Reduction of Dynamics with Lie Group Analysis

@article{Iwasa2012ReductionOD,
  title={Reduction of Dynamics with Lie Group Analysis},
  author={Masatomo Iwasa},
  journal={Advances in Mathematical Physics},
  year={2012},
  volume={2012},
  pages={1-17}
}
  • M. Iwasa
  • Published 2012
  • Mathematics
  • Advances in Mathematical Physics
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… Expand
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References

SHOWING 1-10 OF 19 REFERENCES
Continuous symmetries of difference equations
Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program:Expand
Solution of reduced equations derived with singular perturbation methods.
  • M. Iwasa
  • Mathematics, Physics
  • Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2008
TLDR
It is shown that all of the solutions of the reduced equations constructed with singular perturbation methods are exactly equal to the sum of the most divergent secular terms appearing in the naive expansion. Expand
Renormalization group in difference systems
A new singular perturbation method based on the Lie symmetry group is presented to a system of difference equations. This method yields consistent derivation of a renormalization group equation whichExpand
Lie equations for asymptotic solutions of perturbation problems of ordinary differential equations
Lie theory is applied to perturbation problems of ordinary differential equations to construct approximate solutions and invariant manifolds according to the renormalization group approach of IwasaExpand
Asymptotic symmetries and asymptotically symmetric solutions of partial differential equations
Symmetry methods for differential equations are a powerful tool to attack nonlinear problems, in particular for determining solutions with given symmetries to nonlinear PDEs. Since in realExpand
Lie-Group Approach to Perturbative Renormalization-Group Method
A new Lie-group approach to the perturbative renormalization group (RG) methodis developed to obtain asymptotic solutions of both autonomous and non-autonomous ordinary differential equations.Expand
Asymptotic Methods in the Theory of Nonlinear Oscillations
Abstract : This book is devoted to the approximate asymptotic methods of solving the problems in the theory of nonlinear oscillations met in many fields of physics and engineering. It is intended forExpand
The Bogoliubov renormalization group and solution symmetry in mathematical physics
Abstract Evolution of the concept known in theoretical physics as the renormalization group (RG) is presented. The corresponding symmetry, that was first introduced in quantum field theory (QFT) inExpand
A Method to Construct Asymptotic Solutions Invariant under the Renormalization Group
A renormalization group method with the Lie symmetry is presented for the singular perturbation problems. Asymptotic solutions are obtained as group-invariant solutions under approximate Lie groupsExpand
ASYMPTOTIC SCALING SYMMETRIES FOR NONLINEAR PDES
In some cases, solutions to nonlinear PDEs happen to be asymptotically (for large x and/or t) invariant under a group G which is not a symmetry of the equation. After recalling the geometricalExpand
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