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

Journal of Mathematical Physics , Volume 62; doi:10.1063/5.0031043

Abstract: 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.
Keywords: normal form theory / vector fields / renormalization / brings / manifolds / Explicit / infinitely / vanishing / invariant
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