# Math 53: Projects - FALL 2009

Here's a preliminary list of potential topics. Please read around on chaotic dynamics and keep on the look out for ideas that interest you. All suggestions welcome! However, in the end your topic choice must be approved by me. Projects are not required to have a numerical (computer) component, but I think you'll enjoy investigating this way, and I encourage it.

You must choose a project topic by Fri Oct 30.

A preliminary 1-page project description with a couple of references is due Tues Nov 8.

Projects are presented starting Tues Dec 1.

#### Topics

• Any of the "Lab Visits" in the book: summarise the original research papers, numerically investigate some of their findings
• Billiards: ball bouncing in a billiard table can be chaotic. Write code to show bouncing motion in some simple chaotic table. Measure the Lyapunov exponent numerically, or review billiard theory. (Enough room for more than one different project here)
• Review work on 3- or n-body problems with gravity. This includes the history of Poincare's solution (for which he was given the prize even though the solution was wrong...). Simulate a simple such solar system using ODEs, animate their orbits.
• Figure out the details of 2-3 "Challenges" (proofs) from the book, and write up their proofs using LaTeX typesetting, the standard which makes equations look beautiful. This is a more pure-mathy, exposition project.
• Compare different ways of measuring fractal dimensions, on real-world sets, and discuss Hausdorff vs box-counting.
• Discuss some biological models for synchronization. See book by J. D. Murray, Mathematical Biology (1993, Springer-Verlag).
• Discuss nonlinear oscillations in economic models. You may want to contact Tilman Dette '10 about this.
• Review properties of the Mandelbrot and Julia sets in detail, using the book by Peitgen, Jurgens and Saupe, Chaos and Fractals (1993, Springer). et al
• Write code to find and plot the stable and unstable manifolds of 2D maps, and reproduce some of the plots from Ch. 10. Investigate for other maps.
• Build a mechanical, electrical, chemical, etc (depending on your existing skills) chaotic system and compare against a simple ODE computer simulation. Measure its Lyapunov exponent.
• Find a chaotic toy and analyse it, compare to numerical simulation.
• Try out the idea of Chris Danforth (UVM) and James Yorke of using shadowing to improve ensemble forecasting in a simple 2d map (see their Phys. Rev. Lett. paper of 2006).
• Collect some time-series data, either existing, or from a system such as the dripping tap, and apply techniques of Ch. 13. One example is the stock market. Another is: heart EKG or climate data (e.g. Eric Posmentier (Earth Sciences Dept).
• summarize work of Posmentier, E.S., 1990: Periodic, quasiperiodic, and chaotic behaviour in a nonlinear toy climate model. Annales Geophysicae, 8 , 11, 781-790.
• Use deterministic chaos to numerically generate a music or art piece, review works or performances which have used concepts from our class (this is clearly a more `far-out' project, but I encourage you to consider it)
• More ideas (and some of the above) are here, with references, thanks to James Meiss at Applied Math, UC Boulder.
• Even more ideas at the ChaosBook site here at Georgia Tech.