Canards in modified equations for Euler discretizations

TL;DR

This paper investigates how explicit Euler discretization affects canard phenomena in fast-slow ODEs, revealing differences in stability delays compared to continuous systems through modified equations analysis.

Contribution

It introduces a modified equations approach to analyze and quantify the effects of Euler discretization on canard dynamics, providing new insights into stabilization and destabilization phenomena.

Findings

Euler discretization shortens delay in loss of stability for folds

For transcritical singularities, delay can be arbitrarily prolonged

Modified equations accurately predict discrete-time canard behavior

Abstract

Canards are a well-studied phenomenon in fast-slow ordinary differential equations implying the delayed loss of stability after the slow passage through a singularity. Recent studies have shown that the corresponding maps stemming from explicit Runge-Kutta discretizations, in particular the forward Euler scheme, exhibit significant distinctions to the continuous-time behavior: for folds, the delay in loss of stability is typically shortened whereas, for transcritical singularities, it is arbitrarily prolonged. We employ the method of modified equations, which correspond with the fixed discretization schemes up to higher order, to understand and quantify these effects directly from a fast-slow ODE, yielding consistent results with the discrete-time behavior and opening a new perspective on the wide range of (de-)stabilization phenomena along canards.