C^1 Approximation of Vector Fields on the Renormalization Group Method and its Applications
(繰り込み群によるベクトル場の C^1 近似とその応用)

The renormalization group (RG) method for differential equations is
one of the perturbation methods for obtaining solutions which are
approximate to exact solutions for a long time interval.  In this
work, it is shown that for a given vector field on a manifold and for
the differential equation associated with it, a family of approximate
solutions obtained on the RG method defines a vector field which is
close to the original vector field in C^1 topology under appropriate
assumptions.  Furthermore, some topological properties of the original
vector field such as the existence of an invariant manifold and its
stability are inherited from those for the RG equation.  This fact is
applied to show the presence of the Hopf bifurcation and other global
bifurcation.  The present RG method is also useful for detecting an
approximate center manifold and a flow thereon. This method is viewed
as strict mathematical formulation of the reductive perturbation
method proposed by Kuramoto, and includes the geometric singular
perturbation method proposed by Fenichel as a particular case.  The RG
method applied to time dependent linear equations shows that the
stability of the trivial solutions are inherited from those of the RG
equations. This fact results in the presence of synchronization in
coupled systems.