MAT 351

Newton's Method for z^6-1 created with FractalAsm

Math 351
Differential Equations: Dynamical Systems and Chaos

Professor Brakalova, Spring 2003

Parameter Plane for Newton's method for z*(z-1)*(z-c) createde with FractalAsm

Meeting Time: Mon, Wed, Fri 10:30-11:25, Math Tower, room 5-127 (fifth floor)

This page will be regularly updated. If you are taking this course you should check weekly for any changes.

A description of the course, including grading policies, text, etc.

Welcome to the Dynamical Systems and Chaos web page (Mat351/Spring 03) at SUNY, Stony Brook
The grades for the course are available here .
Everybody should try to be present at the class meetings during the week of May 5th

Presentations on fractals and IFS--May 5th

Presentations on Newton's method for complex functions and the geometry of the M set--May 7th

Presentations on the Feigenbaum's constant and two dimensional dynamics--May 9th

The following links will be used throughout the semester:
FractalAsm web page at Cornell
and Dynamical Systems web page at Boston University

Suggested problems to work on are available here (BU webpage)
A module on the Mandelbrot set and external rays taught at the WISE'03 program is available here (pdf)

Handout on Intro to Iterations is a available here.

Handout on the Doubling and tent Function is a available here.

Handout on Conjugacy and Semiconjugacy is a available here.

Project 1 is available here.

Project 2 (due march 14th) is available here.

Project 3 (due April 30th) will be/is available here. In problem 7 of the project, please prove that S(T(F(z))=S(F(z)), instead of S(M(F(z))=0.

Final project topics, some refernces and assignments are available here.The presentations will start Monda, May 5th. The final project is due Monday, May 19th.

The Midterm Exam is April 7th and 9th.

Week Chapters Topics

1/22 Ch. 1-3 Introduction. Discrete Dynamical Systems. Iterations. Orbits.

1/27 Ch. 4 and 5 Graphical Analysis. Calculus of Fixed and Periodic Points.

2/3 Ch. 4 and 5 Graphical Analysis. Calculus of Fixed and Periodic Points.

2/10 Ch. 6 and 7 Bifurcation and Transition to Chaos

2/17 Ch. 6 and 7 Bifurcation and Transition to Chaos

2/24 Ch. 8 Transition to Chaos.

3/3 Ch. 8 and 9 Transition to Chaos. Symbolic Dynamics

3/10 Ch. 9 and10 Symbolic Dynamics. Chaotic Dynamical Systems.

3/17 Enjoy! Spring Break

3/24 Ch. 11 Sharkovskii's theorem. Period Three Implies Chaos

3/31 Ch. 12 The role of the Critical Orbit. The Schwarzian Derivative

4/7 Ch. 14 Complex Functions.

to be continued...
Homework

date due problems

2/3
p.27, 7d and 11, p. 28 # 15-19, p. 34 #1 c,g and #2 a, p.35 #6, p. 50 #1a,e,g,h,i and #2 a, e

2/21 p. 50 # 3 and 4c,e, p. 67 # 1 a-f

3/10 p. 80 # 1-8

3/28 p. 111-112 # 1-9, p. 130-131 # 1,3,4,6,20,21,22,23

Week	Chapters	Topics
1/22	Ch. 1-3	Introduction. Discrete Dynamical Systems. Iterations. Orbits.
1/27	Ch. 4 and 5	Graphical Analysis. Calculus of Fixed and Periodic Points.
2/3	Ch. 4 and 5	Graphical Analysis. Calculus of Fixed and Periodic Points.
2/10	Ch. 6 and 7	Bifurcation and Transition to Chaos
2/17	Ch. 6 and 7	Bifurcation and Transition to Chaos
2/24	Ch. 8	Transition to Chaos.
3/3	Ch. 8 and 9	Transition to Chaos. Symbolic Dynamics
3/10	Ch. 9 and10	Symbolic Dynamics. Chaotic Dynamical Systems.
3/17	Enjoy! Spring Break
3/24	Ch. 11	Sharkovskii's theorem. Period Three Implies Chaos
3/31	Ch. 12	The role of the Critical Orbit. The Schwarzian Derivative
4/7	Ch. 14	Complex Functions.

date due	problems
Homework
2/3	p.27, 7d and 11, p. 28 # 15-19, p. 34 #1 c,g and #2 a, p.35 #6, p. 50 #1a,e,g,h,i and #2 a, e
2/21	p. 50 # 3 and 4c,e, p. 67 # 1 a-f
3/10	p. 80 # 1-8
3/28	p. 111-112 # 1-9, p. 130-131 # 1,3,4,6,20,21,22,23