Tegan Morrison, School of Mathematics and Statistics, University of Sydney I will discuss results of an asymptotic study of a second Painlevé hierarchy in the limit as the independent variable approaches infinity. The hierarchy is defined by an infinite sequence of non-linear ordinary differential equations, indexed by order, with the classical second Painlevé equation as the first member. Particular attention will be given to the fourth-order analogue of the classical second Painlevé equation. In this case, the general asymptotic behaviour is given to leading-order by two related genus-2 hyperelliptic functions. The fourth-order equation also admits two classes of special asymptotic behaviours which are described by algebraic formal power series. This work builds upon the foundations of established asymptotic results for the classical second Painlevé equation and so I will begin with a survey of these results.