$C^1$ Approximation of Vector Fields Based on the Renormalization Group Method

The renormalization group (RG) method for differential equations is one of the perturbation methods for obtaining solutions which approximate exact solutions for a long time interval. This article shows that, for a differential equation associated with a given vector field on a manifold, a family of approximate solutions obtained by the RG method defines a vector field which is close to the original vector field in the $C^1$ topology under appropriate assumptions. Furthermore, some topological properties of the original vector field, such as the existence of a normally hyperbolic invariant manifold and its stability, are shown to be inherited from those of the RG equation. This fact is applied to the bifurcation theory.