Renormalization Group as a Probe for Dynamical Systems

Abstract

The use of renormalization group (RG) in the analysis of nonlinear dynamical problems has been pioneered by Goldenfeld and co-workers [1]. We show that perturbative renormalization group theory of Chen et al can be used as an effective tool for asymptotic analysis for various nonlinear dynamical oscillators. Based on our studies [2] done on two-dimensional autonomous systems, as well as forced non-autonomous systems, we propose a unified methodology – that uses renormalization group theory – for finding out existence of periodic solutions in a plethora of nonlinear dynamical systems appearing across disciplines. The technique will be shown to have a non-trivial ability of classifying the solutions into limit cycles and periodic orbits surrounding a center. Moreover, the methodology has a definite advantage over linear stability analysis in analyzing centers.