Normal Forms of $C^\infty$ Vector Fields based on the Renormalization Group

TL;DR

This paper extends normal form theory to smooth vector fields using the renormalization group, providing explicit formulas and applications to invariant manifolds and periodic orbits.

Contribution

It introduces a $C^ abla$ normal form theory for smooth vector fields and applies the renormalization group to derive explicit formulas and analyze dynamics.

Findings

Explicit formulas for $C^ abla$ normal forms are derived.

The method proves the existence of infinitely many periodic orbits.

Normal forms reveal invariant manifolds and dynamic structures.

Abstract

The normal form theory for polynomial vector fields is extended to those for $C^\infty$ vector fields vanishing at the origin. Explicit formulas for the $C^\infty$ normal form and the near identity transformation which brings a vector field into its normal form are obtained by means of the renormalization group method. The dynamics of a given vector field such as the existence of invariant manifolds is investigated via its normal form. The $C^\infty$ normal form theory is applied to prove the existence of infinitely many periodic orbits of two dimensional systems which is not shown from polynomial normal forms.