The term project for Math 4/5027 is a major paper on some aspect of nonlinear dynamics involving differential or difference equations. The topic may be analytical or numerical in nature (or both). It may be theoretical or applied.


As a rough guideline, a project for undergraduates should be 15-20 pages long and projects for graduate students should be 25-30 pages long. It must be typed neatly with perfect spelling and grammar, fully documented, well organized, and detailed in explanations and conclusions. It should contain any graphs, diagrams, figures, or data that are needed for a full exposition. This is a good opportunity to learn a mathematical typesetting package such as LaTeX. However, you needn't type formulas and equations; you may write them neatly between lines of typed text.

The project must have a cover page showing the title and author. It must have an introduction that explains the problem and gives relevant background information. The main body should explain in detail the procedures used to solve the problem and present interesting observations that you made. The conclusion must give a concise summary of your results and give possibilities for future work on the problem. The paper must include at least five references that show evidence of library or web research. The paper itself needn't contain original work.

A detailed outline with an annotated bibliography is due no later than March 15. The project is due no later than the last day of classes.

The following list of topics is merely suggestive. Let it give you some ideas, but don't be constrained by it. You should discuss possible topics with me as soon as possible.

1. One source of projects is the book. Some of the Challenge Problems or Lab Visits, if elaborated sufficiently, could be made into projects.

2. A theoretical project could consist of choosing a major theorem, such as the Poincare-Bendixson Theorem, Sharkovsky's Theorem, or the Stable Manifold Theorem, giving a detailed proof, and providing applications or uses of it.

3. Nonlinear oscillators. Forced mechanical or electrical oscillator problems often exhibit a wide variety of behavior that can be uncovered through a combination of numerical and analytical methods.

4. The Lozi map. This is a discrete map in the plane that displays intricate behavior.

5. Fractals. We will not cover Chapter 4 of the book, but there are plenty of opportunities for project topics. A related topic is iterated function systems for generating fractal images.

6. Specific chaotic systems. A project could explore a specific system such as the Lorenz equations, the Rossler system, the Chua circuit, or the B-Z reaction.

7. Control of chaotic systems. There are important methods for guiding a chaotic system into stable states.

8. State space reconstruction. Chapter 13 of the book contains interesting ideas that relate time series to nonlinear systems.

9. Population genetics. Both nonlinear difference and differential equations govern the evolution of gene pools.

10. Economics. Is there chaos in international currency markets or in stock markets?

11. Physiology. Models of the heart have led researchers to conclude that the heart can go into chaotic states.

12. Population Biology. What do the logistic equation or its ODE analog look like for several interacting species?