Normal form theory for reversible equivariant vector fields

Abstract

We give a method to obtain formal normal forms of reversible equivariant vector fields. The procedure we present is based on the classical method of normal forms combined with tools from invariant theory. Normal forms of two classes of resonant cases are presented, both with linearization having a 2-dimensional nilpotent part and a semisimple part with purely imaginary eigenvalues.

Normal forms of bireversible vector fields

“…In [4], a method is given to obtain formal normal forms of reversible equivariant vector fields under the action of Γ based on the classical method of normal forms combined with tools from invariant theory. More specifically, consider a system of ODEs…”

“…From the linearization L of X at the origin, consider the group S as defined in (1) which acts on V by matrix product. The algebraic method given in [4] consists of computing the truncated normal form of (3) at any degree via the computation of the generators of the module of homogeneous reversible equivariants under the group S ⋊ Γ : Theorem 4.7]) Let Γ be a compact Lie group acting linearly on V and consider X : V → V a smooth Γ−reversible-equivariant vector field, X(0) = 0 and L = (dX) 0 . Then (3) is formally conjugate tȯ…”

“…Finally, we re-apply the first step to Γ 1 ⋊ (Z 2 × Z 2 ) replacing Γ 1 in the first step by Γ 1 ⋊ Z 2 . It should be clear that we present this method as a simpler alternative to the algorithm given in [4] for the special class of semi-direct products. Some results hold for the more general case when the fist component Γ 1 of the semi-direct product is any compact Lie group, so we shall present these in this generality.…”

Complete transversals of symmetric vector fields

“…Elphick et al in [12] give an algebraic method for obtaining the normal form introducing an action of a group of symmetries S, namely (1) S = {e sL t , s ∈ R}, so that the polynomial nonlinear terms are equivariant under this action. In [4] we treat formal normal forms of smooth vector fields in the simultaneous presence of symmetric and reversing symmetric transformations. The algebraic treatment shows advantage at once, since the set Γ formed by such transformations has a group structure.…”

“…The normal form of a Γ-reversible-equivariant system inherits the symmetries and reversing symmetries if the changes of coordinates are equivariant under the group Γ. Belitskii normal form has been used by many authors in different aspects; for example, in the analysis of occurrence of limit cycles or families of periodic orbits either in purely reversible vector fields or in reversible equivariant ones (see [19] and references therein). Motivated by these works, in [4] we have established an algebraic result related to those by Belitskii [5] and Elphick [12] in the reversible equivariant context using tools from invariant theory. In this process we have proved that the normal form comes from the description of the reversible equivariant theory of the semidirect product S Γ.…”