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

TL;DR

This paper applies the renormalization group method to asymptotic analysis of vector fields, demonstrating its effectiveness in reducing complex dynamical systems and deriving simplified equations like Ginzburg-Landau.

Contribution

It extends the RG method to vector fields using envelope theory, providing a unified approach for analyzing bifurcations and reducing dynamics in complex systems.

Findings

Derivation of Landau-Stuart and Ginzburg-Landau equations from vector field analysis.

Explicit solution for the Lotka-Volterra system's time evolution.

Construction of Lorenz system's center manifolds using RG method.

Abstract

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.