Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling
type: Item
instance of: scholarly article
title: Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling
zbMATH Open document ID: 1019.65043
DOI: 10.1016/S0377-0427(01)00573-8
published in: Journal of Computational and Applied Mathematics
publication date: 2002-08-22T00:00:00Z
review text: A nonlinear ordinary differential equation \[ {dx\over dt}=f(x,\mu),\quad x\in {\mathbb{R}^n},\quad \mu\in{\mathbb{R}^m}, \] analytic with respect to \(x\) and a parameter \(\mu\), with the equilibrium point \((0,\mu)\) is considered. The aim of this work is to find a possibly simple form of any term homogeneous with respect to \(x\) and \(\mu\) of the Taylor expansion of \(f(x,\mu)\) at the equilibrium point. Such a procedure may be very useful in studying the singularities such as bifurcation, instability and so on of the equation discussed . Here the important tool is the Lie transformation of any homogeneous \(k\)-degree polynomial \(U_k\) \[ U_k\mapsto[U_k,v]=DvU_k-DU_kv, \] where \(v\) is of degree 1. The author develops basic ideas involved in the procedure of finding the so called conventional normal form. The number of variables in the system of linear algebraic equations of the above mentioned procedure is greater than the number of equations. Instead of setting to zero the variables which cannot be determined from the system, the author retains all the variables. It gives him the possibility to use them in the process of further reduction. Such a procedure leads to the new `form' of the normal form, the so called simplest normal form (SNF). Theorem 1 gives formulae connecting any homogeneous term of the SNF with corresponding terms of the nonlinear transformation. The formulae contain time rescaling factors, and are suitable for recursive computation, for example, with use of a kind of computer algebra system as Maple. The case of single zero singularity is discussed more in details.
reviewed by: Krzysztof Moszyński
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1787669
zbMATH Keywords: simplest normal forms
zbMATH Keywords: nonlinear ordinary differential equation
zbMATH Keywords: bifurcation
zbMATH Keywords: instability
zbMATH Keywords: Lie transformation
zbMATH Keywords: conventional normal form
zbMATH Keywords: simplest normal form
zbMATH Keywords: computer algebra system
zbMATH Keywords: single zero singularity
author: P. Yu
describes a project that uses: Maple
MaRDI profile type: Publication
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full work available at URL: s0377-0427(01)00573-8
OpenAlex ID: W2145011665
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description: scientific article; zbMATH DE number 1787669