The normal form theory for polynomial vector fields is extended to those for C∞ vector fields vanishing at the origin. Explicit formulas for the C∞ 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∞ 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.
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