My first paper, which was an abbreviated version of my dissertation, had nothing to do with dynamical systems. In it, I investigated a differential equation developed in 1885 by the French mathematician Boussinesq which he used to describe the motion of shallow waves in a canal. I managed to show that there are an infinite number of solutions to the Boussinesq equation; I used techniques from a field of mathematics called functional analysis.

"The Existence of Many Periodic, Nontravelling Solutions to the Boussinesq Equation", Journal of Differential Equations, 126 no 2, 169-183 (April 1996).

When I arrived at F&M, I steered my research into the field of dynamical systems. Dynamical systems began as a field in its own right in the late 1960's. Most of the pioneers in dynamics are still alive, and there are fundamental questions (such as, "how big is the Mandelbrot set?") which are still unsolved. The fundamental question that I have been working on is, "What is chaos?"&emdash;or more appropriately, "how do the many working definitions of chaos relate to one another?".

This is precisely the question I described in my next papers. "The Role of Transitivity in Devaney's Definition of Chaos" considers a topological criterion for chaos (transitivity) and shows that it can, in certain circumstances, be replaced with a different topological criterion ("blending") which is slightly more intuitive. The baker's transformation described above is in fact blending. What this means is that any two sub-segments of the whole line segment will eventually blend together (which is why bakers knead their dough&emdash;to mix all the ingredients together).

"Periodic Orbits from Non Periodic Orbits on the Interval", with Mario Martelli, Applied Mathematics Letters, 10 no 6, pp. 45-47 (1997).

"The Role of Transitivity in Devaney's Definition of Chaos", American Mathematical Monthly, 102 (1995) 788-793.

The preceding paper describe how one kind of chaos implies another kind of chaos. The next paper I wrote, "A Chaotic Non-mixing Subshift" describes an example of a very strange system which is chaotic according to most definitions, but which is not chaotic by the criterion of "mixing". The best physical analogy I can describe is this: suppose you were to break eggs into the left side of a bowl, pour milk into the middle of the bowl, scoop flour into the right side of the bowl, and then knead. If you kneaded that batter according to the rules of my very-strange-system, then usually the batter would look fine, but every once in a while the eggs would spontaneously separate themselves out from the flour, and vice versa. (There is a lovely off-shoot to this problem, which I describe in the next section.)

"A Chaotic, Non-mixing subshift", Proceedings of the International Conference on Dynamical Systems and Differential Equation, special volume of "Discrete and Continuous Dynamical Systems,"  Southwest Missouri State University.

One of the obvious questions arising from this paper is whether similar examples exist in two dimensions. The paper that I wrote with Ben Shanfelder ('98) was an attempt to investigate 2-d dynamics by looking at a specific example. Ben showed that his "triangular map of the square" is chaotic in all senses, including mixing, and his proof gives indications as to how we might show that all other chaotic triangular maps must be mixing, too.

"Chaotic Results for Triangular Maps of the Square" with Ben Shanfelder (class of '98), submitted to Mathematics Magazine.

Ben's paper is an indication of something I referred to earlier, that chaos theory is an ideal vehicle for including students in professional activities: giving talks at conferences, doing research, reading and even writing scholarly articles. The dynamical systems course that I taught (MAT 490) included an optional research seminar which 11 of the 19 students opted into. This seminar led not only into Ben's paper, but another Hackman project with Laura Ciobanu, an independent study with Tarun Jain, and a presentation, one semester later, by 3 of the students in that class for other Franklin & Marshall undergraduates. My syllabus describes the format for the optional seminar, and also the type of scholarly reading I expected of all the students in that course.

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