Center or Limit Cycle: Renormalization Group as a Probe

TL;DR

This paper introduces a unified renormalization group-based method to identify and classify periodic solutions, such as centers and limit cycles, in various nonlinear dynamical systems, surpassing traditional linear stability analysis.

Contribution

It presents a novel, unified approach using renormalization group theory to detect and differentiate between centers and limit cycles in diverse nonlinear systems.

Findings

Effective classification of centers and limit cycles

Applicable across multiple types of nonlinear systems

Outperforms linear stability analysis in certain cases

Abstract

Based on our studies done on two-dimensional autonomous systems, forced non-autonomous systems and time-delayed 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.