Some Complicated Dynamical Systems

Originally Posted 01-22-2012

Dynamics Notes Updated

I spent some time today designing new tasks in my activity sequence for dynamical systems. The new tasks confront the students with several problems at once:

  • First, the systems have much more complicated behavior than the simple systems we have dealt with so far.
  • Second, they need to use the computer to do some numerical investigation.

The second item is a bit subtle. Of course, using a computer to model precise computations often leads to difficulties. But I have the students using Sage. Sage can often handle exact symbolic computations! I say often, because even when I figured out how to set up my orbit computations to be exact, at some point things got too nasty and it threw back an error message. So, the big message of “the computer might lie” is a bit muddled. If you know how to use it carefully, you can make Sage do exact computations, except when you can’t.

To get through these points, I have chosen the doubling map D: x \mapsto 2x \mod{1} and the tent map where T(x) = 2x if x \in [0,1/2] and T(x) = 2 - 2x if x \in [1/2, 1]. These are much more exciting than our previous systems, and they will help motivate most of the rest of the semester.

Also, I have included some rather open-ended and hopeful tasks asking the students to make up “graphical methods” for plotting orbits. I hope that we get out cobweb plots and phase portraits. I think we will, but the students could honestly not see where I am aiming and have no idea what to try. Fingers Crossed!

Other Achievements

I managed to do some other small bits of business today.

  • I did a bunch of organizational work for our Undergraduate Research Committee. (I am the chair.)

  • I read another chapter of Indra’s Pearls to prepare for a meeting with a student tomorrow.

  • I finished the update to my web page for the new semester. Since the pages only needed the slightest tweaks, this really came down to figuring out how to get the Sphinx Python library working on my computer and figuring out how to make ssh and sftp work on my new OS. (Turns out it is easy. It helps that I am old enough to have used a Unix machine on campus for my first email checker.)

  • I played Contre Jour on the ol’ iPad and completed six or seven levels. (Two of these took me a few days to get right. So, yes, this is an achievement.)

Random Thoughts

  1. I still have linear algebra to prepare for tomorrow. This is becoming a pattern.
  2. Thanks to the magic of Google+, I got alerted to Treesaver. This looks like a possible way to make your own ebook using HTML5. I wonder if it would be worth it to edit and write up my old linear algebra course now that I’m thinking about it again…