Bruce B. Peckham - Spring 2003 Chaos, Fractals, and Dynamics (Math 1234)

Math 1234 Chaos, Fractals, and Dynamics

Spring 2003
Prof. Peckham

Texts:

CHAOS: Making a New Science by James Gleick, Penguin Books 1987.
ITERATION, by Choate, Devaney and Foster

Journal: Each student should keep a journal for the course. The journal should be a brief summary of ACTIVITIES done in the class and OBSERVATIONS/QUESTIONS/IDEAS which have arisen either individually or from the group. There should be an entry for each class, and an entry for each time period between classes. Each entry should have the date of the entry and the date(s) when activities were performed.

Grading

Homework sets: 27%
Quizzes: 25%
- Quiz 1: 50 pts
- Quiz 2: 75 pts
- Quiz 3: 100 pts
- Quiz 4 (Final): 150 pts Optional. Count the greater of Quiz 4 and the average of Q1-Q3.
Participation: 22%
- Attendance: 10%
- Journal 1: 1%
- Journal 2: 2%
- Journal 3: 2%
- Journal 4: 2% Cancelled. Use the average of J1-J3 instead.
- Class participation: 5%
- Paper on Gleick's book: 6%. Changed to more general Final Paper on Class. See below.
Final HW set/ Final Paper: 26%
- Final Problem Set: 15%
- Final Paper: 11%
- Oral: 5% Cancelled. Points counted in paper instead.

Homework sets.

HW 1 ITERATION Lesson 1: Numerical Iteration. Do all Investigations and Further Exploration problems.
HW 2 ITERATION Lesson 2: Financial Models. Do Investigations: Problems 1,2,3,4,5a-i,k only. Do at least one of these problems using a spreadsheet. Include a (labelled) printout of the spreadsheet with your work. Problems 5j,5l, 6,7 may be done for extra credit. Do not do any of the Further Explorations.
HW 3 ITERATION Lesson 3: Behavior over time. Do all Investigations problems.
HW 4 ITERATION Lesson 6: Geometric Iteration. Do all Investigations problems and Futher Exploration problems 3 and 7 only. For problems 3 and 7 do the first two ``triangle bullets'' only.
HW 5 CHAOS Lesson 3: Graphical Iteration. Do all Investigations problems.
HW 6 CHAOS Lesson 4: Measuring Population Growth. Do all Investigations problems 1 - 7, with the following exceptions. In 6a, exclude k=3.554. In 7, read parts a, b, fill in at least one k value in each range in part c except for the last, and for d plot at least one point distinct from everyone else in the class on a graph I will provide. It is recommended that you use the software at http://math.bu.edu/DYSYS/applets/nonlinear-web.html to do problems 4-7, but you could also use a calculator or spreadsheet.
HW 7 CHAOS Lesson 5: Nonlinear Iteration. Do Investigations problems 1 - 7 (3c and 3d for extra credit) and Further Explorations problem 2.
Final HW Set: See Handouts. 4 problems to write up; 1 problem of lab tasks to check off.

Final Paper.

General description: Write a paper discussing what you have learned in this course and how your views about mathematics have evolved throughout the course.

Your paper should include:

Some information about the Gleick book, including some summary information - what the book was about - and some specific information about topics discussed.
A summary of what you learned in the course, through any of the reading, discussions, HW sheets, software, classes, etc. An indication of how you learned is better than just what you learned.
Some indication about how your views about Dynamical Systems, Chaos, Fractals, and more generally Mathematics and Science have evolved since the beginning of the course. Cite specific examples whenever possible.

Suggested length: 2-3 pages. (Approx 1000-1500 words)

Grading:

Content. Are your observations accurate and supported?
Quality of writing (organization: introduction - body - conclusion, presentation, grammar, spelling, ...)
Evidence that you learned something in the course.
Originality. Have you thought about and assimilated ideas from the course?

Assignments and Due Dates
Week	Date Due	Assignment
1	T 1/21/03	INTRO CLASS
1	Th 1/23/03	GLEICK: Read prologue: 1-8; bring observations/questions to class. ITERATION: Continue to work on Lesson 1 investigations and Further Exploration. JOURNAL: Begin journal. Make entries for `Class 1: T 1/21/03'.
2	T 1/28/03	GLEICK: Read Butterfly Effect: 9-29; bring observations/questions to class. ITERATION: Complete HW 1 (Lesson 1: Numerical Iteration) Investigations and Further Exploration. JOURNAL: Entries for `Between Classes 1-2' and `Class 2: Th 1/23/03'.
2	Th 1/30/03	GLEICK: ITERATION: Hand in HW 1 (Lesson 1: Numerical Iteration) Investigations and Further Exploration. JOURNAL: Entries for `Between Classes 2-3' and `Class 3: T 1/28/03'.
3	T 2/4/03	GLEICK: Read Revolution: 33-56; bring observations/questions to class. ITERATION: Work on HW 2 (Lesson 2: Financial Models) Investigations. JOURNAL: Entries for `Between Classes 3-4' and `Class 4: Th 1/30/03'.
3	Th 2/6/03	GLEICK: ITERATION: Read Explanation pages for Lesson 1 (Numerical Iteration) and Lesson 2 (Financial Models). Complete HW 2 (Lesson 2: Financial Models) Investigations problems. JOURNAL: Entries for `Between Classes 4-5' and `Class 5: T 2/4/03'.
4	T 2/11/03	GLEICK: Read Life's Ups and Downs: 57-80; bring observations/questions to class. ITERATION: Hand in HW 2 (Lesson 2: Financial Models) Investigations: Problems 1,2,3,4,5a-i,k only. Do at least one of these problems using a spreadsheet. Include a (labelled) printout of the spreadsheet with your work. Problems 5j,5l, 6,7 may be done for extra credit. Do not do any of the Further Explorations. Read Explanation pages for Lesson 3: Behavior over time. JOURNAL: Entries for `Between Classes 5-6' and `Class 6: Th 2/6/03'. Hand in Journal, completed through entries for Class 6, Th 2/6/03.
4	Th 2/13/03	GLEICK: ITERATION: Hand in HW 3 (Lesson 3: Behavior over time) Investigations problems. JOURNAL: Entries for `Between Classes 6-7' and `Class 7: T 2/11/03'.
5	T 2/18/03	QUIZ on HW 1 - HW 3 (Lessons 1-3). Last 30 minutes of class. GLEICK: ITERATION: Reveiw Lessons 1-3. JOURNAL: Entries for `Between Classes 7-8' and `Class 8: Th 2/13/03'.
5	Th 2/20/03	GLEICK: ITERATION: Read Lesson 6 (Geometric Iteration) Explanation. Try to complete HW 4 (Lesson 6: Geometric Iteration) Investigations problems and Futher Exploration problems 3 and 7 only. For problems 3 and 7 do the first two ``triangle bullets'' only. JOURNAL: Entries for `Between Classes 8-9' and `Class 9: T 2/18/03'.
6	T 2/25/03	GLEICK: Read A Geometry of Nature: 81-118; bring observations/questions to class. ITERATION: Hand in HW 4 (Lesson 6: Geometric Iteration) Investigations problems. CHAOS: Read Explanation pages for Lessons 1-3 (Linear Iteration, Types of Fixed Points, Graphical Iteration) from New ``CHAOS'' book. Begin working on HW 5: (Lesson 3: Graphical Iteration) Investigations problems. JOURNAL: Entries for `Between Classes 9-10' and `Class 10: Th 2/20/03'.
6	Th 2/27/03	GLEICK: ITERATION:Last day to hand in HW 1 (Lesson 1: Numerical Iteration) corrections. CHAOS: Continue to work on HW 5 (Lesson 3: Graphical Iteration) Investigations problems. JOURNAL: Entries for `Between Classes 10-11' and `Class 11: T 2/25/03'.
7	T 3/4/03	GLEICK: Read Strange Attractors: 119-153; bring observations/questions to class. ITERATION: Last day to hand in Quiz 1 corrections.. CHAOS: Continue to work on HW 5 (Lesson 3: Graphical Iteration) Investigations problems and Further Exploration problems 1ab and 3abc. JOURNAL: Entries for `Between Classes 11-12' and `Class 12: Th 2/27/03'.
7	Th 3/6/03	GLEICK: ITERATION: CHAOS:Hand in HW 5 (Lesson 3: Graphical Iteration) Investigations and Further Exploration problems. JOURNAL: Entries for `Between Classes 12-13' and `Class 13: T 3/4/03'.
8	T 3/11/03	GLEICK: ITERATION: CHAOS: Review for Quiz 2. JOURNAL: Entries for `Between Classes 13-14' and `Class 14: Th 3/6/03'.
8	Th 3/13/03	QUIZ 2. On HW 1 - HW 5, including all material in the Explanation pages for Lessons 1-3, 6 from Iteration book and Lessons 1-3 from Chaos book. 80-90% of the quiz will be on HW's 4 and 5. Last 40 minutes of class. GLEICK: ITERATION: Last day to hand in HW 2 (Lesson 2: Geometric Iteration) corrections and HW 3 (Lesson 3: Behavior over time) corrections. JOURNAL: Entries for `Between Classes 14-15' and `Class 15: T 3/11/03'. Hand in Journal, completed through entries for Class 15, T 3/11/03.
.	Spring Break	.
9	T 3/25/03	GLEICK: Read Universality: 155-187; bring observations/questions to class. ITERATION: CHAOS: Read Lesson 4: Measuring Population Growth Explanation pages. Begin Investigations problems. JOURNAL: Entries for `Between Classes 15-16' and `Class 16: Th 3/13/03'.
9	Th 3/27/03	GLEICK: ITERATION: CHAOS: Go over Quiz 2. Work on HW 6 (Lesson 4: Measuring Population Growth) Investigations problems. JOURNAL: Entries for `Between Classes 16-17' and `Class 17: T 3/25/03'.
10	T 4/1/03	GLEICK: Read The Experimenter. Bring observations/questions to class. ITERATION: CHAOS: Continue working on HW 6. Begin work on HW 7. JOURNAL: Entries for `Between Classes 17-18' and `Class 18: Th 3/27/03'.
10	Th 4/3/03	GLEICK: ITERATION: Last day to hand in Corrections for HW 4 (Lesson 6: Geometric Iteration). CHAOS: Continue to work on HW's 6 and 7. JOURNAL: Entries for `Between Classes 18-19' and `Class 19: T 4/1/03'.
11	T 4/8/03	GLEICK: Read Images of Chaos. Bring observations/questions to class. CHAOS: Continue working on HW's 6 and 7. JOURNAL: Entries for `Between Classes 19-20' and `Class 20: Th 4/3/03'.
11	Th 4/10/03	GLEICK: CHAOS: JOURNAL: Entries for `Between Classes 20-21' and `Class 21: T 4/8/03'.
12	T 4/15/03	GLEICK: Read pp 250-255, 267-272, 278-280, 289-292, 303-307, 315-317 Bring observations/questions to class. CHAOS: HW 6 due. OTHER: Last day to hand in Quiz 2 corrections. JOURNAL: Entries for `Between Classes 21-22' and `Class 22: Th 4/10/03'.
12	Th 4/17/03	CHAOS: Finish HW 7. JOURNAL: Hand in Journal Entries through Tues. April 15 class.
13	T 4/22/03	CHAOS: HW 7 due. JOURNAL:
13	Th 4/24/03	CHAOS: Orbit diagram. Mac lab tasks in Montague 239.
14	T 4/29/03	Quiz 3
14	Th 5/1/03	Final Paper draft or outline due.
15	T 5/6/03	FINAL PROBLEM SET DUE
15	Th 5/8/03	Final Paper due. All corrections due. Optional FINAL EXAM

Correction policy for HW's and Tests/Quizzes

Homework sets. General policy: Homework corrections will be graded and counted one point for every two points corrected.
- Exception: HW 1 corrections will be counted at full credit.
Tests and Quizzes. General policy: Test/Quiz corrections will be graded and counted one point for every three points corrected.
- Exception: Quiz 1 corrections will be counted at one point for every two points corrected.

Calculator/Spreadsheet hints for Lesson 2 and beyond.

Iterating is "easy" on spreadsheets and on most calculators. To illustrate, we will explain how to do problem 3a from Lesson 2 in Choate, Devaney and Foster. The problem asks you to calculate the amount in a savings account if you deposit $1500 initially, earn 12% annual interest compounded monthly (1% per month), and make monthly deposits of $225. The iteration rule for this problem is: B_n+1=B_n(1.01)+225. (You should know how to obtain this.) The initial seed is B₀=1500.

On a spreadsheet. Type the initial seed (1500) in any cell (say it is in cell C4). In the cell below it (C5 in my example), type =C4*1.01 + 225. This is the iteration formula!!! Now put the cursor at the lower right hand corner of the cell for which you just typed this formula (C5). The cursor should change to a small plus. Now select, drag down any number of rows (say 10) and release. This copies the formula from cell C5, but incrementing the cells referred to in the formula by one. For example, the formula "copied" to cell C6 is =C5*1.01 + 225. This is exactly what you need to iterate!! It is useful to make sure the cell labels are visible and to print out your results.
On a Calculator. This works on a TI-85 and probably many other calculators. Type in your initial seed: 1500 and press enter. Type x 1.01 + 225 and press enter. The calculator should respond with Ans* 1.01 + 225. Pressing enter again takes the new answer and performs the same operations on it. Successively pressing "enter" continues the "iteration" process.

This page (http://www.d.umn.edu/~bpeckham/www) is maintained by Bruce Peckham (bpeckham@d.umn.edu) and was last modified on Thursday, 01-May-2003 15:14:42 CDT.