Renormalization group in difference systems

TL;DR

This paper introduces a Lie symmetry-based singular perturbation method for difference equations, deriving a renormalization group equation that describes long-term behavior, including bifurcations, in symplectic maps.

Contribution

It presents a novel Lie symmetry-based approach to derive renormalization group equations for difference systems, linking them to Hamiltonian dynamics.

Findings

Renormalization group equations are derived using Lie symmetries.

The method applies to 2-D symplectic maps, revealing Hamiltonian structure.

Conditions for Poincaré-Birkhoff bifurcation are established.

Abstract

A new singular perturbation method based on the Lie symmetry group is presented to a system of difference equations. This method yields consistent derivation of a renormalization group equation which gives an asymptotic solution of the difference equation. The renormalization group equation is a Lie differential equation of a Lie group which leaves the system approximately invariant. For a 2-D symplectic map, the renormalization group equation becomes a Hamiltonian system and a long-time behaviour of the symplectic map is described by the Hamiltonian. We study the Poincar\'e-Birkoff bifurcation in the 2-D symplectic map by means of the Hamiltonian and give a condition for the bifurcation.