Global Asymptotics of the Second Painlev\'e Equation in Okamoto's Space

We study the solutions of the second Painlev\'e equation in the space of initial conditions first constructed by Okamoto, in the limit as the independent variable, x, goes to infinity. Simultaneously, we study solutions of the related equation known as the thirty-fourth Painlev\'e equation. By considering degenerate cases of the autonomous flow, we recover the known special solutions, which are either rational functions or expressible in terms of Airy functions. We show that the solutions that do not vanish at infinity possess an infinite number of poles. An essential element of our construction is the proof that the union of exceptional lines is a repellor for the dynamics in Okamoto's space. Moreover, we show that the limit set of the solutions exists and is compact and connected.