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

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.

References in corpus (2)

• Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method
• Stability of an [N/2]-dimensional invariant torus in the Kuramoto model at small coupling

Cited by in corpus (1)

• Dynamical renormalization and universality in classical multifield cosmological models