# Quantifying the Tangling of Trajectories Using the Topological Entropy

## TL;DR

This paper introduces an efficient method to estimate the lower bounds of topological entropy in 2D mappings, enabling detailed analysis of flow complexity and mixing efficiency with reduced computational costs.

## Contribution

The authors develop an adaptive, cost-effective technique to compute topological entropy and its spatial distribution in 2D flows, improving analysis of flow mixing and braiding.

## Key findings

- Method accurately estimates topological entropy in complex flows.
- Adaptive resolution reduces computational costs significantly.
- Variation in entropy reveals spatial differences in mixing efficiency.

## Abstract

We present a simple method to efficiently compute a lower limit of the topological entropy and its spatial distribution for two-dimensional mappings. These mappings could represent either two-dimensional time-periodic fluid flows or three-dimensional magnetic fields, which are periodic in one direction. This method is based on measuring the length of a material line in the flow. Depending on the nature of the flow, the fluid can be mixed very efficiently which causes the line to stretch. Here we study a method that adaptively increases the resolution at locations along the line where folds lead to high curvature. This reduces the computational cost greatly which allows us to study unprecedented parameter regimes. We demonstrate how this efficient implementation allows the computation of the variation of the finite-time topological entropy in the mapping. This measure quantifies spatial variations of the braiding efficiency, important in many practical applications.