Symbolic computation of normal forms for semi-simple cases
type: Item
instance of: scholarly article
title: Symbolic computation of normal forms for semi-simple cases
zbMATH Open document ID: 0944.65132
DOI: 10.1016/S0377-0427(98)00222-2
author: Qinsheng Bi
author: P. Yu
published in: Journal of Computational and Applied Mathematics
publication date: 2000-09-14T00:00:00Z
review text: A nonlinear system of ordinary differential equations is considered \[ \dot x=Ax+F(x),\quad x\in \mathbb{R}^{n},\quad F(x):\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}. \] Here \(A\) is an \(n\times n\) matrix with semi-simple eigenvalues, the nonlinear function \(F\) is analytic, and \(F_{i}=0,\frac{\partial F_{i}}{\partial x_{j}}=0\) at \(x=0,\forall i,j=1,\cdots,n.\) Efficient symbolic computation programs are developed for computing explicit normal forms and associated nonlinear transformations based on the coefficients of the original differential system. This approach combines the center manifold theory and normal form theory into one step to simultaneously obtain closed form expressions with the aid of the suggested iterative procedure. It does not require the computation of large-dimensional inverse matrices and therefore reduces computation time. The computer language Maple is used to facilitate the application of the method. A number of illustrative examples is presented.
Mathematics Subject Classification ID: 65P30
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 37M20
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1331060
zbMATH Keywords: normal forms
zbMATH Keywords: semi-simple eigenvalues
zbMATH Keywords: center manifold
zbMATH Keywords: symbolic computation
zbMATH Keywords: Maple
zbMATH Keywords: numerical examples
reviewed by: Boris V. Loginov
describes a project that uses: Maple
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1331060