba, “Extension and unification of singular perturbation methods for ODEs based on the renormalization group method,” SIAM Journal on Applied Dynamical Systems, vol. 8, no. 3, pp. 1066–1115, 2009.
- D. Levi and P. Winternitz, “Continuous symmetries of difference equations,” Journal of Physics, vol. 39, no. 2, pp. R1–R63, 2006.
- L.-Y. Chen, N. Goldenfeld, and Y. Oono, “Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory,” Physical Review E, vol. 54, no. 1, pp. 376–394, 1996.
- S.-I. Goto, Y. Masutomi, and K. Nozaki, “Lie-group approach to perturbative renormalization group method,” Progress of Theoretical Physics, vol. 102, no. 3, pp. 471–497, 1999.
- A. H. Nayfeh, Method of Normal Forms, Wiley Series in Nonlinear Science, John Wiley & Sons, New York, NY, USA, 1993.
- J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983.
- N. N. Bogoilubov and C. M. Place, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Break, 1961.
- G. I. Hori, “Theory of general perturbations with unspecified canonical variables,” Publications of the Astronomical Society of Japan, vol. 18, no. 4, pp. 287–296, 1966.
- C. K. R. T. Jones, “Geometric singular perturbation theory,” in Dynamical Systems, vol. 1609 of Lecture Notes in Math., pp. 44–118, Springer, Berlin, Germany, 1995.