基于集成的拓扑熵计算（E-tec）
Ensemble-based Topological Entropy Calculation (E-tec)
摘要：
拓扑熵测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合的集体运动，同时考虑轨迹如何相互“编织”。本着这种精神，我们引入了基于集合的拓扑熵计算（E-tec），这是一种通过考虑“橡皮筋”（分段线性曲线）的演变来推导二维系统拓扑熵下限的方法。缠绕在数据点周围并随着其轨迹而演变。拓扑熵在下面被这个带的指数增长率所限制。我们使用计算几何的工具来跟踪数据点撞击和变形时橡皮筋的演变。因为我们保持了关于轨迹相对于彼此的配置的信息，所以更新带配置是在本地执行的，这使得E-tec比一些竞争方法更具计算效率。在这项工作中，我们验证和说明了许多功能的E-TEC上的混沌盖驱动空腔流。特别是，我们证明了收敛的E-TEC的近似相对于轨迹的数量（合奏大小）和轨迹的持续时间。
英文摘要：
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.
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