Simplified renormalization group method for ordinary differential equations

Hayato Chiba

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

The renormalization group (RG) method for differential equations is one of the perturbation methods which allows one to obtain invariant manifolds of a given ordinary differential equation together with approximate solutions to it. This article investigates higher order RG equations which serve to refine an error estimate of approximate solutions obtained by the first order RG equations. It is shown that the higher order RG equation maintains the similar theorems to those provided by the first order RG equation, which are theorems on well-definedness of approximate vector fields, and on inheritance of invariant manifolds from those for the RG equation to those for the original equation, for example. Since the higher order RG equation is defined by using indefinite integrals and is not unique for the reason of the undetermined integral constants, the simplest form of RG equation is available by choosing suitable integral constants. It is shown that this simplified RG equation is sufficient to determine whether the trivial solution to time-dependent linear equations is hyperbolically stable or not, and thereby a synchronous solution of a coupled oscillators is shown to be stable.

Original languageEnglish
Pages (from-to)1991-2019
Number of pages29
JournalJournal of Differential Equations
Volume246
Issue number5
DOIs
Publication statusPublished - Mar 1 2009
Externally publishedYes

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Linear equations
Ordinary differential equations
Renormalization Group
Ordinary differential equation
Differential equations
Invariant Manifolds
Higher Order
Approximate Solution
Indefinite integral
First-order
Coupled Oscillators
Perturbation Method
Theorem
Error Estimates
Linear equation
Vector Field
Trivial
Sufficient
Differential equation

All Science Journal Classification (ASJC) codes

  • Analysis

Cite this

Simplified renormalization group method for ordinary differential equations. / Chiba, Hayato.

In: Journal of Differential Equations, Vol. 246, No. 5, 01.03.2009, p. 1991-2019.

Research output: Contribution to journalArticle

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