We give a comprehensive review of the renormalization group method for global
and asymptotic analysis, putting an emphasis on the relevance to the classical
theory of envelopes and the existence of invariant manifolds of the dynamics
under consideration. We clarify that an essential point of the method is to
convert the problem from solving differential equations to obtaining suitable
initial (or boundary) conditions.
We mention that the notion of envelopes is also useful for constructing
global and asymptotic behavior of wave functions of quantum systems such as the
ones with the quartic potential or double-well potential.Comment: Talk presented at RIMS (Kyoto) Workshop on Geometrical Methods for
Asymptotic Analysis 1997.5.20 -- 5.23. LaTex, 15 page
