Normal form theory for reversible equivariant vector fields

TL;DR

This paper introduces a new approach to derive normal forms for reversible equivariant vector fields by integrating invariant theory, addressing both non-resonant and resonant cases with specific symmetry properties.

Contribution

It presents an alternative method using invariant theory to compute normal forms for reversible equivariant vector fields, including explicit formulas for non-resonant and resonant cases.

Findings

Normal forms derived for non-resonant cases with Z_2 symmetry.

Normal forms derived for resonant cases with Z_2 symmetry.

Method effectively incorporates symmetries into the normal form computation.

Abstract

We give an alternative method to obtain normal forms of reversible equivariant vector fields. We adapt the classical method using tools from invariant theory to establish formulae that take symmetries into account as a starting point. Normal forms of two classes of non-resonant and resonant cases are presented, both under a Z_2-action with linearization having a 2-dimensional nilpotent part and a semisimple part with purely imaginary eigenvalues.