# Math 53: Projects - FALL 2011

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, since I don't want you to attempt something impossible. Projects are not required to have a numerical (computer) component, but I think you'll enjoy investigating this way, and I encourage mixing theory, proofs, background, and computer experiment. You will enjoy reading through the previous projects from this course (see main page links), and getting inspired. Maybe take one of these further, or study a related question?

You should choose a tentative project topic by Tues Oct 24.

You can team up, but I prefer 2 as the largest group size.

A preliminary 1-page project plan description with a couple of references (which you have looked at!) is due between Tues Nov 1 and Fri Nov 4.

Projects are presented starting Tues Nov 29 - it's a fun mini-conference.

#### 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). Very similar is chaotic scattering from the exterior of a set of balls, related to the cover of Ott's book. How accurately can you predict the future ray path here?
• Review work on 3- or n-body gravitating problems. 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. There are some crazy n-body periodic orbits out there.
• 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.
• Explore some other chaotic maps, such as 'kicked rotator', as their parameter is varied. Connects to KAM theorem (present overview of this).
• Dig deeper into, and simulate, different types of bifurcations, Hopf, etc. (Use Strogatz Nonlinear Dynamics and Chaos book).
• 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.
• Understand and present 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 find out what happens when S and U touch! 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.
• Chaotic dynamics in fluid flow, eg chaotic water wheel, or more real-world situations.
• Chaotic dynamics in systems of 2 or more neurons with Hodgkin-Huxley ODE models.
• 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).
• Quantum chaos: either study PDE analog of billiards above (the drum problem), or quantized torus maps (matrix algebra problem). Harder.
• Collect some time-series data, either existing, or from a system such as the dripping tap, and apply techniques of Ch. 13 to search for underlying dynamics. One example is the stock market. Another is: heart EKG or climate data (e.g. Eric Posmentier (Earth Sciences Dept, Dartmouth).
• 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 music or and 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.