copied 15 Feb 1999 from  http://amath-www.colorado.edu:80/appm/courses/7100/

J Meiss: Projects for Applied Math 7100

You should choose a project by Feb. 14. Please come to my office to discuss your choice. Proposals, in the form of 1-2 page description of project with some references are due March 21. The written project is due on the last day of class (May 5).

 Below I have some brief descriptions of some possible projects. You are by no means limited to these; however, your topic must be approved by me. Please come discuss with me these topics or your own ideas.

Your grade will reflect your ability to synthesize material from a number of sources--do not simply copy from a single book or article. Your project may, but need not, involve computer investigations, either in the form of a program you write, or the use of an existing program such as MacMath or Maple, IDL, etc. Your project may, but need not, involve constructing a physical model of a chaotic system, or analysis of the data of such a system (such as the stock market).

1. Phase Space Reconstruction

Discuss the Takens theory of phase space embedding and how it can be used in experiments to show the presence or absence of chaos. Please be careful in stating the theory behind the method! See the paper: Sauer, T., J. A. Yorke, et al. (1991). "Embedology." J. Stat. Phys. 65: 95-116.

2. Forecasting

Nonlinear Forecasting, as introduced by Sidorovich and Farmer, is a current hot topic. How does it work? Apply it to some representative data. What are the fundamental limitations on prediction implied by chaos? Data for trials is available at the Sante Fe Institute, e.g. < ftp://ftp.cs.colorado.edu/pub/Time-Series/TSWelcome.html>

3. Economics

Explain some possible applications of Dynamical Systems to Economics. In particular, discuss why some applications to Microeconomics seem to fail. Explain why the so called "Invisible Hand" might be an idea that has no reality. This corresponds to the belief that an equilibrium allocation (zero excess demand) can be viewed as a stable fixed point solution of a suitable Dynamical System (with the introduction of a tatonnement). A good reference is Saari, Donald. (1995) "Mathematical complexity of simple economics," Notices Amer. Math. Soc. 42, 222--230.

4. Chaos and Finance

Some people have proposed some interesting and controversial applications of dynamical systems to the research in capital markets. Some of the authors claim to have new points of view on how to deal with the prediction of stock prices. At the center of the controversy is the question of the validity of the efficient market hypothesis, and the existence of positive Lyapunov exponents in the historical stock data. The following reference might be useful: E.Peters, Chaos and order in the Capital Markets, (Wiley, 1991).

5. Newton's Method

Show how Newton's method, in the complex domain, can have chaotic behavior. Investigate this method, or other root finders for some example functions. Numerical experiments are expected. See Benziger, H. E., S. A. Burns, et al., "Chaotic complex dynamics and Newton's method," Phys. Lett. A 119, 442 (1987). Professor Curry and Lora Billings are great references on this problem as well.

6. Hilberts 16th Problem

Hilbert conjectured that the number of limit cycles for a polynomial differential equation on the plane is finite. Remarkably this is still an open conjecture, though partial results have been obtained. Lookup up some of these results. Investigate models on the computer determining the number of limit cycles numerically.

7. Periodic Orbit finding

Write symbolic or computational procedures to construct bifurcation diagrams and/or periodic orbit finding routines. Apply these to one and two dimensional maps. There are an existing programs, AUTO and DSTOOL which do this (for ode's) that you might get up and running. Investigate the most efficient methods for following orbits in bifurcation diagrams, especially in the neighborhood of bifurcation points. See, e.g., Seydel, R. (1991). "Tutorial on Continuation." Int. J. Bifurcation and Chaos 1, 3-11.

8. Spectrum of Dimensions

What are the definitions of dimension for fractals? Compute the fractal dimension using some of these techniques for some chaotic systems. Investigate the "f(a)" statistics. Are there any methods which actually compute the Hausdorff dimension?

9. Periodic Orbits and Symbolic Dynamics

Recently Cvitanovic and colleagues (Artuso, R., E. Aurell, et al. (1990). "Recycling of Strange Sets I: Cycle Expansions." Nonlinearity 3: 325-360, and "Recycling of Strange Sets II: Applications." Nonlinearity 3: 361-386).have been using periodic orbits has analytical windows onto more complicated dynamics. The theory requires some use of symbolic dynamics.

10. Hamiltonian Chaos: Period doubling

Investigate period doubling for area preserving maps (see Greene, J. M., R. S. MacKay, et al. (1981). "Universal Behaviour in Families of Area-Preserving Maps." Physica D 3, 468-486). How does the limit differ from the Feigenbaum case we study in class? Analyze the renormalization group operator for this case.

11. Catastrophe Theory

Discuss the ideas and basic theory of catastrophes. You might investigate the catastrophe theory controversy between Smale, and Thom and Zeeman (see me to get Zeeman's recent lecture on this).

12. Computer Assisted Proofs

Computers have been used to prove a number of important results in dynamical systems. They were used by Guckenheimer to prove the Feigenbaum renormalization scheme(see also Stephenson, J. and Y. Wang (1990). "Numerical Solution of Feigenbaum's Equation." Appl. Math. Notes 15, 68-78.), by a number of authors (such as de la Llave, R. and D. Rana (1990). "Accurate Strategies for Small Divisor Problems." Bulletin of the American Mathematical Society 22, 85-90.) to prove versions of the KAM theorem), by MacKay (MacKay, R. S. and I. C. Percival (1985). "Converse KAM: Theory and Practice." Comm. Math. Phys. 98: 469-512.) to prove that the standard map has no invariant circles above a certain parameter, by Yorke to prove that certain systems have a transversal homoclinic point (Hammel, S. M., J. A. Yorke, et al. (1988). "Numerical Orbits of Chaotic Processes Represent True Orbits." Bull. AMS 19, 465-469.), and are therefore chaotic, etc. Investigate the use of the computer in proof strategies, using one or more of these examples.

13. Hamiltonian Chaos: Invariant Circles

Discuss the Greene residue theory for the breakup of invariant circles in area preserving maps. How does the golden-mean become the most robust frequency? Investigate MacKay's renormalization theory for these systems. See MacKay, R. S. (1983). "A Renormalisation Approach to Invariant Circles in Area-Preserving Maps." Physica D 7. 283-300 and MacKay, R. S. (1993). Renormalisation in Area-Preserving Maps. Singapore, World Scientific.

14. Kolmogorov-Arnold-Moser (KAM) Theorem

Discuss the KAM theorem for Hamiltonian flows (we'll briefly discuss the map case in class). Discuss its history, and the consequences for the "ergodic hypothesis" of Boltzmann. (The proof is long and difficult, requiring lots of advanced mathematics). See, e. g. de la Llave, R. (1994). "Introduction to KAM Theory," University of Texas, or Celletti, A. and L. Chierchia (1988). "Construction of Analytic KAM Surfaces and Effective Stability Bounds." Communications in Mathematical Physics 118, 119-161.

15. Chemical Reaction Modeling

Consider a model for a chemical reaction, such as the Brusselator, or Belousov- Zhabotinsky reaction (See Strogatz, S. (1994). Nonlinear Dynamics and Chaos. Reading, Addison-Wesley. for some infor on these). Study phase portraits and bifurcation phenomena.

16. Chemical Patterns

There have been interesting experiments recently on patterns arising from simple chemical reactions. Could these explain the Leopard's spots and the Zebra's stripes? See Swinney, H. L. (1993). "Spatio-temporal patterns: Observation and analysis." in Time series prediction: Forecasting the future and understanding the past. A. S. Weigend and N. A. Gershenfeld. (Reading, MA, Addison Wesley) 557-567.

17. Mixing

The problem of mixing has many Industrial applications. The interesting thing is that mixing has been studied for a long time by people in Dynamical Systems. Your task is to distinguish between mathematical mixing (part of ergodic theory) and physical mixing. Think about some possible industrial applications. Interesting applications to fluid mechanics are in Ottino, J. M. (1989). The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge, Cambridge Univ. Press, and Aref, H. (1984). "Stirring by Chaotic Advection." J. Fluid Mech. 143, 1-21.

18. Biological Modeling

Discuss some biological models for synchronization (fireflies), population dynamics, the swimming of eels, etc. See the book Mathematical Biology by J.D. Murray for possibilities.

19. Nonlinear Circuits

Build a nonlinear oscillator to demonstrate the Period doubling route to chaos. Designs can be found in Pecora, L. and Carroll, T. Nonlinear Dynamics in Circuits, (SingaporeWorld Scientific, 1995).

20. Controlling Chaos

Chaotic systems by their nature are unpredictable. It is interesting that this very property can be used to make them controllable. This hot topic was initiated by Ott, E., C. Grebogi, et al. (1990). "Controlling Chaos." Physical Review Letters 64, 1196-1199. Erik Bollt did his Ph.D. thesis on this area as well, see Bollt, E. and J. D. Meiss (1994). "Controlling Transport Through Recurrences." Physica D 81, 280-294.