Stephen P. Shipman
Professor and Director of Graduate Studies
Department of Mathematics
Louisiana State University
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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
Office Hours: Monday 9:30-11:30 and Wednesday 1:30-3:30 or by appointment

For a PDF version of the basic course information on this page, click here: 2030syl.pdf

Course Synopsis

A Communication-Intensive Course

This is a certified Communication-Intensive (C-I) course which meets all of the requirements set forth by LSU's Communication across the Curriculum program, including
  • 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.
Students interested in pursuing the LSU Distinguished Communicators certification may use this C-I course for credit. For more information about this student recognition program, visit CxC.


A First Course in Chaotic Dynamical Systems, by Robert Devaney.

Basic 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, 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

Cobweb diagram JAVA applet by Dave Richeson at Dickinson College
Period JAVA doubling applet from UIUC
Cobweb diagram JAVA applet from Math Insight
Robert Devaney's applets

Communication-intensive activities

There are four types of communication-intensive activities:

Type 1: (writing) 3-part mathematical exposition. Proving a mathematical statement involves a creative process, in which one explores and creates structures with the aim of elucidating why a statement is true or not, and an organizational process, in which one unravels the creative process to produce a step-by-step logically coherent argument that starts with certain axioms and ends with the desired statement. When communicating a proof to another person, one wants to retain elements of the creative process that highlight the essential ideas behind the truth of a statement without sacrificing mathematical rigor. This activity involves three parts. (1) Explaining the creative process an a mathematically meaningful (correct use of math objects) but informal (nontechnical) way. This is an account of the reason why the proposition is true that elucidates the essential elements of the argument. (2) Extracting from the creative process a formal step-by-step proof, in which each step consists of a well-defined statement and its justification based on previous statements and axioms in the proof. (3) A prose version of the proof meant to convey to a human being both the underlying ideas and the full logically coherent argument. This part retains all of the mathematical precision of part (2).

Type 2: (writing and speaking) 4-part mathematical invention. The mathematical statements that serve as the starting point of a Type-1 activity have their genesis in a more primitive creative process. We observe general patterns or phenomena that seem interesting and worth investigating further. Careful mathematical investigations require precise definitions, statements, and justifications. This activity involves (1) identifying and explaining a mathematical phenomenon in clear but mathematically informal language, providing examples and non-examples; (2) defining the ideas, objects, and phenomena in a mathematically rigorous way; (3) making a mathematical statement; and (4) proving the statement.

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 ΣN, 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. …

Type 3: (writing) Peer review. When communicating an idea, it is not always easy to imagine your audience's state of mind, especially when the idea seems clear to you. Feedback from a peer has many benefits: (1) it is an excellent means of ascertaining how effective your communication is; (2) it motivates the writer to higher standards; and (3) it helps the reviewer to read mathematical ideas and arguments critically and learn how to improve her or his own writing. In this activity, a student writes a detailed and serious review of another student's mathematical exposition, the expositor writes a rebuttal, and there is a final discussion.

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.


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.
The speaking requirements of a C-I course include the video project as well as short presentations by students of definitions or proofs in class.

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)


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 = si if 0≤i≤√j,
t(j)i = 1 if i>√j.
Prove that t(j) → s as 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 nth 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 Hc, defined by Hc(x) = cx(x-1) for all real numbers x. Prove that the family Hc undergoes a period-doubling bifurcation at c=1.

Video project

In groups of three, students will produce a video presentation of a Type-2 assignment; it will count for 20-25% of the C-I portion of the course. The video will be 30 minutes long and suitable for posting online. The topic should arise from questions that the group has pondered during the course. The presentation will be divided evenly among the members of the group and will include a clear but nontechnical exposition of the concepts, a rigorous development including definitions and conjectures, and then precise mathematical statements and their proofs. The video may be produced in LSU's CxC Studio 151.
  • 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

The final exam is on Saturday, December 14, from 3:00 to 5:00 PM.


Evaluation of performance in the course is based on scores on the assignments and the final exam as follows:
Eight Communication-Intensive activities: 50%
Other problems: 30%
Final exam: 20%
Grading scale: A---at least 90%; B---at least 80%; C---at least 70%; D---at least 60%.

Communication-intensive assignments of Type 1-4 will be graded in a feedback-improvement loop. Each assignment will be submitted multiple times (ideally thrice). Each submission will receive two scores. The green score is an evaluation of the work based on my expectations up to that point in the course, and the red grade indicates the score the assignment would receive if it were the final submission. The green score is permanent, but the red one is replaced by the new red score on the next submission of the same assignment. The final submission receives only a red score, which is permanent. The red grade provides feedback about the quality of the work and helps indicate how much improvement is needed. The green score motivates students to do their best, while not penalizing them for going through the learning process.

Ethical Conduct

Students may discuss problems with each other and other people and consult other literature; however, all work that is turned in must ultimately be that of the submitter alone. If a student receives aid on an assigned problem from discussions with people or other sources, he or she must begin from scratch in writing the solution so that the result is the product of his or her own understanding alone. Students must abide by the LSU Code of Student Conduct. (x=shipman)  Privacy Statement here.  Provide Website Feedback here.  Accessibility Statement here.