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arXiv · 1912.10286

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

Maximilian Engel·Hildeberto Jardón-Kojakhmetov
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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.

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Maximilian Engel, Hildeberto Jardón-Kojakhmetov. 2019-12-21. Extended and symmetric loss of stability for canards in planar fast-slow maps. https://doi.org/10.1137/20m1313611

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