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

The one-parameter family of second order nonlinear difference equations each of which is given by $$x_{n-1}x_nx_{n+1}=x_{n-1}+(x_n)^{\beta-1}+x_{n+1} \qquad(\beta\in\mathbb{N})$$ is explored. Since the equation above is arising from seed mutations of a rank 2 cluster algebra, its solution is periodic only when $\beta\leq3$. In order to evaluate the dynamics with $\beta\geq4$, algebraic entropy of the birational map equivalent to the difference equation is investigated; it vanishes when $\beta=4$ but is positive when $\beta\geq5$. This fact suggests that the difference equation with $\beta\leq4$ is integrable but that with $\beta\geq5$ is not. It is moreover shown that the difference equation with $\beta\geq4$ fails the singularity confinement test. This fact is consistent with linearizability of the equation with $\beta=4$ and reinforces non-integrability of the equation with $\beta\geq5$.