# Math 53: Projects - FALL 2015

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?

Because of the short quarter, you need to start early and take initiative to explore things, do background reading.

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

You should fix a project topic by Tues Oct 20.

A preliminary 1-2 page project plan description with a couple of references (which you have read some of!) is due Tues Oct 27.

Projects are presented Nov 17-18 - it's a fun mini-conference. (This is only 8 days after Midterm 2, so you can't leave it all to after this exam. Plan ahead.)

#### Topics

• Understand and summarize Chapter 12 on period-doubling cascades. How accurately can you compute Feigenbaum's universal constant giving the asymptotic ratio by which the cascade shrinks when it doubles?
• Study the "recurrence plot" as a data analysis tool for dynamical systems. Test it out on a bunch of discrete and continuous systems that you simulate. See http://www.recurrence-plot.tk/ and the Marwan et al Phys Rep 2007 paper. (from Nishant Malik, postdoc in our dept).
• Ecology applications. Start with A. Hastings, T. Powell, "Chaos in Three-Species Food Chain," Ecology, 72(3), p 896-903 (1991). And find papers that cite it.
• Hopf bifurcations, applications and analysis. See Strogatz Nonlinear Dynamics and Chaos book.
• Synchronization of coupled nonlinear oscillators: fireflies, neurons, and metronomes. Simulate numerically and find theoretical results. See Strogatz.
• Any of the "Lab Visits" in the book: summarise the original research papers, numerically investigate some of their findings. These haven't been chosen enough so please look through them.
• 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?
• Oscillons and spatio-temporal chaos. Numerical simulation, localized repeating structures, and summarize some of the theory.
• 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 (see Vanderbei). What are their stabilities?
• 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).
• Discuss some biological models for synchronization. See book by J. D. Murray, Mathematical Biology (1993, Springer-Verlag).
• Nonlinear dynamics in economic models (see Futoma and Southworth's 2001 project; there is much more to be done here).
• 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, write a numerical simulation using Hamiltonian dynamics and Matlab's built-in ODE solver.
• Speaking of Hamiltonian dynamics, model and understand rigid body motion about the axis of intermediate moment of intertia, ie understand this.
• Chaotic dynamics in fluid flow, eg chaotic water wheel, or more real-world situations.
• Chaotic dynamics in the brain: 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 (economic, financial, social media data), or from a system such as the dripping tap, and apply techniques of Ch. 13 (time-delay embedding) to search for underlying dynamics. Other examples: heart EKG or climate data (e.g. Eric Posmentier (Earth Sciences Dept, Dartmouth). This type of project has the down-side of being vague or not finding anything interesting.
• 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. Study and implement the techniques of Diana Dabby for chaotic music generation (see Jack Sisson's nice 2007 project).
• Look through old Math 53 projects and think or ask how to extend them.
• 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.