Quantifying the tangling of trajectories using the topological entropy

Candelaresi, S. , Pontin, D.I. and Hornig, G. (2017) Quantifying the tangling of trajectories using the topological entropy. Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(9), 093102. (doi: 10.1063/1.5000812)

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.

• Item Type: Articles
• Additional Information: All the authors acknowledge financial support from the UK’s STFC (Grant No. ST/K000993).
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Candelaresi, Dr Simon
• Authors: Candelaresi, S., Pontin, D.I., and Hornig, G.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Chaos: An Interdisciplinary Journal of Nonlinear Science
• Publisher: AIP Publishing
• ISSN: 1054-1500
• ISSN (Online): 1089-7682
• Published Online: 05 September 2017
• Copyright Holders: Copyright © 2017 The Authors
• First Published: First published in Chaos: An Interdisciplinary Journal of Nonlinear Science 27(9): 093102
• Publisher Policy: Reproduced under a Creative Commons License