The renormalization-group method applied to asymptotic analysis of vector fields

Abstract

The renormalization group method of Goldenfeld, Oono and their collaborators is applied to the 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 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 (time-dependent) Ginzburg-Landau equations. It is confirmed that this method is actually a powerful theory for the reduction of dynamics as is the reductive perturbation method. Some examples for ordinary differential 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 given explicitly, while the center manifolds of the Lorenz equation are constructed in a simple way using the RG method.