Reversible-equivariant systems and matricial equations

Marco A Teixeira,Ricardo M Martins

doi:10.1590/s0001-37652011000200003

Marco A Teixeira, Ricardo M Martins

Open Access

https://doi.org/10.1590/s0001-37652011000200003

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Abstract

This paper uses tools in group theory and symbolic computing to classify the representations of finite groups with order lower than, or equal to 9 that can be derived from the study of local reversible-equivariant vector fields in <img border=0 width=32 height=32 src="../../../../img/revistas/aabc/v83n2/carr.jpg" align=absmiddle>4 . The results are obtained by solving matricial equations. In particular, we exhibit the involutions used in a local study of reversible-equivariant vector fields. Based on such approach we present, for each element in this class, a simplified Belitskii normal form.

Highlights

The presence of involutory symmetries and involutory reversing symmetries is very common in physical systems, for example, in classical mechanics, quantum mechanics and thermodynamic
The work on differential equations with symmetries stay restricted to hamiltonian equations
We study some possible linearizations for symmetries and reversing symmetries, around a fixed point, and employ this to simplify normal forms for a class of vector fields

Summary

INTRODUCTION

The presence of involutory symmetries and involutory reversing symmetries is very common in physical systems, for example, in classical mechanics, quantum mechanics and thermodynamic (see Lamb and Roberts 1996). We study some possible linearizations for symmetries and reversing symmetries, around a fixed point, and employ this to simplify normal forms for a class of vector fields. Using tools from group theory and symbolic computing, we exhibit the involutions used in a local study of reversible-equivariant vector fields. Based on such approach we present, for each element in this class, a simplified Belitskii normal form. It is worth pointing out that properties of reversing symmetry groups are a powerful tool to study local bifurcation theory in presence of symmetries, see for instance Knus et al (1998). The authors are grateful to the referees for many helpful comments and suggestions

STATEMENT OF MAIN RESULTS

SETTING THE PROBLEM

PROOF OF THEOREM A

CASE D3

CASE D4

CONCLUSIONS