**Ma 4.
Introduction to Mathematical Chaos**

**
**

**MW 13:00 - 14:25
159 Sloan**

**Instructor:****
Anton Gorodetski **

**Office:** 282 Sloan

**Phone:** 395-4350

**E-mail:** asgor@caltech.edu

**Office Hours:** by appointment

**Teaching Assistant:**** Vaibhav
Gadre**

**Office:** 260 Sloan

**E-mail:**
vaibhav@caltech.edu

**Office Hours:** Tuesday, 3-4 pm

The analysis of how a complex system changes over time is known as dynamics. The interaction between only a few simple equations can produce enormously complex behavior, such as the population of different species within an ecosystem or the orbits of the planets within our solar system. Some complex systems can appear totally random, and are said to be chaotic. Such systems are not only highly unpredictable, but the slightest nudge can set them off on a wildly different course, a feature known as the 'butterfly effect'.

In this course we will study the most interesting examples and the most basic features of dynamical systems:

- Iterations of maps. Periodic points. Examples from "real world".
- Rotations of a circle. Equidistribution Theorem.
- One-dimensional dynamics. Sharkovsky Theorem. Rotation number.
- Expanding maps of a circle. Conjugacy, semiconjugacy. Transitivity and topological mixing.
- Diffeomorphisms of surfaces. Hadamar-Perron Theorem. Invariant manifolds.
- Symbolic dynamics. Smale's horseshoes. Topological Markov chains.
- Fractals. Fractal dimensions. Cantor sets. Fractals as invariant sets in dynamics.
- Fractals in complex dynamics.

__Recommended texts:__

**
1 **
B.Hasselblatt, A.Katok, *"A
first course in Dynamics",*

**
2 **
R.Devaney,
*"An introduction to chaotic dynamical systems",*

**
3 **
M.Brin,
G.Stuck, * "Introduction to dynamical systems",*

**
4 **
B.Hasselblatt, A.Katok, *"Introduction to the modern theory of
Dynamical Systems".*

Additional references will be given for a few topics not covered by these books.

__ Grading:__ Weekly homework 50%, midterm exam 20%, final exam 30%.

__ Collaboration Policy:__ You may discuss homework problems with
other students, but solutions should be written up individually in your own
words. Take-home exams must be your own work.

- Homework #1
- Homework #2
- Homework #3
- Homework #4
- Midterm (time limit 3 hours; due Wednesday May 2, 1:00 pm, 2007)
- Homework #5
- Homework #6
- Homework #7
- Final (time limit 3.5 hours; due Wednesday May 30, 1:00 pm, 2007)

Famous dynamical systems in motion