Langfield Peter, Krauskopf Bernd, Osinga Hinke M
Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand.
Chaos. 2014 Mar;24(1):013131. doi: 10.1063/1.4867877.
We consider the FitzHugh-Nagumo model, an example of a system with two time scales for which Winfree was unable to determine the overall structure of the isochrons. An isochron is the set of all points in the basin of an attracting periodic orbit that converge to this periodic orbit with the same asymptotic phase. We compute the isochrons as one-dimensional parametrised curves with a method based on the continuation of suitable two-point boundary value problems. This allows us to present in detail the geometry of how the basin of attraction is foliated by isochrons. They exhibit extreme sensitivity and feature sharp turns, which is why Winfree had difficulties finding them. We observe that the sharp turns and sensitivity of the isochrons are associated with the slow-fast nature of the FitzHugh-Nagumo system; more specifically, it occurs near its repelling (unstable) slow manifold.
我们考虑菲茨休 - 纳古莫模型，它是一个具有两个时间尺度的系统的示例，对于该系统，温弗里无法确定等时线的整体结构。等时线是吸引周期轨道的盆地中所有点的集合，这些点以相同的渐近相位收敛到该周期轨道。我们使用基于合适的两点边值问题延续的方法，将等时线计算为一维参数化曲线。这使我们能够详细展示吸引盆如何由等时线叶状分布的几何结构。它们表现出极高的敏感性并具有急转弯，这就是温弗里难以找到它们的原因。我们观察到等时线的急转弯和敏感性与菲茨休 - 纳古莫系统的快慢性质有关；更具体地说，它发生在其排斥（不稳定）慢流形附近。
