WHAT IS A DYNAMICAL SYSTEM? Dynamical systems is a branch of mathematics that attempts to understand processes in motion. Such processes occur in all branches of science. For example, the motion of planets is a dynamical system, one that has been studied for centuries. Some other systems are the stock market, the world's weather, and the rise and fall of populations.

Some dynamical systems are predictable, whereas others are not. The reason for this unpredictable behavior has been called ``chaos.'' One of the remarkable discoveries of the modern mathematics is that very simple systems-even as familiar as quadratic functions-may be chaotic and behave as unpredictably as the stock market or as wildly as a turbulent waterfall.

ABOUT THIS COURSE: The aim of the course is to give a first introduction to the subject. The emphasis will be on mathematical ideas and concepts.

This is a course in discrete dynamical systems, which is basically iteration, or composing a function with itself over and over. We are interested in the long-term behavior of a system. Often complexity the system calls for qualitative reasoning as opposed to looking for specific analytic solutions. Continuous dynamical systems, which arise from differential equations, are closely related to, and have many common features with, discrete dynamical systems. They will be studied in Math 436.

The plan is to cover most of the book, except Chapter 14 (fractals), since we will study fractals in Math 436 in more detail. One of the main themes will be the dynamics of the quadratic family Q_c(x) = x²+c depending on the parameter c, first when c and x are real, and later when c and x are complex. We will study how the long-term behavior of the system changes from stable and predictable to chaotic.

MATH 436: The tentative plan is to give an introduction to three large areas: (1) fractals, (2) higher-dimensional dynamics (hyperbolicity, homoclinic points, horseshoes), and (3) continuous dynamical systems (including the Lorenz attractor).

PREREQUISITES: Introductory analysis (Math 327 or equivalent). In particular: e-d definitions of convergence and continuity; infinite series; sets and functions; proofs by induction, contradiction, and contraposition; countable and uncountable sets. For Math 436: linear algebra and differential equations, equivalent to Math 307-309.

TESTS, HOMEWORK: There will be a midterm exam on Friday, February 7, on Chapters 3-6. It will be ``open notes.'' The final exam will be ``take-home;'' it will be given on the last day of instruction and due by the end of the finals week. I will assign homework to be turned in weekly, on Mondays (or on Wednesdays when Monday is a holiday). There will be additional optional homework for extra credit, with theoretical and computing options. Students can work together on the homework but everyone should write his/her own solutions.

PROJECT: Students may elect to do an optional project. This project may be theoretical, computational, or applied. A theoretical project may be to study a more advanced topic (with rigorous proofs) related to the course. A computational project may be to do some computer experiments related to the course (with a discussion of relevant mathematics). An applied project may be to report on a dynamical system in physics, engineering, biology, or another field. There are no hard boundaries between three kinds of projects: for instance, one can combine the applied and computational, or theoretical and applied, in one project. Students may collaborate on a project (up to three persons in a group). If you are interested in doing a project, talk to me; the topic should be approved not later than the end of the 4th week. The first draft should be turned in by the end of the 8th week, and the final draft is due on the last day of instruction.

GRADING SCHEME: A (without the project): Homework 40%, Midterm 20%, Final 40%; B (with the project): Homework 30%, Midterm 15%, Final 30%, Project 25%.

COMPUTING: The explosive growth of dynamical systems theory that we are witnessing today would be impossible without computers. I will not use computers in the classroom but it is highly recommended that students do computer experiments relevant to the course. You can use PCs in the MSCC lab, located in the basement of Communications, in rooms B-022 and B-027. You can see information about the lab by looking at the Web pages: http://www.ms.washington.edu

Some handouts will include Mathematica code. You can use Mathematica, Mathlab or other software if you prefer.

TENTATIVE SCHEDULE:

Week 1: Ch. 1-4 (introduction, orbits, graphical analysis)

Week 2: Ch. 5 (fixed and periodic points), start Ch. 6 (bifurcations)

Week 3 (short): end Ch. 6, Ch. 12 (light coverage: Schwarzian derivative)

Week 4: Ch. 7 (quadratic family, Cantor set), Ch. 8 (transition to chaos)

Week 5: start Ch. 9 (symbolic dynamics), MIDTERM on Ch. 3-6

Week 6: end Ch. 9 (conjugacy), Ch. 10 (chaos).

Week 7 (short): Ch. 11 (Sharkovskii's Theorem)

Week 8: Ch. 13 (Newton's method), Ch. 15 (complex functions)

Week 9: Ch. 16 (Julia sets)

Week 10: Ch. 17 (Mandelbrot set)

ADDITIONAL READING LIST:

R. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, 1989. (This text treats many of the same topics on a more advanced level.)

R. A. Holmgren, A First Course in Discrete Dynamical Systems, Springer-Verlag, 1994. (Covers roughly the same ground as our course, but is written from a different point of view. More careful than Devaney in some proofs and gives more background material from real analysis.)

I. Stewart, Does God Play Dice?: the mathematics of chaos, Blackwell, 1989. (Good and readable discussion of the implications of modern dynamical systems and chaos theory on the development of science. Recommended as companion reading.)

K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos. An introduction to Dynamical Systems, Springer-Verlag, 1997. (Parts of this book will be used as a required Text in Math 436. It has ``Lab visits'' and ``Challenges'' which may be used as a basis for the project.)

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