Normal forms of C∞ vector fields based on the renormalization group

The normal form theory for polynomial vector fields is extended to those for C-infinity vector fields vanishing at the origin. Explicit formulas for the C-infinity normal form and the near identity transformation that 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-infinity 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.