  The renormalization group method of Goldenfeld, Oono and their collaborators
is applied to asymptotic analysis of vector fields. The method is formulated on
the basis of the theory of envelopes, as was done for scalar fields. This
formulation actually completes the discussion of the previous work for scalar
equations. It is shown in a generic way that the method applied to equations
with a bifurcation leads to the Landau-Stuart and the (time-dependent)
Ginzburg-Landau equations. It is confirmed that this method is actually a
powerful theory for the reduction of the dynamics as the reductive perturbation
method is. Some examples for ordinary diferential equations, such as the forced
Duffing, the Lotka-Volterra and the Lorenz equations, are worked out in this
method: The time evolution of the solution of the Lotka-Volterra equation is
explicitly given, while the center manifolds of the Lorenz equation are
constructed in a simple way in the RG method.
