Math 5535 --- Final Project
due Monday, December 20 at high noon

You should complete a project dealing with dynamical systems theory or its applications. This handout contains a few suggestions but its main purpose it to give you an idea of the possibilities. It would be even better if you make up you own project. If you have an idea and want to check it out first, just stop by my office.

Projects typically divided into three basic types: papers about math, papers about applications and computer programs. I use the following criteria in coming up with grades. For math papers: correctness and depth (50), use of references (20), exposition (30). For papers about applications: use of references (30), analysis/critique (30), exposition (40). For computer projects: design and accuracy (50), use of program (30), exposition (20). For math projects, depth means going beyond the math of the course and getting into the details of your topic. For papers on applications, you need to look at several references; then analyze what you find, perhaps illustrating the points with your own examples. For computer projects, you should write the program for a specific purpose and then really use it for that purpose. It should be flexible enough to allow you to handle several examples.

Mathematical Theory.

1. A project in this area might by based on one of the challenge sections at the end of a chapter of the text. These are presented as step by step investigations where you have to fill in all the details. You could supplement that with some outside reading about your topic.

2. The Henon map has been studied extensively both with computers and theoretically. In section 2.9 of the book by Devaney cited below, there is a sequence of problems leading you through some of the results.

• Devaney, R., An Introduction to Chaotic Dynamical System, Section 2.9, Addison-Wesley (1989).
• Henon, M., A two-dimensional mapping with a strange attractor, Comm. Math. Phys.,50 (1976), 69-77.
• Devaney, R. and Nitecki, Z., Shift automorphism in the H\'enon mapping, Comm.Math.Phys. 67 (1979), 137-148.
• Robinson, C., Dynamical Systems -- Stability, Symbolic Dynamics and Chaos, Section 7.4.1, CRC Press (1995).

3. The theory of homeomorphisms of the circle could be explored, including the
theory of rotation numbers and Denjoy's theorem.

• Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems and
  Bifurcations of Vector Fields, Section 6.2, Springer Verlag (1983).
• Robinson, C., Dynamical Systems -- Stability, Symbolic Dynamics and Chaos, Section 2.8, CRC Press (1995).
• Denjoy, A., Sur les courbes d\'efinies par les equations differentielles a la surface du tor, Jour.Math. 17(IV), (1932), 333-375.

4. Look into the theory of linear maps of a torus. These are torus maps determined by integer matrices. See challenge 2 in the book. In addition to the dynamics, the algebraic theory behind these matrices is quite interesting. Also see:

• Devaney, R., An Introduction to Chaotic Dynamical System, Section 2.4, Addison-Wesley (1989).
• Adler, R., Tresser, C. and Worfolk, P., Topological conjugacy of linear endomorphisms of the 2-torus, Trans. AMS, 349 (1997), 1633.
• Adler, R. and Weiss, B., Similarity of automorphisms of the torus, Memoirs Amer. Math. Soc.98(1970).

Applications.

Write a survey paper describing an application of dynamical systems theory. Here are some ideas.

5. Circle maps are widely used as models of biological rhythms. The first reference below is a book which contains many more references.

• Glass, L. and Mackey, M.C., From Clocks to Chaos: The Rhythms of Life, Princeton University Press (1988).
• Glass, L. and Mackey, M.C., A simple model for phase locking of biological oscillators, Jour.Math.Biol. 7 (1979), 339-352.
• Guevara, M.R., Glass, L., and Shrier, A., Phase locking, period doubling bifurcations and irregular dynamics in periodically stimulated cardiac cells, Science214 (1981), 1350-53.
• Whitelaw et. al., Hiccups as a dynamical disease, Chaos 5 (1995), 14.

6. Fractal geometry has been applied to many fields. Look into one area and describe how the theory is applied. Beware of hype. Try to judge whether the application is believable or not.

• Mandelbrot, B., The Fractal Geometry of Nature, Freeman (1982).
• Barnsely, M., Fractals Everywhere, Academic Press (1988).
• Barnsely, M. and Sloan, D., A better way to compress images, Byte 13, 215-233 (1988).
• http://www.iterated.com
• Schroeder, M., Chaos, Fractals, Power Laws, Freeman (1991).
• Bassingthwaighte, J.B., Liebovitch, L.S., West, B.J., Fractal Physiology, Oxford Univ. Press (1994).

7. Two-dimensional maps provide models for the ecology of two species in the same environment or one species with a time-delayed growth law. For example, the delayed logistic map x(n+2)=ax(n+1)*(1-x(n)) is a second order difference equation which can be made into a map of the plane,F(x,y)=(y,ay(1-x)) by setting y(n)=x(n+1). Here are some references about this and other two-dimensional models.

• Maynard-Smith, J., Mathematical Ideas in Biology, Cambridge Univ. Press (1971).
• Pounder, J.R. and Rogers, T.D., The geometry of chaos: dynamics of a nonlinear second-order difference equation, Bull. Math. Biol. 42 (1980), 551-597.
• Beddington, J.R., Free, C.A., and Lawton, J.H., Dynamic complexity in preditor prey models framed in difference equations, Nature 255 (1975), 58-60.
• May, R.P., Regulation of populations of nonoverlapping generations of microparasites: a purely chaotic system, Amer. Nat.125 (1985),573-584.

Experiment.

8. Write a program which can compute and display stable and unstable manifolds for diffeomorphisms of the plane. Assuming you have subroutines for computing F, F^{-1} and the derivative DF, the program could first locate a fixed point (perhaps using Newton's method), then find the multipliers and eigenvectors. If the fixed point is a saddle, you can use the eigenvectors to approximate the local stable manifold and then extend the manifold by iteration. Use the program to study how the stable and unstable manifolds of the Henon map change with parameters.

9. Write a computer program for computing Lyapunov exponents for 2D or higher maps. See section 5.2 of the text.

10. Write a program which can display strange attractors for diffeomorphisms of the plane or attractors of iterated function systems. Write it so you can zoom in on them to show the fractal structure. Amenities would include interactive choice of zoom region, number of computed points, etc. You could also color the basin of attraction of the attractor. You could combine this with a simple dimension computing program as below.

11. Write a program for estimating the box-counting dimension of a set of data points in the plane, space or in n-dimensional space. Apply it to some experimental data set or to a set of points in a strange attractor or an attractor of an iterated function system. For experimental data, try embedding the data in spaces of different dimensions. How does the answer depend on the box sizes used? How much data do you need to get consistent answers?

• Grassberger, P. and Procaccia, I., Characterization of strange attractors, Phys. Rev. Lett. 50, 346-349 (1983).
• Grassberger, P., An optimized box-assisted algorithm for fractal dimensions, Phys. Lett. A 148, 63-68 (1990).