  In this paper, we find all possible asymptotic behaviours of the solutions of
the second Painlev\'e equation $y''=2y^3+xy +\alpha$ as the parameter
$\alpha\to\infty$ in the local region $x\ll\alpha^{2/3}$. We prove that these
are asymptotic behaviours by finding explicit error bounds. Moreover, we show
that they are connected and complete in the sense that they correspond to all
possible values of initial data given at a point in the local region.
