A family of integrable and non-integrable difference equations arising from cluster algebras

Abstract

The one-parameter family of second order nonlinear difference equations each of which is given byis explored. Since the equation above is arising from seed mutations of a rank 2 cluster algebra, its solution is periodic only when β ≤ 3. In order to evaluate the dynamics with β ≥ 4, the algebraic entropy of the birational map equivalent to the difference equation is investigated; it vanishes when β = 4 but is positive when β ≥ 5. This fact suggests that the difference equation with β ≤ 4 is integrable but that with β ≥ 5 is not. It is moreover shown that the difference equation with β ≥ 4 fails the singularity confinement test. This fact is consistent with linearizability of the equation with β = 4 and reinforces non-integrability of the equation with β ≥ 5.