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arXiv · 2009.12771

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

Hayato Chiba
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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.

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Hayato Chiba. 2020-09-27. Normal Forms of $C^\infty$ Vector Fields based on the Renormalization Group. https://doi.org/10.1063/5.0031043

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