Extended and symmetric loss of stability for canards in planar fast-slow maps

TL;DR

This paper investigates how different discretization schemes affect the delayed loss of stability in fast-slow dynamical systems near singularities, revealing that Kahan-Hirota-Kimura discretization preserves symmetry in the delay.

Contribution

It demonstrates that Kahan-Hirota-Kimura discretization maintains symmetry in the delayed loss of stability for various singularities, unlike Runge-Kutta methods.

Findings

Kahan-Hirota-Kimura discretization preserves symmetry in delay.

Runge-Kutta discretization can produce arbitrarily long delays.

Symmetry in delay aligns with continuous-time behavior.

Abstract

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.