I’m happy to report that AIP Chaos published our paper on Finite-Time Braiding Exponents (co-authored with Jean-Luc Thiffeault):
MB. and Thiffeault, J.-L. Finite-time braiding exponents. Chaos: An Interdisciplinary Journal of Nonlinear Science 25, 087407 (2015).
In this paper, we use braid group generators to represent the entangling of trajectories of a dynamical system, and compute the Finite-Time Braiding Exponent (FTBE). FTBE should represent a finite-time, finite-information version of the topological entropy of the flow, and we show evidence that this is really the case.
Mean FTBE and top. entropy correlate, and almost-match as number of strands is increased (Fig. 10 from the paper)
In addition to publishing a paper, we have also released the new version of our MATLAB toolbox braidlab (v.3.2) which was used for all computations in the paper. You can see the release notes on our GitHub repository, as well as download source and compiled versions of the toolbox. Please let us know if you’re using braidlab and, especially, submit any problems with it to our issues page.
Jean-Luc Thiffeault and I have just uploaded our paper on Finite-Time Braiding Exponents (FTBE) to arXiv. In the paper we study how closely topological entropy of a mixing dynamical system can be approximated by sampling only finitely many trajectories from the flow.
Based on numerical results (obtained using braidlab 3.1), we find that FTBEs of finitely-many trajectories converge to topological entropy as number of trajectories is increased. The paper further explores robustness and dependence of FTBEs on time step, number of trajectories, and their length.
UPDATE (2012-12-21): The paper [link] has appeared in AIP Chaos under DOI 10.1063/1.4772195 with one of images making the front page (shown).
UPDATE (2012-12-03): The final draft (arXiv:1206.3164) has been accepted by AIP Chaos and is due to be published in Dec 2012, in Focus issue on 50 years of chaos.
UPDATE (2012-10-14): The revised version of the manuscript is available at arXiv now.
(2012-06-14) A review paper on applications of Koopman operator analysis to dynamical systems, co-authored with Ryan Mohr and Igor Mezić has been submitted for publication to AIP Chaos. The preprint is available on arXiv: “Applied Koopmanism“.
2012-03-16: Dr. Mihai Putinar and I have completed a paper on entropy optimization of singular measures. The pre-print is available on arXiv:
Conditioning moments of singular measures for entropy optimization. I
2012-11-05 UPDATE: The paper has been officially published by Indagationes Mathematicae: DOI: 10.1016/j.indag.2012.05.008
Dr. Igor Mezić and I have just submitted a revised version of the paper to Physica D. The pre-print is available on arXiv:
Geometry of the ergodic quotient reveals coherent structures in flows
UPDATE: The paper is now published at DOI: 10.1016/j.physd.2012.04.006.