Discrete Dynamical Systems, a.k.a. "Proofs Bootcamp"
A Communication-Intensive (CxC) Course

Math 2030-1
Louisiana State University
Fall Semester, 2013

Prof. Stephen Shipman

Place: Room 232 of Lockett Hall
Time: Monday, Wednesday, and Friday from 11:30 to 12:20

Office: Room 314 of Lockett Hall
Telephone: 225/578-1674
Email: shipman@math.lsu.edu
Office Hours: Monday 9:30-11:30 and Wednesday 1:30-3:30 or by appointment

Course Synopsis

A Communication-Intensive Course

• instruction and assignments emphasizing informal and formal writing and speaking;
• teaching of discipline-specific communication techniques;
• use of draft-feedback-revision process for learning;
• practice of ethical and professional work standards;
• 40% of the course grade rooted in communication-based work; and
• a student/faculty ratio no greater than 35:1.

Textbook

Basic Course Description

Dynamical systems with discrete time and in one spatial dimension; complex dynamics; quadratic maps; chaos; structural stability; bifurcation theory.

Prerequisite

The prerequisite for this course is Math 1552, the second semester of calculus.

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 definitions of concepts and proving of theorems will be the main objective 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. In addition, computer experiments may be assigned for the purpose of thoughtful exploration of ideas in discrete dynamics.

Links to web sites and applets on discrete dynamical systems

Communication-intensive activities

Students may work in groups of up to three people. Here are some possible topics, but students are encouraged to come up with something on their own.

• Conditions for a neutral fixed point to be attracting/repelling
• Dynamics in Σ_N, the set of sequences from {0,1,2,…,N-1} (Chaos, periodic orbits, density, etc.…)
• P. 113, Remark 2 on two different symbolic representations of elements of the Cantor set
• Condition for an orbit to alternate about a fixed point
• Attracting/repelling n-cycles
• Stuff on fractals (coming up …)
• Etc. …

Here are some guiding questions for your evaluation of your peer's work.

Part 1

Is the statement to be proved described clearly?

Does the author make clear what are the crucial ideas that show why the statement to be proved is true?

Does the language communicate the main mathematical arguments clearly while avoiding overly technical or detailed arguments or notation?

Part 2

Is the statement to be proved clearly delineated and well defined?

Is it clear where the proof begins and ends?

Is each statement justified based on facts established in previous steps of the proof, assumptions, lemmas, or theorems?

If a justification is based on a lemma, is the lemma clearly stated and proved or (if it's difficult) argued convincingly?

Are mathematical terms used correctly?

Is the conclusion to be proved stated in the last line of the proof?

Part 3

Is the logically coherent argument, which is expected in Part 2, retained?

Is the language mathematically precise?

Is the prose written in a natural and flowing way?

Is the grammar correct?

Type 4: (speaking) Oral mathematical exposition. Whether at a technical level or at the level of general exposition, oral communication of mathematics requires carefully identifying ideas and their logical connections. (1) When explicating a technical proof orally, careful choice of words and well formed sentences are of utmost importance. The live nature of oral communication requires an even firmer understanding of the arguments than does written communication, and it requires more practice than one might imagine. (2) In a typical conference presentation, one needs to find a way to communicate the underlying ideas behind why a statement is true or why a method works, and know which details are important and which are not. In this activity, both aspects of communicating mathematics are practiced by presenting orally an activity of Type 1 or Type 2.

Assignments

C-I activities. There will be several assignments of the types described above.

• Four assignments of Type 1.
• One assignment of Type 2 and 4 (see the video project described below)
• One assignment of Type 3.

Routine problems will be assigned periodically. These will be held to high standards of mathematical logic but will not be subject to the structure of the C-I assignment types discussed above.

In-class participation. Periodically, some problems from an assignment will be discussed in class before the due date of the assignment. These problems will be submitted at the beginning of class and students' work will be shared with the rest of the class for constructive criticism. Students should be ready to write a definition on the chalk board on the spot any time.

Due_date	Type	Problems to do
Wed., Aug. 28	routine	Ch. 3: 7, 10 for discussion
Fri., Aug. 30	routine	Ch. 3: 15-19 for discussion
Wed., Sept. 4	routine	Ch. 3: 7, 10, 11, 15-19; Ch.4: 1
Mon., Sept. 9	Type 1 (#1-v1)	Problem 1 below in the style
Fri., Sept. 13	routine	Ch. 5: 1, 4
Mon., Sept. 16	Type 1 (#1-v2)	Second draft of Type 1 (#1). To be evaluated by fellow students.
Fri., Sept 20	routine	Problem 2 below. Do rigorous step-by-step proofs. These will be discussed on Friday.
Wed., Sept 25	routine	Problem 3 below
Fri., Sept 27	Type 1 (#2-v1)	Problem 3 below and problem 18g from Chapter 9.
Mon., Sept 30	Type 3 (v1)	Peer review of Type 1 (1-v2)
Wed., Oct. 2	routine	Ch. 10: 2, 3, 6, 8, 17. Answer the questions and give convincing reasons for your answers.
Fri., Oct. 4	Type 1 (#3-v1)	Ch. 10: 6, 17. For each, state the proposition in mathematically precise language and then prove it. Particularly, for #17, you need to convert Devaney's vague statement into a mathematically precise one.
Mon., Oct. 7	Type 1 (#2-v2)	Second draft of Type 1 (#2).
Wed., Oct. 9	routine	Ch. 10: Prove that the shift map σ acting in Σ is a chaotic dynamical system.
Fri., Oct. 11	Type 1 (#3-v2)	Second draft of Type 1 (#3).
Wed., Oct. 16	routine	Ch. 9 18i, 19
Fri., Oct. 18	Type 1 (#2-v3)	Third draft of Type 1 (#2).
Oct. 21-25	Type 2,4	Talk to me in groups about your ideas for a video presentation, described below.
Wed., Oct. 23	Type 3	Give a copy of your peer review of Type 1 (#1) to the author.
Fri., Oct. 25	Type 1 (#3-v3)	Third draft of Type 1 (#3).
Fri., Nov. 1	Type 2,4	Turn in a plan of your video project, described below.
Wed., Nov. 6	Type 1 (#4-v1)	Problem 4 below.
Fri., Nov. 15	Type 2,4	Turn in a detailed written exposition of your video project, including your definitions, statements, and proofs.
Wed., Nov. 20	Type 1 (#4-v2)	Second and final version of Type 1 (#4)
Fri., Nov. 22	Type 2,4	Have your oral presentation of your video project ready to present.
Wed., Nov. 27	Type 1 (#1-v3)	Third version of Type 1 (#1)
Nov. 25-Dec. 6	Type 2,4	Produce a 30-min. video presentation in CxC Studio 151.
Thurs., Dec. 12	Deadline to submit	Video presentation (see below)

Problems

1. Prove that the tent map (T defined on page 28) admits exactly nine 6-cycles.

2. For the following, use my definitions of H:C→C, the itinerary map S, and the shift map σ. C is the Cantor set.

a. Prove that SοH = σοS. (SοH means S composed with H. I couldn't find a good HTML symbol for the usual composition circle.)

b. Prove that, if the orbit of x∈C under H is periodic, then the orbit of S(x) under σ is periodic.

3. Let s be an element of Σ. For each positive integer j, define an element t^(j) of Σ by

t^(j)_i = s_i if 0≤i≤√j,

t^(j)_i = 1 if i>√j.

Here is a good response for part 1 of a Type-1 assignment, which asks for a precisely but non-technically communicated account of why the proposition is true.

… To prove convergence of t^(j) to s in Σ, one has to show that, if j is sufficiently large, the distance between t^(j) and s is smaller than any prescribed "error". What is meant by "distance" is made precise by the metric in Σ, which places t^(j) and s within 1/2ⁿ of each other if they agree up to their n^th entry. From the definition of t^(j), one sees that this is accomplished if √j > n. This shows how to relate the largeness of j to the "error" ε= 1/2ⁿ in order to have d(t^(j), s)<ε. Indeed, one need only make j>n².

4. Let a family of discrete dynamical systems on the real line be be given by a family of iterating functions H_c, defined by H_c(x) = cx(x-1) for all real numbers x. Prove that the family H_c undergoes a period-doubling bifurcation at c=1.

Video project

• Make sure that the viewer can see you explaining the stuff -- not just your voice dubbed over a presentation.
• If you do your video in the CxC Studio 151, the side-by-side option is good. It shows you presenting at the screen as well as a full-screen view of the presentation so you can see details.
• Look into the camera and don't cover your writing with your body.

Final Exam

Evaluation

Eight Communication-Intensive activities: 50%

Other problems: 30%

Final exam: 20%

Ethical Conduct