## Math 1234 Chaos, Fractals, and DynamicsCourse Homepage

### Homework sets.

1. HW 1 ITERATION Lesson 1: Numerical Iteration. Do all Investigations and Further Exploration problems.
2. 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.
3. HW 3 ITERATION Lesson 3: Behavior over time. Do all Investigations problems.
4. 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.
5. HW 5 CHAOS Lesson 1: Linear Iteration. Investigations 6. Further Explorations 1 (4 orbits only), 2, 3.
CHAOS Lesson 2: Types of Fixed Points. Investigations 8,9,10.
CHAOS Lesson 3: Graphical Iteration. Do ALL Investigations problems; #8 is Extra Credit.
6. 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; do part d on a graph I will provide for the class; part f is for extra credit. It is recommended that you use the software at http://math.bu.edu/DYSYS/applets/nonlinear-web.html to do problems 4-7. For parameter values not available on the website (as well as ones that are) you could use a calculator or spreadsheet or "Chaos software" on the Mac laptop that I bring to class.
7. Final HW Set CHAOS Lesson 5: Nonlinear Iteration. Do Investigations problems 1,2,4,5,6,7 and Further Explorations problem 2. Extra Credit: Investigations 3a, 8. You can use a graphing calculator or other graphing software to obtain the graph for problem 3a. Only the part of the graph near the origin is necessary to do 3a.
In addition, do the following extra problems:
1. Use spreadsheet and an iteration rule to determine the annual (simple) interest rate corresponding to
1. 10% interest compounded monthly (provide labelled printout)
2. 10% interest compounded daily (no printout necessary)
2. Reproduce the following figures found in the Gleick book. You may either provide a printout, or show me the figures in class. Printouts in black and white are OK. The coloring may not be exactly as in Gleick.
1. The orbit diagram: pages 71, 74, 75. Suggested software: Boston University Dynamical Systems Website/Java applets/Orbit Diagram : http://math.bu.edu/DYSYS/applets/OrbitDiagram.html. The default should already be the "Logistic Map", which is x->Cx(1-x)!! Select the Iterate button to draw the orbit diagram for this map. Zoom in selecting the left mouse button, dragging, and releasing. Get a better picture by in creasing the number of "Iterations".
2. The Mandelbrot Set: the set and one enlargement shown on the second color plate between pages 114 and 115. Suggested software: Boston University Dynamical Systems Website/Java applets/The Mandelbrot-Julia Set Applet : http://math.bu.edu/DYSYS/applets/Quadr.html. The Mandelbrot set is on your right. To zoom in on a rectangular piece of it, position the mouse at any corner of the rectangle, press and hold the left mouse button, drag the mouse to the opposite corner of your rectangle, and release the left mouse button. The coordinates of the lower right corner and the upper left corner of the triangle you just selected should be displayed in the lower right part of the window. Now select the Compute button, and you will see the part of the Mandelbrot set in the rectangle you just selected. To see a more accurate version, increase the Max. Iteration number from 50 to 100 or 500.
3. The Henon map (a two-dimensional iteration rule) is given by: (x,y) -> (a-by-x2, x). Compute the first 4 iterates of the seed (0,0) using parameters a=1.4, b=-0.3. (If you continued to iterate 1000 times or so, and plotted all points of the orbit (in the x-y plane), you would begin to see the famous Henon attractor. (See p. 151 in Gleick.)
4. Extra credit: The complex quadratic family is defined by the iteration rule:
z -> z2+c, where z and c are complex numbers. Compute the first 4 iterates of the seed z0 = 0+0i using the parameter c=-0.12+0.73i. Can you guess the fate of this orbit?

### Assignments and Due Dates

Class Date Due Assignment
1 T 1/18/05
INTRO CLASS
2 Th 1/20/05
GLEICK: Read prologue: 1-8; bring observations/questions to class.
ITERATION: Read Introduction and Lesson 1 explanation. Work on Lesson 1 investigations and Further Exploration.
JOURNAL: Begin journal. Make entry for `Class 1: T 1/18/03'.
3 T 1/25/05
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 1/20/03'.
4 Th 1/27/05
GLEICK:
ITERATION: Hand in HW 1 (Lesson 1: Numerical Iteration) Investigations and Further Exploration. Read Explanation pages for Lesson 2.
JOURNAL: Entries for `Between Classes 2-3' and `Class 3: T 1/25/05'.
5 T 2/1/05
GLEICK: Read Revolution: 33-56; bring observations/questions to class.
ITERATION: Work on HW 2: Investigations for Lesson 2: Financial Models.
JOURNAL: Entries for `Between Classes 3-4' and `Class 4: Th 1/27/05'.
6 Th 2/3/05
GLEICK:
ITERATION: Hand in "revised" Further Explorations from HW 1. (HW 2 DUE DATE MOVED TO TUES. 2-8-05.)
JOURNAL: Entries for `Between Classes 4-5' and `Class 5: T 2/1/05'.
7 T 2/8/05
GLEICK: Read Life's Ups and Downs: 57-80; bring observations/questions to class.
ITERATION: Hand in HW 2. Read Explanation pages for Lesson 3: Behavior over time. Start working on HW 3.
JOURNAL: Entries for `Between Classes 5-6' and `Class 6: Th 2/3/05'.
8 Th 2/10/05
GLEICK:
ITERATION: Continue working on HW 3.
JOURNAL: Entries for `Between Classes 6-7' and `Class 7: T 2/8/05'. Hand in Journal, completed through entries for Class 7, T 2/8/05.
9 T 2/15/05
GLEICK:
ITERATION: Hand in HW 3 (Lesson 3: Behavior over time) Investigations problems. Reveiw Lessons 1-3.
JOURNAL: Entries for `Between Classes 7-8' and `Class 8: Th 2/10/05'.
10 Th 2/17/05

GLEICK: Read A Geometry of Nature: 81-118; bring observations/questions to class.
ITERATION: Continue review of Lessons 1-3. Read Lesson 6 (Geometric Iteration) Explanation. Start HW 4.
JOURNAL: Entries for `Between Classes 8-9' and `Class 9: T 2/15/05'.
11 T 2/22/05

GLEICK:
ITERATION: QUIZ on HW 1 - HW 3 (Lessons 1-3); first 30 minutes of class. Continue with Lesson 6, Geometric Iteration.
JOURNAL: Entries for `Between Classes 9-10' and `Class 10: Th 2/17/05'.
12 Th 2/24/05
GLEICK:
ITERATION: Hand in HW 4 (Lesson 6: Geometric Iteration) Investigations problems.
JOURNAL: Entries for `Between Classes 10-11' and `Class 11: T 2/22/05'.
13 T 3/1/05
GLEICK: Read Strange Attractors: 119-153; bring observations/questions to class.
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: (Lessons 1-3: Graphical Iteration)
JOURNAL: Entries for `Between Classes 11-12' and `Class 12: Th 2/24/05'.
14 Th 3/3/05
GLEICK:
ITERATION:
CHAOS: Continue to work on HW 5 (Lessons 1-3: Graphical Iteration).
JOURNAL: Entries for `Between Classes 12-13' and `Class 13: T 3/1/05'.
15 T 3/8/05
GLEICK: Read Universality: 155-187; bring notes on observations/questions to class.
ITERATION:
CHAOS: Continue working on HW 5 (Lessons 1-3: Graphical Iteration).
JOURNAL: Entries for `Between Classes 13-14' and `Class 14: Th 3/3/05'.
16 Th 3/10/05
GLEICK:
ITERATION:
CHAOS: Finish and hand in HW 5 (Lessons 1-3: Graphical Iteration) If caught up, begin work on HW 6.

JOURNAL: Entries for `Between Classes 14-15' and `Class 15: T 3/8/05'.
17 T 3/15/05
GLEICK:
ITERATION:
CHAOS: Prequiz/Review for Quiz 2.
JOURNAL: Entries for `Between Classes 15-16' and `Class 16: Th 3/10/05'.
18 Th 3/17/05
GLEICK:
ITERATION: Last day to hand in Corrections on HW's 1, 1a, 2, 3, 4, Quiz 1.
CHAOS: 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.

JOURNAL: Entries for `Between Classes 16-17' and `Class 17: T 3/15/05'. Hand in Journal entries #14-#34 completed through entries for Class 15, T 3/15/05.
. 3/19-3/27/05Spring Break
19 T 3/29/05
GLEICK:
ITERATION:
CHAOS: Continue working on HW 5. Begin work on HW 6.
JOURNAL: Entries for `Between Classes 17-18' and `Class 18: Th 3/17/05'.
20 Th 3/31/05
GLEICK: Read The Experimenter. Bring observations/questions to class.
ITERATION:
CHAOS: HW 5 due. Continue to work on HW 6.
JOURNAL: Entries for `Between Classes 18-19' and `Class 19: T 3/29/05'.
21 T 4/5/05
GLEICK:
CHAOS: Redo "hard" problems from HW 5. Work on HW 6. Miniquiz 1: redo of problems 7 and 9 from Quiz.
JOURNAL: Entries for `Between Classes 19-20' and `Class 20: Th 3/31/03'.
22 Th 4/7/05
GLEICK:
CHAOS: Rest of HW 5 due. Miniquiz 2: more graphical iteration.
JOURNAL: Entries for `Between Classes 20-21' and `Class 21: T 4/5/03'.
23 T 4/12/05
GLEICK:
CHAOS: Miniquiz 3. More Graphical iteration. Describe "all" orbits.
OTHER:
JOURNAL: Entries for `Between Classes 21-22' and `Class 22: Th 4/7/03'.
24 Th 4/14/05
GLEICK: Read Images of Chaos. Bring observations/questions to class.
CHAOS: HW 6 due. This includes your 2 parameter values on the class "orbit diagram."
JOURNAL: Entries for `Between Classes 22-23' and `Class 23: Th 4/12/03'.
25 T 4/19/05
GLEICK:
CHAOS: Miniquiz 4: escape to infinity? Definition of Chaos.
JOURNAL: Entries for `Between Classes 23-24' and `Class 24: Th 4/14/03'.
26 Th 4/21/05
GLEICK: Read pp 250-255, 267-272, 278-280, 289-292, 303-307, 315-317 Bring observations/questions to class.
CHAOS:
JOURNAL: Entries for `Between Classes 24-25' and `Class 25: Th 4/19/03'.
27 T 4/26/03
Final Paper draft or outline due.
GLEICK:
CHAOS: Miniquiz 5: Graphical Iteration: fate of ALL orbits.
JOURNAL: Entries since Spring Break due. (Optional. If not handed in, your journal score will be the grade determined by your first two journal sets.)
28 Th 4/28/05
GLEICK: How to compute some of the figures in GLEICK.
CHAOS:Quiz 3. Lessons 1-5 in Chaos, and Miniquiz topics; focus on use of graphical iteration.
Work on final problem set.
29 T 5/3/05
GLEICK: CHAOS: FINAL PROBLEM SET DUE
30 Th 5/5/05
Final Paper due.
All corrections due.

### 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.
• Tests and Quizzes. General policy: Test/Quiz corrections will be graded and counted one point for every three 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: Bn+1=Bn(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.

```1. Class #1, Tuesday Jan. 18, 2005.  Recorded Wednesday Jan 19, 2005.

Activities:
....
Observations/Questions/Ideas/Extensions:
....

2. Between classes #1 and #2, Tues. Jan 18 - Thurs. Jan 20, 2005.
Recorded Fri. Jan. 21, 2005.

Activities:
....
O/Q/I/E:
....
```

### 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: 3-5 pages. (Approx 1500-2500 words)