Non-standard discretizations of a Hamiltonian system related to the fourth Painlevé equation

We consider non-standard discretizations of a cubic, non-autonomous Hamiltonian system, studied by the authors in an earlier article, related to the fourth Painlevé equation (a class of integrable, second-order non-linear, non-autonomous differential equations). Discretizing a differential system is a non-unique process and we employ different methods such as the (symmetric) Kahan method and a more general class of discretizations by introducing further parameters to the equations, with the aim of reducing the algebraic entropy of the system, which is a measure of complexity of the solutions of the system. Integrability of discrete systems is usually associated with zero algebraic entropy. Using the concept of singularity confinement, we can identify the cases which have reduced, though non-zero, algebraic entropy as compared with generic discretization.