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.
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>
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,
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
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
325-360, and "Recycling of Strange Sets II: Applications."
3: 361-386).have been using periodic orbits has analytical windows
onto more complicated dynamics. The theory requires some use of symbolic
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,
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."
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.
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."
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.