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Chaos and Fractals

Chaos and Fractals

In the last few decades, mathematicians and scientists have made important discoveries that reveal hidden orders within phenomena that appear to be chaotic.  The mathematics behind this is known as dynamical systems theory, or more informally, chaos theory

Much of the mathematics of chaos theory involves analyzing the geometry of fractals, figures that exhibit self-similarity across all measurement scales.  Images of fractals are some of the most beautiful found in mathematics, often reminiscent of complicated natural objects. 

In this course, we will begin by analyzing and simulating simple chaotic dynamical systems, through computer simulation, laboratory work, observation, and mathematical analysis.  We will observe the period-doubling route to chaos, both experimentally and in action, and thus arrive at the notion of universality in chaos.  We will also examine how chaos theory has been put into practice in industry and medicine. 

Chaos theory often lives on fractals.  We will create fractal images using chance and linear systems, using chaotic attracting sets, and using the geometry of self-similarity.  We will also develop the concept of the fractal dimension as a measure of the 'jaggedness' of a fractal.  We will see how fractal geometry is used in applications in gaming, economics, and science. 

The student who is well-prepared for this course will be comfortable with algebraic manipulation.  Knowledge of trigonometry, exponential functions and logarithms would be useful as well.  Calculus, or at least pre-calculus, would be excellent background.  This course will involve a lot of computer work, so a willingness to learn to work with spreadsheet programs (like Excel) and computer algebra systems (like Maple) will be expected.  There will be a significant laboratory component to this course.

Charlie Jacobson is an associate professor of mathematics at Elmira College.  He earned his Ph.D. in mathematics at Northwestern University in 1989, where he studied dynamical systems under John Franks.  He has been a faculty member at Elmira College since 1990.  During his career, he has developed many technology-based approaches to mathematical instruction, and helped design the College's innovative mathematics computer classroom, Watson 302.  His current research interest is in the statistical analysis of bipartite data.  Charlie is also involved in Elmira College's Spring Term Travel Program, and has led students on study tours of Australia since 2000. 

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