Week of

Topics

Assignments

Sep 8

  Chapter 3. Orbits
  Chapter 4. Graphical Analysis
 Types of Orbits, The Doubling Function, Orbit Analysis, The Phase Portrait

Sep 15

  Chapter 5. Fixed and Periodic Points: 5.1-5.5
  Chapter 6. Bifurcations: 6.1-6.3

  HW #1
 Page 26: Nos. 1,3,5,6,7 A-F, 11,12,13.
 Page 34: Nos. 1 A,C,F; 4 A,B,D; 5.
 due Friday Sep. 19th

Sep 22

  Chapter 7. The Quadratic Family: 7.1, 7.2
  First and second bifurcations in the quadratic family.
  Periodic points of x^2-2.
  Nonescaping points for c<-2.

 HW #2
 Page 50: Nos. 1 A,B,F,J; 2 A,C,E; 5,7.
 due Friday Sep. 26th

Sep 29

  Chapter 7. The Quadratic Family: 7.2, 7.3
  Non-escaping points of x^2-c, for c<-2.
  Middle thirds Cantor set.

no homework due this week

Oct 6

  Chapter 9. Symbolic dynamics: 9.1-9.4
  Sequence space, shift map.
  Itineraries. Conjugating x^2-c to the shift map for c<-2.

 HW #3
 Ch.6, Page 67: Nos. 1 c,d,e,i; 3-5, 8,9, 14,15
 Ch.7, Page 81: Nos. 16, 17
 due Monday Oct. 6th (note the change of date)

Oct 13

  Chapter 10. Chaos: 10.1-10.3  
Midterm 1: Friday, October 17, 6-8pm
Medical Sciences Buiding, room MS 3163
Click for a sample exam 1 and sample exam 2

 HW #4
 Ch.7, Page 80: 2, 9-15
 Ch.9, Page 111: 1,4,7,10-12,18 B,E,G,I.
 due Wednesday Oct. 15th

Oct 20

  Chapter 10. Chaos: 10.1-10.3
  Shift map is chaotic, semiconjugaies, x^2-2 is chaotic

no homework this week

Oct 27

  Chapters 11. Sarkovskii's Theorem
 1.1 Period 3 implies every other period
 1.2 Sarkovski's Theorem (not a complete proof), examples

 HW #5
 Ch.10, Page 131: 2-5, 10-13, 15,17, 21,24
 due Wednesday Oct. 31st

Nov 3

  Chapters 11. Sarkovskii's Theorem
  Period 3 window in the quadratic family, Subshifts of finite type.
 Chapter 12. The role of the critical point
 The Schwarzian derivative, critical point and basins of attraction

 HW #6
 Ch.11, Page 151: 2,4,6,8,10,11,15
 due Monday, November 10th
(note change of the date)

Nov 10

  Chapters 12.The role of the critical point.
  The Schwarzian derivative, critical point.
 Overview of one dimensional dynamics. Zero/full measure sets, examples This part is not in the book

  Further reading: This survey article of Mikhail Lyubich describes in more details some of the main results in 1-dimensional dynamics: The Quadratic Family as a Qualitatively Solvable Model of Chaos (it is quite hard to understand, and uses many terms which are not familiar to you. If I find something more accesible I will post it here.)

  no homework this week

Nov 17

  Chapters 14. Fractals.
 Cantor set, Sierpinski Triangle, Koch snowflake
 Iterated Function Systems, Fractal dymension
 Check out this beautiful list of fractals.
  Further reading about fractals: Hausdorff and Minkowski dimension.
This is a more advanced exposition of the dimension concepts I introduced in the class. There are many examples and problems.

 HW #7
 Ch.12, Page 161: 1-8
 due Monday, November 17th