Normal forms, resonances, and meandering tip motions near relative equilibria of Euclidean group actions
type: Item
instance of: scholarly article
title: Normal forms, resonances, and meandering tip motions near relative equilibria of Euclidean group actions
zbMATH Open document ID: 0957.37026
author: B. Fiedler
author: D. Turaev
published in: Archive for Rational Mechanics and Analysis
publication date: 2001-01-30T00:00:00Z
review text: Motivated by applications to PDEs equivariant with respect to the affine action of the Euclidean group \(E(2)\), which model the dynamics of spiral waves, the authors study finite dimensional smooth differential systems equivariant with respect to an action of noncompact, finite dimensional Lie groups, near relative equilibria. Introducing some special coordinates respecting the action, the system has a triangular form with the base system independent of other variables. Relative equilibria of the whole system are equilibria of this base system. After that they apply the normal form method to transform the system locally near relative equilibria to some normal form eliminating resonances up to any finite order. All transformations preserve skew product structure and equivariance. Then this normal form is applied to the case of an action of the Euclidean group \(SE(N)\). One of the main points is the interpretation of the normal form results for the initial system since normal form transformations can mix important variables in the group \(SE(N)\), namely, rotations \(R\in SO(N)\), and drift \(S\in \mathbb{R}^n\). This point is clarified in the important case of the group \(SO(2)\). Here the notions of tip motion of the spiral and meandering spiral obtain the precise meaning. Also bifurcations that can occur in the base system are examined and how they can effect to the motion of the group variables in the case of resonances between Hopf and rotation frequencies. They also discuss phenomena in the whole system related with the presence of homoclinic orbits in the base system.
Mathematics Subject Classification ID: 37C85
Mathematics Subject Classification ID: 37G40
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 37C80
Mathematics Subject Classification ID: 37C29
Mathematics Subject Classification ID: 37G05
zbMATH DE Number: 1287915
zbMATH Keywords: equivariant system
zbMATH Keywords: Lie group action
zbMATH Keywords: Palais coordinates
zbMATH Keywords: normal form method
zbMATH Keywords: resonance
zbMATH Keywords: bifurcation
zbMATH Keywords: relative equilibrium
zbMATH Keywords: homoclinic
zbMATH Keywords: heteroclinic
zbMATH Keywords: tip motion
reviewed by: L. M. Lerman
MaRDI profile type: Publication
full work available at URL: s002050050126
OpenAlex ID: W2021369835
DOI: 10.1007/S002050050126
description: scientific article; zbMATH DE number 1287915