School of Natural Sciences, University of California, Merced, Merced, California 95343, USA.
Department of Physics, Mount Holyoke College, South Hadley, Massachusetts 01075, USA.
Chaos. 2019 Jan;29(1):013124. doi: 10.1063/1.5045060.
Topological entropy measures the number of distinguishable orbits in a dynamical system, thereby quantifying the complexity of chaotic dynamics. One approach to computing topological entropy in a two-dimensional space is to analyze the collective motion of an ensemble of system trajectories taking into account how trajectories "braid" around one another. In this spirit, we introduce the Ensemble-based Topological Entropy Calculation, or E-tec, a method to derive a lower-bound on topological entropy of two-dimensional systems by considering the evolution of a "rubber band" (piece-wise linear curve) wrapped around the data points and evolving with their trajectories. The topological entropy is bounded below by the exponential growth rate of this band. We use tools from computational geometry to track the evolution of the rubber band as data points strike and deform it. Because we maintain information about the configuration of trajectories with respect to one another, updating the band configuration is performed locally, which allows E-tec to be more computationally efficient than some competing methods. In this work, we validate and illustrate many features of E-tec on a chaotic lid-driven cavity flow. In particular, we demonstrate convergence of E-tec's approximation with respect to both the number of trajectories (ensemble size) and the duration of trajectories in time.
拓扑熵衡量动力系统中可区分轨道的数量，从而量化混沌动力学的复杂性。在二维空间中计算拓扑熵的一种方法是分析系统轨迹的集体运动，同时考虑轨迹之间的“交织”方式。本着这种精神，我们引入了基于集合的拓扑熵计算（E-tec）方法，该方法通过考虑围绕数据点缠绕并随其轨迹演化的“橡皮筋”（分段线性曲线）来推导出二维系统拓扑熵的下界。拓扑熵受该带的指数增长率的限制。我们使用计算几何的工具来跟踪橡皮筋的演化，因为数据点会撞击和变形它。由于我们维护了关于轨迹彼此之间配置的信息，因此更新带的配置是局部进行的，这使得 E-tec 比一些竞争方法更具计算效率。在这项工作中，我们在混沌驱动腔流上验证和说明了 E-tec 的许多特性。特别是，我们证明了 E-tec 逼近的收敛性，这与轨迹的数量（集合大小）和轨迹在时间上的持续时间都有关系。
