鸭翼和曲率：非光滑pinching逼近
Canards and curvature: nonsmooth approximation by pinching
摘要：
在多时间尺度（奇摄动）动力系统中，鸭式是沿着吸引和排斥不变流形演化的违反直觉的解。在二维空间中，鸭翼会导致周期性振荡，其振幅和周期以高度非线性的方式增长：它们相对于控制参数缓慢变化，除了指数小的值域之外，它们增长得非常快。这种突然的增长，被称为鸭式爆炸，已经在许多应用中遇到，从化学到神经动力学，航空航天工程和生态学。鸭翼最初是用非标准分析来研究的，后来用匹配渐近、不变流形理论和参数blow-up等标准技术证明了相同的结果。最近，鸭式行为已经与分段光滑动力系统中的不连续表面联系起来。本文提供了一个新的视角，鸭式飞机爆炸的非标准分析，可以重新铸造到分段光滑动力系统的框架。将指数坐标标度应用于奇摄动常微分方程系统。缩放就像一个透镜，解决了所有时间尺度上的动态问题。然后分析了局部曲率的变化对鸭式布局爆炸的影响。不同的时间尺度占主导地位的区域被超曲面分开，这些被捏在一起，以获得一个分段光滑的系统，其中曲率的变化表现为不连续引起的分叉。该方法被用来分类鸭在任意尺寸，并得出的参数值，鸭形成小周期（鸭头）或大周期（鸭头）。
英文摘要：
In multiple time-scale (singularly perturbed) dynamical systems, canards are counterintuitive solutions that evolve along both attracting and repelling invariant manifolds. In two dimensions, canards result in periodic oscillations whose amplitude and period grow in a highly nonlinear way: they are slowly varying with respect to a control parameter, except for an exponentially small range of values where they grow extremely rapidly. This sudden growth, called a canard explosion, has been encountered in many applications ranging from chemistry to neuronal dynamics, aerospace engineering and ecology. Canards were initially studied using nonstandard analysis, and later the same results were proved by standard techniques such as matched asymptotics, invariant manifold theory and parameter blow-up. More recently, canard-like behaviour has been linked to surfaces of discontinuity in piecewise-smooth dynamical systems. This paper provides a new perspective on the canard phenomenon by showing that the nonstandard analysis of canard explosions can be recast into the framework of piecewise-smooth dynamical systems. An exponential coordinate scaling is applied to a singularly perturbed system of ordinary differential equations. The scaling acts as a lens that resolves dynamics across all time-scales. The changes of local curvature that are responsible for canard explosions are then analysed. Regions where different time-scales dominate are separated by hypersurfaces, and these are pinched together to obtain a piecewise-smooth system, in which curvature changes manifest as discontinuity-induced bifurcations. The method is used to classify canards in arbitrary dimensions, and to derive the parameter values over which canards form either small cycles (canards without head) or large cycles (canards with head).
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