Category Archives: Dynamical systems

FTBE paper published in Chaos and braidlab v.3.2 released

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.

A graph from the AIP Chaos paper on FTBE.

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.

SIAM DS15 Snowbird meeting

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.

Two mode decompositions: POD and DMD

This is a short overview of the Proper Orthogonal and Dynamic/Koopman Mode Decompositions, which are commonly used in analysis of velocity fields of fluid flows. While I worked with the theoretical side of Koopman modes, I never implemented the numerical code myself; I wrote these notes up a I was teaching myself the basics of numerics of these decompositions, and consequently used the notes for two lectures. The notes are based References at the end of the post. Caveat lector: Notes may contain gross oversimplifications — the emphasis was on understanding and not on precision. I welcome your corrections and comments below. (You can always stop by my Van Vleck office if you’re in Madison to discuss any part of this).

UPDATE: I have now posted my own implementation of several algorithms for Koopman mode decompositions. [GitHub]

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Madison Math Circle: Mathematics of epidemics

Last night I had an amazing time at Madison Math Circle, hanging out with a bunch of bright school kids, who were undaunted by either diseases or mathematics. Since I had quite a few questions after the lecture, I decided to post an overview so anyone interested can go over it themselves. For those of you who attended and have further questions, or even wish to explore mathematical dynamics on your own, feel free to leave a comment below or e-mail me at and I’ll try to answer all your questions and provide some guidance for future.
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Ergodic quotient in a finite-time, expanding context

I am currently attending 10th AIMS Conference on dynamics in Madrid, where I participated in the session on transport barriers in unsteady fluid flows. One of the talks in the session suggested that the ergodic quotient techniques for detecting coherent sets would not apply to finite-time setting, i.e., when ergodic averages are replaced by finite-time averages. This is, of course, of interest in cases when ergodic averages do not exist, so finite-time average is a potential resolution of this problem. I decided to investigate the potentially-problematic example given (2d linear saddle) using the code I made available at [github]. I found that in that case, ergodic quotient algorithm still works, even away from the asymptotic averaging limit.
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Earth Wind/Ocean Map

This page , created by Cameron Baccario, hosts a beautiful visualization of atmospheric and oceanic currents in close-to-real time. Go to the “About” page to see the explanation of what to look for at each pressure levels, what types of projections you could use, and what other data, like temperature and precipitation, you can overlay to help your understanding of Earth’s processes.

Basic readings in ergodic theory

For those of you who would like to read more about the basics of applied ergodic theory that I talked about in 703 today, there are a few accessible papers and books below the fold. You are always very welcome to send me an e-mail or drop by my office (VV517) if you want to talk more about any of the papers or the topic in general.

Please e-mail me if any of the links below the fold don’t work, or if you have trouble locating the pdfs for the papers.

Here’s a simple demonstration of Arnold’s Cat Map. V.I. Arnold, a cool cat himself, figured out, decades before the age of the internet, that putting an image of a kitteh in your research vastly increases its appeal.

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