Reversible equivariant linear systems
type: Item
instance of: scholarly article
title: Reversible equivariant linear systems
zbMATH Open document ID: 0954.37016
author: J. S. W. Lamb
author: R. M. Roberts
published in: Journal of Differential Equations
publication date: 2001-02-12T00:00:00Z
review text: The authors consider reversible equivariant systems of the form \[ \frac{dx}{dt} = L(x), \] where \(L: {\mathbb R}^n \mapsto {\mathbb R}^n\) is linear. Let \(G\) be the group of symmetries and reversing symmetries (which act linear) of the system. Throughout of the paper it is assumed that \(G\) is a compact Lie group. In the case when \(G\) contains reversing symmetries the subgroup \(H\) of symmetries is a normal subgroup of index two. If \((V, \rho)\) is a representation of \(G\) a linear map \(L: V \rightarrow V\) is said to be \(\sigma\)-reversible if \[ L(\rho(g)v) = \sigma(g) \rho(g) L(v) \] for all \(g\) in \(G.\) Here \(\sigma: G \rightarrow \{-1,1\} \subset GL(1;{\mathbb R})\) is the one-dimensional representation of \(G,\) given by \(\sigma(g) = 1\) if \(g \in H\) and \(\sigma(g) = -1\) if \(\sigma \in G \setminus H.\) The main result (Theorem 3.10) of the paper is a classification of the possible structures of the spaces of \(\sigma-\)reversible maps. The authors give this in terms of both real, complex and quaternion-type representations of \(G.\) Along with the theorem the authors consider normal form theory and eigenvalue movements. Some aspects of the linear and nonlinear cases are discussed in \textit{V. Arnold} [Supplementary chapters to the theory of ordinary differential equations (Nauka, Moscow) (1978)] and different techniques, results and examples for the nonlinear case are discussed in the collection [Bifurcations and periodic orbits of vector fields (D. Schlomiuk, ed., NATO ASI Ser., Kluwer Acad. Publ., Dordrecht) (1993; Zbl 0780.00040)]. The paper under review consists of 3 Sections. In Section 1 the authors give some definitions, examples, summarize the main results and briefly discuss some applications. Examples illustrate the aspects of the local dynamics systems under investigation. A discussion of the changes in the local dynamics in the presence of nonlinear terms is contained in \textit{J. S. W. Lamb} and \textit{H. W. Capel} [Chaos …
Mathematics Subject Classification ID: 37C80
Mathematics Subject Classification ID: 37G40
Mathematics Subject Classification ID: 37G05
zbMATH DE Number: 1389123
zbMATH Keywords: reversible equivariant linear system
zbMATH Keywords: symmetry
zbMATH Keywords: linear time-reversal symmetry
zbMATH Keywords: irreducible representation of compact Lie group
zbMATH Keywords: local bifurcation
zbMATH Keywords: normal form
reviewed by: Nikolaj M. Glazunov
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3632
OpenAlex ID: W2041926611
cites work: Q5606652
cites work: Q3704110
cites work: Conjugacy classes in linear groups
cites work: Q5675842
cites work: Q4849844
cites work: Time-Reversibility and Particle Sedimentation
cites work: Singularities and groups in bifurcation theory. Volume II
cites work: Versal deformations and normal forms for reversible and Hamiltonian linear systems
cites work: Normal forms and unfoldings of linear systems in eigenspaces of (anti)-automorphisms of order two
cites work: Reversing symmetries in dynamical systems
cites work: Local bifurcations on the plane with reversing point group symmetry
cites work: Stability of Equilibria for a Class of Time-Reversible, $D_n \times O(2)$-Symmetric Homogeneous Vector Fields
cites work: Versal unfoldings of equivariant linear Hamiltonian vector fields
cites work: Normal forms for linear Hamiltonian vector fields commuting with the action of a compact Lie group
cites work: Periodic solutions near equilibria of symmetric Hamiltonian systems
cites work: Q5579608
cites work: Q4126536
cites work: Q4861425
cites work: Reversible systems
cites work: Q4146925
DOI: 10.1006/JDEQ.1999.3632
Recommended article: Reversible-equivariant systems and matricial equations
Recommended article: Q3363805
Recommended article: Smooth equivalence and linearization of reversible systems
Recommended article: Reversible linear reversible systems and their versal deformations
Recommended article: Reversible subsets of certain linear transformations
Recommended article: Reversible linear differential equations
Recommended article: Equivalences of Linear Functional Systems
Recommended article: Transformations of linear systems under external equivalence
Recommended article: Underlying algebraic framework of equivalence relations on linear systems
Recommended article: Versal deformations and normal forms for reversible and Hamiltonian linear systems
description: scientific article; zbMATH DE number 1389123