High-School Level Introduction to Dynamical Systems

Question

In one month I'll be giving a talk to motivated high schools students on a topic of my choice from dynamical systems and/or ergodic theory.

I'm having trouble coming up with a topic compelling enough to keep their interest, yet elementary enough to be comprehensible. In particular I need to keep clear of concepts of measure and multivariate calculus, unless I can somehow compartmentalize the necessary measure concepts in some neat way (e.g. Lebesgue measure on a manifold is probably okay).

I have some inclination towards giving a talk on dynamical billiards on polygons (very easy in terms of prerequisites), but I thought I would ask anyway to hear some different ideas. For instance, can anybody think of a high-school friendly topic in celestial mechanics?

Answer 1

I would look at the following items.

Books:

• "A First Course In Chaotic Dynamical Systems: Theory And Experiment (Studies in Nonlinearity)" by Robert L. Devaney

• National Academy of Sciences: Science At The Frontier

• Nonlinear Dynamics and Chaos by Strogatz has a few dynamical systems on Celestial Dynamics (see problems 6.5.7 through 6.5.10), but I am not sure they are appropriate.

• It would be great if you can build an electronic circuit, analyze it with mathematics and then measure stuff with DMMs and O'scopes and the like. maybe the Van Der Pol Oscillator or a Harmonic Oscillator.

Web Sites:

• The Dynamical Systems and Technology Project at Boston University

• Discrete Dynamical Systems

• Drexel University Math Forum - for school kids

Answer 2

I'd talk about the chaos game for generating images of IFS fractals or Julia sets. See the books Fractals Everywhere and Fractals for the Classroom for instance.

Answer 3

If you're happy to just show some fun stuff (rather than really explaining the deep mathematics), then you could show movies of $n$-body choreographies:

http://melusine.eu.org/syracuse/swf/1-nbody/

Or talk about solutions to the $n$-body problem which go to infinity in finite time:

http://www.ams.org/notices/199505/saari-2.pdf

Answer 4

Talking about celestial mechanics, please see this pedagogical blurb: http://www.whydomath.org/node/space/index.html

I would be hard pressed to find a more spectacular success of the dynamical systems theory in the real-world.