Most people are at least vaguely
familiar with the term "chaos", perhaps largely due to the movie Jurassic
Park. The movie is by no means a mathematical discussion
but one of the key figures is a mathematician. With the exception
of the dinosaurs, one of the most memorable scenes was Ian Malcolm's enthusiasm
in discussing the "butterfly effect". Paraphrasing Professor Malcolm,
a butterfly flapping its wings in Beijing could have a profound influence
on the (future) weather in Central Park.
Crudely stated, chaos is the unpredicatable final result given some small change in initial conditions. However, chaos can be placed on a firm mathematical footing. Chaos is a part of the larger field of study known as difference equations.
Try this simple experiment: consider some function and apply a random initial value. Take your output and re-apply it as a new input to the function. Generate a sequence by using this iterative process. Now generate a new sequence using the same function with a slightly different initial input. It would seems reasonable to expect sequences starting off similiar (with the same function) would stay similiar.
The actual results can be surprising and counter-intuitive. We may have sequences that start close and stay close or we may have one sequence that is totally unpredictable. We may find a sequence that is stable or periodic. The undergraduate Difference Equations course, Mth 442, at the University of Rhode Island, explores these and other exciting ideas.
This site was created by Dr. M. Kulenovic and the students enrolled in Mth 442 during the Spring Semester of 2000. As part of the course requirements, students prepared a webpage dealing with different aspects of difference equations. This site will expand as future Mth 442 students add their projects to these pages.
Our site was created to give a computer presentation which anyone with some mathematical background could find beneficial. Listed below are the names of students contributing projects and a link to their page:
|Gregg Arpin||Henon's Difference Equation|
|Cathy Ann Clark||Solitons|
|Peter Dobratz||Java Applets|
|Chad Griep||Riccati Difference Equation|
|Krzysztof Mailloux||Rational Difference Equation|
|Nicole O'Connell||Lyness' Type Difference Equations|
|Uriah Petit||Mira's Difference Equations|
|Desire Preira||Logistic Difference Equation|
|Stephanie Seran||Lotka-Volterra Difference Equations|
|Victor Taveras||Arnold's Cat Difference Equation|
Click here to go to Home Page for Difference
Equations at URI