Discrete Dynamical Systems
Math 2030-1
Louisiana State University
Spring Semester, 2008
Place: Room 235 of Lockett Hall
Time: Monday, Wednesday, and Friday from 1:40 to 2:30
Instructor: Prof. Stephen Shipman
Office: Room 314 of Lockett Hall
Telephone: 225/578-1674
Email: shipman@math.lsu.edu
Office Hours: Monday, Wednesday, and Friday 10:40--12:00 or by appointment
For a PDF version of the basic course information on this page,
click here: 2030syl.pdf
Notice that the room for the course has been changed to room 235, where we will have a computer at our disposal.
Textbook
A First Course in Chaotic Dynamical Systems, by Robert L. Devaney
Course Description
Dynamical systems with discrete time and in one spatial dimension; complex dynamics; quadratic maps; chaos; structural stability; bifurcation theory.
Course Content
The content of the course will span the entire book, with some
subsections being omitted. The subject lends it self very
well to a fruitful interplay between computational experiment
and mathematical theory. Some of the key ideas are the following:
-
Concepts: Discrete dynamical systems on the real line; fixed points, orbits, and their stability; chaos and the Cantor set; dependence on parameters.
Related mathematical ideas: Iteration of functions and its
graphical analysis, calculus of fixed points, countability
and uncountable sets.
Computer experiments: Cobweb diagrams, computation of attracting
orbits, changing parameters and the transition to chaos.
-
Concepts: Symbolic dynamics for the analysis of
real-valued discrete systems.
Related mathematical ideas: Sequence spaces, metric and homeomorphism, conjugacy of dynamical systems.
Computer experiments: Feigenbaum's constant: order in
the transition to chaos.
-
Concepts: Periodic orbits and Sarkovskii's
Theorem on their existence.
Related mathematical ideas: Continuity and the Intermediate-Value Theorem, ordering of sets, the Fibonacci sequence.
-
Concepts: Basin of attraction and the critical orbit.
Related mathematical ideas: The Schwarzian derivative.
Computer experiments: Computing basins of attraction.
-
Concepts: Fractals and their dimension.
Related mathematical ideas: Randomly iterated maps in 1D and 2D.
Computer experiments: Iterated function systems with a random
choice of function at each step.
-
Concepts: Complex dynamics, the Julia set and
fractals; chaos, Cantor sets in the complex plane, repellors,
critical orbits.
Related mathematical ideas: Algebra and calculus of complex functions, polar decomposition, complex square roots.
Computer experiments: Computation of the filled Julia sets and
critical orbits.
-
Concepts: The Fundamental Dichotomy of behavior of
complex systems, the critical orbit, the Mandelbrot set.
Related mathematical ideas: The Boundary-Mapping Principle of complex-analytic functions.
Computer experiments: Finding periods represented by different
parts of the Mandelbrot set.
Links to web sites on discrete dynamical systems
Orbit diagram java applet
Cobweb diagrams
Davaney's java applets
Devaney's Mandelbrot set explorer
Maple worksheets
For sample Maple worksheets, click
here.
Assignments
| Due date
|
Chapter
|
Problems to do |
| Wed., Jan. 23
| Chapter 3
| 1, 2, 3, 7ah, 10, 11ac, 14, 17
|
| Wed., Jan. 23
| Chapter 4
| 1 b,g; 3 b,g; 5; 7
|
| Fri., Feb. 1
| Chapter 5
| Experiment 5.6. You may work in a group of up to three people.
|
| Fri., Feb. 1
| Chapter 5
| 1 a,f,j; 2 c,f; 4 b,c
|
| Fri., Feb. 15
| Chapter 6
| Experiment 6.4. You may work in a group of up to three people, and you do not need to do the part entitled "Notes and Questions".
|
| Fri., Feb. 15
| Chapter 6
| 1 a,b,c; 6, 7, 8, 9. For problem 1, include a bifurcation diagram for each part.
|
| Mon., Mar. 10
| Chapter 9
| 1, 4, 7, 8, 9 Solutions
|
| Fri., Apr. 4
| Chapter 10
| 1, 3, 6, 8, 14, 17
|
Fri., Apr. 25
| Chapter 11
| 7 (include "except 1" at the end), 9, 10, 11, 12
|
Evaluation
Evaluation of performance in the course is based on scores on the assignments and the final exam as follows:
-
Assignments: 70%
Final exam: 30%
Grading scale: A--at least 90%; B--at least 80%; C--at least 70%; D--at least 60%.
Code of Ethical Conduct
All work turned in must be that of the student except when an assignment is done in group collaboration. In this case, it must be made as clear as possible on the work turned in who the collaborators were and which part of the assignment was done by the submitter.
Students must abide by the LSU
Code of Student Conduct .