Finite-time braiding exponents

Chaos
Marko Budišić, Jean-Luc Thiffeault

Abstract

Topological entropy of a dynamical system is an upper bound for the sum of positive Lyapunov exponents; in practice, it is strongly indicative of the presence of mixing in a subset of the domain. Topological entropy can be computed by partition methods, by estimating the maximal growth rate of material lines or other material elements, or by counting the unstable periodic orbits of the flow. All these methods require detailed knowledge of the velocity field that is not always available, for example, when ocean flows are measured using a small number of floating sensors. We propose an alternative calculation, applicable to two-dimensional flows, that uses only a sparse set of flow trajectories as its input. To represent the sparse set of trajectories, we use braids, algebraic objects that record how trajectories exchange positions with respect to a projection axis. Material curves advected by the flow are represented as simplified loop coordinates. The exponential rate at which a braid stretches loops over a finite time interval is the Finite-Time Braiding Exponent (FTBE). We study FTBEs through numerical simulations of the Aref Blinking Vortex flow, as a representative of a general class of flows having a single invariant compo...Continue Reading

References

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Citations

Apr 3, 2016·Chaos·Sulimon SattariKevin A Mitchell
Dec 23, 2015·Scientific Reports·Nicolas FrancoisMichael Shats
Dec 3, 2016·Chaos·C K Taylor, Stefan G Llewellyn Smith
Mar 3, 2018·Chaos·Lei Fang, Nicholas T Ouellette
Mar 9, 2019·Scientific Reports·Tímea Haszpra, Mátyás Herein
Apr 3, 2020·Chaos·Melissa YeungMathieu Desbrun
Feb 3, 2019·Chaos·Eric RobertsKevin A Mitchell
Aug 22, 2021·Proceedings of the National Academy of Sciences of the United States of America·Jinghui LiuNikta Fakhri

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