The Renormalization-Group Method Applied to Asymptotic Analysis of Vector Fields

The renormalization group method of Goldenfeld, Oono and their collaboratorsis applied to asymptotic analysis of vector fields. The method is formulated onthe basis of the theory of envelopes, as was done for scalar fields. Thisformulation actually completes the discussion of the previous work for scalarequations. It is shown in a generic way that the method applied to equationswith a bifurcation leads to the Landau-Stuart and the (time-dependent)Ginzburg-Landau equations. It is confirmed that this method is actually apowerful theory for the reduction of the dynamics as the reductive perturbationmethod is. Some examples for ordinary diferential equations, such as the forcedDuffing, the Lotka-Volterra and the Lorenz equations, are worked out in thismethod: The time evolution of the solution of the Lotka-Volterra equation isexplicitly given, while the center manifolds of the Lorenz equation areconstructed in a simple way in the RG method.Comment: The revised version of RYUTHP 96/1. Submitted to Prog. Theor. Phys. (Kyoto) in Feb., 1996. 28 pages. LATEX. No figure