Chaos and Fractals
|Years Of Operation||2007-2009|
|Sites Offered||EST, SCZ|
From the CTY Course Catalog (2007):
Can order be revealed in chaos? Can a small action in one part of the world lead to catastrophic consequences in another? For mathematicians the quest to structure the unpredictable dates back to the 1890s and the renowned French mathematician Henri Poincaré. Chaos theory today is an important area of study with applications in meteorology, astronomy, and the arts.
In this course, students investigate the mathematical foundations of chaotic dynamical systems and fractals. They begin by exploring the fundamental process of iteration in functions. Next they examine more advanced concepts including bifurcations, strange attractors, and sensitive dependence related to the “butterfly effect” (which suggests that the flap of a butterfly’s wings in Brazil could theoretically cause a chain of events leading to a tornado appearing in Texas). Finally, students explore the underpinnings of fractal images, pictures which exhibit some degree of self-similarity at any scale. Applications of iteration then lead students to the discovery of fractals such as the important Cantor Set or the Mandelbrot Set.
Through these topics students gain a more formal understanding of the basic principles of chaos theory, an area of math often studied only by advanced undergraduate and graduate students.