**Instructor**: Dr. Tom Carter

DBH-287 667-3175

tom@csustan.edu

http://cogs.csustan.edu/~tom

**Texts:** The required texts for the class are:

* Chaos: Making a New Science*, James Gleick

* Mathematics and the Unexpected*, Ivar Ekeland

* Applied Chaos Theory - A Paradigm for Complexity*, A. B. Cambel

There are also many other books which are worth looking at -- I will
talk about a variety of them during the semester.

**Purpose of Course and Objectives:** The
world is filled with systems of various degrees of complexity, which behave
in various ways. Scientific/mathematical approaches to understanding
systems always involve approximations, simplifications and modelling. The
primary traditional paradigm has been to assume that systems are *linear*,
but this assumption severely limits the extent to which we can
understand systems, and the kinds of systems we can understand. Discarding
the assumption of linearity opens new possibilities. In this course, we
will develop a general understanding of the traditional *linear*
approach, and then explore various nonlinear approaches and systems. Among
the topics we will explore are phase space maps, Poincare sections, basins
of attraction, strange attractors, fractals and fractal dimension,
catastrophe theory, discrete approaches, and some of what is currently
called complexity theory. The course will of necessity have a relatively
high mathematical content, but I will work hard to provide the background
for the material and to explain things in comprehensible ways. I don't
necessarily expect you to understand everything we do, but I do expect you
to work hard and apply yourself to understanding what you can. The
laboratory component (which we will discuss in class later) will help you
develop practical experience with nonlinear systems.

**Grading:**

The grades for this course
will be based on three components: written homework/in class quizzes, a
midterm, and a project/paper. Each of the components will be weighted
approximately equally.

Also, at the beginning of each class period, I want you to hand in a brief response to
the readings for the day. The response should contain a two or three point summary of
the important ideas of the section, and two or three points which you found confusing or
weak or deserving of additional coverage.

The work you do for this course will be your own. You are not to submit other people's
work and represent it as your own. However, I do expect and encourage you to work
collaboratively with others during the course.

**General Summary of Material to be
Covered:**: (MU = Mathematics and the Unexpected, C = Chaos, ACT = Applied Chaos
Theory)

- Overview of nonlinear systems, ideas of chaos theory. (C, Ch. 1-4)
- Classical mechanics, linear systems, classical determinism. (C, Ch. 5-11, MU, Ch. 1)
- Introduction to nonlinear systems, determinism vs. predictability. (MU, Ch. 2, ACT, Ch. 1,2)
- Geometric approaches to dynamical systems, phase space, Poincare sections. (ACT, Ch. 3)
- Catastrophe theory and related topics. (MU, Ch. 3)
- Iterated maps, logistics equation, discrete vs. continuous systems. (ACT, Ch. 5, 6, 7)
- Fixed points, stable sets, attractors, bifurcations. (ACT, Ch. 4, and MU, appendices)
- Fractals, definition of chaos, strange attractors. (ACT, Ch. 9, 10, 11)
- Fractal dimensions, related measures (ACT, Ch. 9)
- Entropy and information theory (ACT, Ch. 8)
- Diagnostics and control of chaotic systems (ACT, Ch. 11)
- Complex adaptive systems.