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Discrete Dynamical Systems
Math 2030-1
Louisiana State University
Spring Semester, 2008
Place: Room 112 of Lockett Hall
Time: Monday, Wednesday, and Friday from 11:40 to 12:30
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 10:00--11:30 or by appointment
For a PDF version of the basic course information on this page,
click here: 2030syl.pdf
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.
The prerequisite for this course is Math 1552.
Course Content
The subject of dynamical systems lends it self very well to a fruitful interplay between computational experiment and mathematical theory. This makes the subject an ideal medium for learning to read and write rigorous mathematics. Learning the rigorous definition of concepts and proving of theorems will be the main objective of the course; this will be emphasized primarily in the second part of the course. The emphasis throughout is on sound logical thinking and communication, and students' work on assignments and the final exam will be held to high standards in this area. Some computer experiments will be assigned for the purpose of thoughtful exploration of ideas in discrete dynamics.
We will not expect to cover most of the material in the textbook. Instead, we will focus on three topics:
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the "quadratic family" of dynamical systems,
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bifurcations of the quadratic family,
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symbolic dynamics.
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., Sept. 10
| Chapter 3
| 1, 2, 3, 7ah, 10, 11ac, 14, 17
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| Mon., Sept. 22
| Chapter 4
| 1bg, 3bg, 5
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| Mon., Sept. 22
| Chapter 5
| Experiment 5.6. You may work in a group of up to three people.
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| Wed., Oct. 8
| Chapter 5
| 3, 4bcd
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| Fri., Oct. 17
| Problems below
| 1, 2
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| Fri., Oct. 24
| Problems below
| Redo 1, 2
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| Fri., Oct. 31
| Chapter 6
| Experiment 6.4. You may work in a group of up to three people. (And redo problems 1 and 2 below.)
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| Fri., Nov. 14 [First draft]
| Chapter 9
| 1, 3 (with clear explanation); 5, 7, 9 (with rigorous proof)
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Mon., Nov. 24
| Chapter 9
| 13, 16, 18af
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Problems
1. Prove the following statement: Let real numbers c and x be given, with c>1/4. Then Qnc(x) → ∞ as n → ∞.
2. Consider the two families of iterating functions Fλ(x) = λ(x3 - x) and Gλ(x) = -λ(x3 + x). Each of these undergoes a bifurcation at λ=1, about the fixed point x=0. Figure out what ilk of bifurcation is occurring for each of these families, and prove your assertion rigorously for one of the two.
(I've worked out a proof of a bifurcation for a different family here.)
Exams
There will be no exams in the course except the final exam, which will take place on Thursday, December 11, from 10:00 to noon.
Evaluation
Evaluation of performance in the course is based on scores on the assignments and the final exam as follows:
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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 .
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