# Discretized Fast-Slow Systems with Canards in Two Dimensions

## TL;DR

This paper investigates how a specific discretization scheme, Kahan's method, preserves the delicate canard phenomena in two-dimensional fast-slow systems, ensuring the occurrence of maximal canards similar to continuous systems.

## Contribution

It demonstrates that Kahan's discretization preserves the structure of maximal canards in quadratic fast-slow systems, extending continuous-time results to discrete schemes.

## Key findings

- Kahan discretization preserves maximal canards in quadratic systems.
- The analysis uses blow-up and Melnikov methods for discrete maps.
- Maximal canards occur between attracting and repelling manifolds in the discretized setting.

## Abstract

We study the problem of preservation of maximal canards for time discretized fast-slow systems with canard fold points. In order to ensure such preservation, certain favorable structure preserving properties of the discretization scheme are required. Conventional schemes do not possess such properties. We perform a detailed analysis for an unconventional discretization scheme due to Kahan. The analysis uses the blow-up method to deal with the loss of normal hyperbolicity at the canard point. We show that the structure preserving properties of the Kahan discretization for quadratic vector fields imply a similar result as in continuous time, guaranteeing the occurrence of maximal canards between attracting and repelling slow manifolds upon variation of a bifurcation parameter. The proof is based on a Melnikov computation along an invariant separating curve, which organizes the dynamics of the map similarly to the ODE problem.