Ensemble-based Topological Entropy Calculation (E-tec)

TL;DR

E-tec is a computational method that estimates the lower bound of topological entropy in 2D dynamical systems by tracking a deforming rubber band around trajectories, offering efficiency and convergence validation.

Contribution

The paper introduces E-tec, a novel ensemble-based approach using computational geometry to efficiently estimate topological entropy in 2D systems.

Findings

E-tec provides a lower bound on topological entropy.

E-tec converges with increasing ensemble size and trajectory duration.

E-tec is more computationally efficient than some existing methods.

Abstract

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.