Last week I traveled to Boston to attend the American Physical Society Division of Fluid Mechanics annual meeting. It was, as usual, a great experience – it’s always a pleasure to see what new things people have thought of since we last talked.
My talk was on the first day, at 8.52a, which made it difficult for some to attend. The slides to the presentation are below, along with updated references. Many thanks to Nick Ouellette whose laptop saved my talk from the claws of Macbook freeze.
In addition, I managed to visit a couple of friends in the area, and saw what MERL and MIT ENDLab look like from inside. Thank you Piyush and Margaux for hosting me.
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.
I am spending this week at my all-time favorite SIAM DS Snowbird meeting. As this is my 5th straight DS meeting, it really feels like coming home. I’ll be speaking about my work with Jean-Luc Thiffeault on Finite-Time Braiding Exponent (arXiv) on Wednesday at 3p (in Ballroom 2). My talk is in the MS92: Topological Fluid and Mass Dynamics, organized by Stefanella Boatto and Mark Stremler. The slides of the presentation are embedded below. Additionally, I am co-organizing a two-session minisymposium (MS111 and MS124) with Jean-Luc Thiffeault and Sanjeeva Balasuriya on Thursday. Come by and see what our speakers have done on control of fluids and things inside fluids.
Update: The video of the talk is available on SIAM website.
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.
The latest full release of braidlab is out! Braidlab is a MATLAB toolbox which incorporates algorithms for analyzing braid groups of punctured disks in both theoretical and applied contexts. It was primarily written by Jean-Luc Thiffeault, but Michael Allshouse and I have contributed code to it as well. Feel free to direct your questions either at JLT or at me.
You can find a good summary of release updates on the main release-3.0 announcement, but here’s my list of favorites:
- We moved the repository to GitHub. This means that you can (and should) use our Issue Tracker to let us know what went wrong or what you would like to see included in future releases. We are also present on MATLAB Central.
- Installation from source should now work on Matlab 2014b without any special configuration. There are two known installation issues which are out of our reach: conflict of “mex” command with a LaTeX command, and lack of GMP libraries on your system. Make sure you read the installation guidelines in the manual first to see how to resolve these (and other) known problems.
- “Data braids” now have a broader support. This type of braids is useful if you are trying to represent physical trajectories, which have an independent variable, e.g., time, attached to them.
- More functions are implemented as MEX C++ code, which means that they ultimately run faster (some of them have even been parallelized!)
To install, go to the release page, scroll down, download the pre-packaged binaries. If you want to build from source, you can either download the source or even clone our git repository to stay up to date with the latest developments.
In all cases: let us know if braidlab works on your end and if you find it useful.
Yesterday, I have given a presentation at AIMS Conference in Madrid, Spain. Braid Dynamics of Non-Periodic Trajectories [pdf]. Click on “Read more” in this post for the abstract.