离散几何奇异扰动理论
Discrete Geometric Singular Perturbation Theory
摘要：
我们提出了一种映射诱导的离散多尺度动力系统的数学形式，它与已建立的连续时间快慢系统的几何奇异扰动理论相平行。我们确定限制映射对应的“快”和“慢”迭代下的地图。通过对临界不动点流形$S$上的快速极限映射的乘子的谱隙要求，定义了正规双曲的概念.我们提供了一组Fenichel的扰动定理，通过重新制定预先存在的结果，使他们适用于附近的紧，正常双曲子流形的$S$。详细讨论了临界流形$S$、局部稳定/非稳定流形$W^{s/u}_{loc}(S)$和$W^{s/u}_{loc}(S)$的叶状结构通过稳定/不稳定纤维的持久性.所得到的离散几何奇异扰动理论（DGSPT）的实际效用在应用中被证明。首先，我们使用DGSPT识别奇异几何对应的兴奋性，松弛，混沌和非混沌突发在基于地图的神经模型。其次，我们得出的结果与非平凡的时间尺度分离和欧拉离散对应的几何和动力学的快，慢常微分方程。最后，我们证明了具有快速旋转的快-慢ODE系统产生了快-慢Poincaré映射，其几何和动力学可以使用DGSPT详细描述。
英文摘要：
We propose a mathematical formalism for discrete multi-scale dynamical systems induced by maps which parallels the established geometric singular perturbation theory for continuous-time fast-slow systems. We identify limiting maps corresponding to both 'fast' and 'slow' iteration under the map. A notion of normal hyperbolicity is defined by a spectral gap requirement for the multipliers of the fast limiting map along a critical fixed-point manifold $S$. We provide a set of Fenichel-like perturbation theorems by reformulating pre-existing results so that they apply near compact, normally hyperbolic submanifolds of $S$. The persistence of the critical manifold $S$, local stable/unstable manifolds $W^{s/u}_{loc}(S)$ and foliations of $W^{s/u}_{loc}(S)$ by stable/unstable fibers is described in detail. The practical utility of the resulting discrete geometric singular perturbation theory (DGSPT) is demonstrated in applications. First, we use DGSPT to identify singular geometry corresponding to excitability, relaxation, chaotic and non-chaotic bursting in a map-based neural model. Second, we derive results which relate the geometry and dynamics of fast-slow ODEs with non-trivial time-scale separation and their Euler-discretized counterpart. Finally, we show that fast-slow ODE systems with fast rotation give rise to fast-slow Poincaré maps, the geometry and dynamics of which can be described in detail using DGSPT.
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