Bifurcation of an invariant torus from an equilibrium under commensurability conditions
type: Item
instance of: scholarly article
title: Bifurcation of an invariant torus from an equilibrium under commensurability conditions
zbMATH Open document ID: 1076.34043
author: Yu. V. Usachev
published in: Differential Equations
publication date: 2005-06-17T00:00:00Z
review text: This paper considers a \((2n+r)\)-dimensional autonomous ordinary differential equation with a parameter which generates a bifurcation when its value crosses through the value zero. The matrix of the linear approximation has \(n\) pairs, \(n>1\), of pure imaginary eigenvalues, and the zero eigenvalue with the multiplicity \(r\). The pure imaginary eigenvalues are supposed to be distinct and commensurable. The author studies the bifurcation of an \(s\)-dimensional torus (\(s<n\) being defined by two relations satisfied by the eigenvalues) from the equilibrium supposed to be at the origin of the phase space when the parameter is equal to zero. For this study the autonomous ordinary differential equation is normalized in a classical way (in particular used by Malkin in the past). A theorem is proved, and an example of a seven-dimensional equation is given.
reviewed by: Christian Mira
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 37G05
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 2176613
zbMATH Keywords: ordinary differential equations
zbMATH Keywords: nonlinear dynamics
zbMATH Keywords: vector fields
zbMATH Keywords: bifurcation
MaRDI profile type: Publication
full work available at URL: b:dieq.0000017926.26138.8c
OpenAlex ID: W2075732003
DOI: 10.1023/B:DIEQ.0000017926.26138.8C
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description: scientific article; zbMATH DE number 2176613