ABSTRACT In this work we consider two-dimensional critical manifolds in planar fast-slow systems near fold and so-called canard (‘duck’) points. These higher-dimension, and lower-codimension, situation is directly motivated by the case of hysteresis operators limiting onto fast-slow systems as well as by systems with constraints. We use geometric desingularization via blow-up to investigate two situations for the slow flow: generic fold (or jump) points, and canards in one-parameter families. We directly prove that the fold case is analogous to the classical fold involving a one-dimensional critical manifold. However, for the canard case, considerable differences and difficulties appear. Orbits can get trapped in the two-dimensional manifold after a canard-like passage thereby preventing small-amplitude oscillations generated by the singular Hopf bifurcation occurring in the classical canard case, as well as certain jump escapes.

中文翻译：

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Duck Traps：平面系统中的二维临界流形

摘要 在这项工作中，我们考虑了靠近折叠和所谓鸭（'鸭'）点的平面快-慢系统中的二维临界流形。这些较高维和较低维的情况是由滞后算子限制到快慢系统以及具有约束的系统的情况直接激发的。我们通过爆炸使用几何去奇异化来研究慢流的两种情况：通用折叠（或跳跃）点和单参数族中的鸭翼。我们直接证明折叠情况类似于涉及一维临界流形的经典折叠。然而，对于鸭子案，出现了相当大的差异和困难。