  Motivated by the normal form of a fast-slow ordinary differential equation
exhibiting a pitchfork singularity we consider the discrete-time dynamical
system that is obtained by an application of the explicit Euler method.
Tracking trajectories in the vicinity of the singularity we show, how the slow
manifold extends beyond the singular point and give an estimate on the
contraction rate of a transition mapping. The proof relies on the blow-up
method suitably adapted to the discrete setting where a key technical
contribution are precise estimates for a cubic map in the central rescaling
chart.
