Reversible equivariant linear systems
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 Solitons Fractals 5, No. 2, 271-293 (1995; Zbl 0843.58097)]. Along with \(\rho\) the authors consider the representation of \(\sigma \rho\) and call it the \(\sigma\)-dual of \(\rho.\) If \(\sigma \rho\) is isomorphic to \( \rho\) as a representation of \(G\) the authors say that \(\rho\) is \(\sigma\)-self-dual. Both cases \(\sigma\)-self-duality and not \(\sigma\)-self-duality are considered. The purely reversible normal form theory over \({\mathbb R}\) has been developed by \textit{I. Hoveijn} [J. Differ. Equ. 126, No. 2, 408-442 (1996; Zbl 0855.34043)]. Section 2 contains a summary of results the authors need from real linear algebra and representation theory. Detailed statements, proofs and examples that illustrate some of the statements are given in Section 3.
irreducible representation of compact Lie group
scientific article; zbMATH DE number 1389123
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