Renormalization group in difference systems

Details Renormalization group in difference systems Renormalization group in difference systems

2008-01-21

arxiv.org/abs/0801.3156v3

Nonlinear Sciences

Chaotic Dynamics

Accepted to J. Phys. A, 7 pages

Scientific paper

10.1088/1751-8113/41/8/085204

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.

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