# Discretized Fast-Slow Systems near Pitchfork Singularities

## TL;DR

This paper analyzes how discretized fast-slow systems behave near pitchfork singularities, showing the extension of the slow manifold and estimating transition contraction rates using a discrete blow-up method.

## Contribution

It adapts the blow-up technique to discrete systems and provides precise estimates for the dynamics near singularities, advancing understanding of numerical discretizations of fast-slow systems.

## Key findings

- Slow manifold extends beyond the singularity in discretized systems.
- Transition mapping exhibits quantifiable contraction rates.
- The discrete blow-up method effectively analyzes singular behavior.

## Abstract

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.