SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2530-2566, January 2020.
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.

中文翻译：

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平面快慢图中Canards的扩展对称对称损失

SIAM应用动力系统杂志，第19卷，第4期，第2530-2566页，2020年1月。
我们研究通过在连续时间内离散平面快慢系统获得的快慢图。我们专注于描述快速慢系统中由于缓慢通过奇异点而引起的所谓的延迟延迟稳定性。这种延迟的稳定性损失可能与卡纳德溶液的存在有关。在这里，我们考虑三种奇异类型：跨临界，干草叉和折叠。首先，我们表明，在明确的Runge-Kutta离散化条件下，由于缓慢通过跨临界或干草叉奇异点而导致的稳定性损失的延迟可以任意长。相反，我们证明了在Kahan-Hirota-Kimura离散化方案下，与所有三个奇点有关的延迟稳定性损失在线性化近似中完全对称，完全符合连续时间设置。