论文标题
离散的几何奇异扰动理论
Discrete Geometric Singular Perturbation Theory
论文作者
论文摘要
我们为由地图引起的离散多尺度动力系统提出了一种数学形式主义，该系统与已建立的连续时间快速慢系统的几何几何奇异扰动理论相似。我们确定与地图下“快速”和“慢”迭代相对应的限制地图。正常双曲线的概念是由沿关键的固定点歧管$ s $的快速限制图的乘数的光谱差距要求定义的。我们通过重新制定现有的结果来提供一组类似fenichel的扰动定理，以便它们将其应用于紧凑的，通常是夸张的$ s $ $ s $。临界流形$ s $，本地稳定/不稳定的歧管的持久性详细描述了稳定/不稳定纤维的$ W^{s/u} _ {s/u} _ {s/u} _ {s/u} _ {s/u} _ {s/u} _ {s/u}的叶子的持久性。在应用中证明了所得离散的几何奇异扰动理论（DGSPT）的实际实用性。首先，我们使用DGSPT来识别与基于地图的神经模型中兴奋性，松弛，混乱和非骨质爆发相对应的奇异几何形状。其次，我们得出结果，将快速慢速ODE的几何形状和动力学与非平凡的时间尺度分离及其Euler-Distretized对应物相关联。最后，我们表明，具有快速旋转的快速慢性ODE系统产生了快速慢的庞加莱地图，可以使用DGSPT详细描述其几何形状和动力学。
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.
