AMATH 502 Final Project Details, Winter 2014
Description and Requirements
The idea is for you to apply the new techniques and ideas you learn in AMATH 502 to a problem that interests you. This will also give you a chance to more deeply explore the topic(s) that are most important to you.
Most importantly, your project must go beyond the material covered in class. Some sample topics are listed at the bottom of this page.
You should complete this project on your own, but it's okay (encouraged!) to ask others for help if / when you need it.
You will be required to submit a proposal for your project topic (1 page max, singled spaced, 12 pt font) and a write-up of your project (5 pages maximum -- single spaced, 12 pt font -- not including appendices, which are described below) . This write-up should be (loosely) arranged like a scientific paper. This means that you will first describe why the problem you are addressing is important to you (and hopefully others!), and then explain how you solved this problem. You will show the results of your work -- that demonstrate the solution to the problem -- and then discuss the implications of those results for the field. On the main course website, there are a few examples of scientific papers, if you want to see examples of this type of writing.
All projects must involve a technical component. This means that you will need to do calculations and / or numerical simulations. You cannot simply review literature on your topic. Please include the details of any calculations, and / or the computer code(s) used for any simulation(s), as an appendix at the end of the paper that you submit. This materials in this appendix will not count towards the 5 page limit.
Finally, your paper must cite at least 5 scholarly sources (books, scientific papers, etc.). This is meant to encourage you to explore the scholarly literature a little bit, and to get practice at explaining the context of your work. Google scholar is a good place to start, as far as finding papers.
Topic proposals due on Friday, Feb 21 by 3pm.
Write-ups (5 pages max.) due on Friday, March 14 by 3pm.
Sample Project Topics
Note: these are samples only, in part to give you ideas, and in part to give some idea of the appropriate scope for your project (not too big, not too small, etc.). You are welcome to choose these topics, although you are also welcome to do whatever you'd like: I'm looking forward to seeing all the great things you come up with!
0) We discussed 3 basic types of bifurcations in class, each of which has conditions on the derivatives of f(x,u). Looking in Glendinning for guidance, derive "higher-order" bifurcations. Using the implicit function theorem, prove that, given appropriate derivative conditions (which you will determine), this bifurcation will exist.
1) As we'll soon learn in class, chaotic systems have trajectories that are strongly dependent on initial conditions. This means that, even though they are deterministic, trajectories that are initially seemingly "nearby" quickly end up very far apart in state space. So, if you used clock times (for example) as initial conditions for a dynamical system, and asked what system state those initial conditions resulted in, then two subsequent trajectories would differ greatly from each other. This can be used to make a pseudo-random number generator, which is a (deterministic) computer program that outputs (seemingly) random numbers. These random numbers are used A LOT in scientific computing, among other things. Explore the distribution of outputs from your RNG, and choose your dynamical system (or apply a function to its outputs) so that the system produces uniformly distributed random numbers over the interval [0,1]. How random are your random numbers?
2) In class, to illustrate the properties of two-dimensional dynamical systems, we'll consider the "Romeo and Juliet" problems, in which Romeo's and Juliet's feelings for each other are dynamical functions of time. Depending on how each party responds to the other, their relationship can have extremely rich dynamical properties, including cycles of love / hate, etc. In his paper "Dynamical Models of Love" (Nonlinear Dynamics, Psychology, and Life Sciences, Vol. 8, No. 3, July, 2004 ), Sprott extends these models to non-linear dynamical equations, and to love triangles, to understand better the possible (complicated!) dynamics of inter-personal relationships. In the same spirit, extend these models even further, to better understand how the dynamics of how we respond to one another act to shape our relationships. Obvious possibilities include the love rhombus or pentagon, although I expect that you can come up with much more creative ways to apply our dynamical systems toolkit!
3) Critical slowing down (yeast) and populations going extinct. In a great recent paper ( Generic Indicators for Loss of Resilience Before a Tipping Point Leading to Population Collapse Science 2012), Dai and colleagues explore the dynamics of populations of yeast in a dish. As they vary parameters of the environment, their dynamics undergo a saddle-node bifurcation, where on one side of the bifurcation, a stable population exists, and on the other, it goes extinct. Moreover, as the parameters near the bifurcation point, a clear signature is observed, in the ``critical slowing" of the population growth! The supplemental materials for the paper give the dynamical equations for the population size. These depend on the life cycle of their organism (yeast). For another species with a different -- possibly more complicated -- life cycle (say, Bigg's Killer Whales), study the bifurcations, and find the signature(s) of healthy vs. endangered populations.
4) Towards the end of class, we'll have learning about fractal dimensions, and the famous paper (by Mandelbrot himself) "How long is the coast of Britain"? The answer to this question is complicated by the fact that the coastline of England is 1.25 dimensional (we'll soon learn what that means!). What is the dimensionality of the shoreline of Puget Sound? To answer this, you'll need to (a) get some digitized data for locations on the shores of PS, and (b) apply the methods from this Mandelbrot paper.
5) In a dripping faucet, the time intervals between subsequent drops is unpredictable; this is a chaotic system where the dynamical state depends sensitively on the initial conditions. This paper, and in particular Refs. 18 and 19, discusses the problem in detail. Design and perform (with some guidance from the literature) an experiment to measure the return map of a leaky faucet, and analyze it to explain the chaotic-ness.
6) In this paper from Greg Stephens, he and Bill Bialek analyze the probability distribution of english-language words. Their references give some ideas for where to get digitized data with lots of english-language words: ripe for our mining! Use these data to estimate the fractal dimension of the English language (or at least it's n-letter words, for some appropriate n). Some measures that we'll learn about in class -- like the correlation dimension -- are probably a good start.
7) In this paper, Song and colleagues discuss the fractal dimension ("fractality") of natural network structures. Conveniently for us, the Stanford Network Analysis Platform (SNAP: https://snap.stanford.edu/) team has assembled large amounts of real-world network data, like the graph formed by email communication between people. Use the tools of Song et al. to estimate fractal dimension of your favorite network / graph.
8) In class, we'll briefly discuss the double pendulum, as a simple example of a chaotic system. What about the triple pendulum? Use your knowledge of physics to derive equations of motion for a triple pendulum, and analyze the result: is the system chaotic? Calculate the Lyapunov Exponents for the system (numerically, of course) to answer this question.
9) When oscillators are coupled (even very weakly), they tend to synchronize. For example, take a look at this youtube video ( "Synchronization of Metronomes"). Interestingly, the same can be true for chaotic systems. Investigate these phenomena using analytical arguments, simulations, and / or experiments