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We study fast-slow maps obtained by discretization of planar fast-slow systems in continuous time. We focus on describing the so-called delayed loss of stability induced by the slow passage through a singularity in fast-slow systems. This delayed loss of stability can be related to the presence of canard solutions. Here we consider three types of singularities: transcritical, pitchfork, and fold. First, we show that under an explicit Runge--Kutta discretization the delay in loss of stability, due to slow passage through a transcritical or a pitchfork singularity, can be arbitrarily long. In contrast, we prove that under a Kahan--Hirota--Kimura discretization scheme, the delayed loss of stability related to all three singularities is completely symmetric in the linearized approximation, in perfect accordance with the continuous-time setting. [ABSTRACT FROM AUTHOR]

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 precise estimates for a cubic map in the central rescaling chart make a key technical contribution. [ABSTRACT FROM AUTHOR]