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University of Wisconsin-Madison
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Math 415
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Fall 2009 |
About the course: Math 415 is an introduction to nonlinear dynamical systems including stability,
bifurcations and chaos. The course will give underlying mathematical ideas, and
will also emphasize applications from many scientific fields.
Course Content:
1. One-dimensional maps and difference equations: linear and nonlinear
problems, graphical solutions, bifurcations, chaos.
2. First-order differential equations (one-dimensional flows): linear and
nonlinear equations, graphical solutions, bifurcations.
3. Two-dimensional flows: phase plane, stability of fixed points, periodic
solutions, and limit cycles. Introduction to bifurcation theory, local and
global bifurcations. Tools for studying global behavior of flows: Lyapunov
functions, Poincare-Bendixson Theorem, gradient flows.
4. Three-dimensional flows: Lyapunov exponents, Poincare sections, strange
attractors, chaos.
Lectures:
There will be three 50 minute lectures per week: MWF 8:50-9:40 in room B123 Van
Vleck.
Text: The
textbook for the course is Nonlinear Dynamics and Chaos, by Stephen Strogatz.
Office hours: MWF 8:30am - 8:50am, 9:40am-10:10am, and by
appointment. For appointments, please talk to me before or after class.
Grading: The
grade will be based on homework and the exams as follows: Homework, Quizzes,
and Class Participation: 25% Midterms: 2x25% Final Exam: 25%.
Homework:
Homework will be due weekly on Monday, before the lecture.
Quizzes:
There will be weekly quizzes in the lecture room, usually on Mondays. Your
lowest two quiz scores will be discarded; there will be no make-up quizzes.
Exams: There
will be two 50-minute midterm exams (Midterm 1 on Monday Oct 12, and Midterm 2 will be
on Wednesday Nov 18) and a 2-hour final exam on Dec 22 from 12:25 to 2:25pm (final exam
room: B123
Van Vleck). No make-up midterm exams will be given, unless proof of
extraordinary circumstance is provided at least two weeks before exam day.
According to University policy, no early final exams will be given for any
reason.
Homework assignments:
(The “*” denotes the exercises I think are
most important.)
Weeks 1-2 (due Monday Sept 14 just before the
lecture):
1. Write up and hand in exercises
2.1.2, 2.1.3*, 2.1.5, 2.2.1, 2.2.3, 2.2.5, 2.2.7*, 2.2.8*, 2.2.9*, 2.2.10*,
2.2.13, 2.3.2*, 2.3.3*, 2.3.4, 2.4.2, 2.4.5, 2.4.6, 2.4.8*, 2.5.1*, 2.5.2*,
2.5.3*, 2.5.4, 2.5.6, 2.6.1*, 2.6.2 (on pages 36-41 in the textbook). Also,
please read and make sure you would be able to solve all the other exercises in
these sections. Some exercises have answers at the end of the book; compare
them with your answers.
2. Reading: please read sections 1.0-1.3 and 2.0-2.8 in the textbook.
Week 3 (due Monday Sept 21 just before the lecture):
1. Write up and hand in exercises 2.7.1,
2.7.2, 2.7.3, 2.7.4, 2.7.6*, 2.8.1*, 2.8.2*, 3.1.1*, 3.1.3*, 3.1.4, 3.1.5 (on
pages 42,79,80 in the textbook). Also, please read and make sure you would be
able to solve all the other exercises in sections 2.7 and 3.1. Some exercises
have answers at the end of the book; compare them with your answers.
Bonus problems:
if you have experience with programming, try to solve some of: 2.8.3, 2.8.4,
2.8.5, 2.8.6, 2.8.7. Note that 2.8.6, 2.8.7 have partial answers at the end of
the book.
2. Reading: please read sections 3.0-3.7 in
the textbook.
Week 4 (due Monday Sept 28 just before the lecture):
1. Write up and hand in exercises 3.2.1,
3.2.2*, 3.3.1*, 3.3.2*, 3.4.1, 3.4.11*, 3.4.16*. Also, make sure that you know
how to use the software dfield, pplane, and 3D-XplorMath.
2. Reading: please read sections 5.0-5.3 in
the textbook.
Week 5 (due Monday Oct 5 just before the lecture):
1. Write up and hand in exercises 5.1.1*,
5.1.2*, 5.1.5, 5.1.7, 5.1.9*, 5.2.1*, 5.2.2, on pages 140-143.
2. Reading: please read sections 5.3, 6.0-6.3
in the textbook.
Week 8 (due Wednesday Oct 28 just before the lecture):
1. Write up and hand in exercises 6.1.1*,
6.1.3, 6.1.7*, 6.2.2, 6.3.1*, 6.3.5, 6.3.10*, 6.4.1, 6.4.4*, 6.4.6*, on pages
181-185. (Hint: use PPLANE to check some of your answers.)
2. Reading: sections 6.1-6.5 and 6.7 in the
textbook.
Week 9 (due Monday Nov 2 just before the lecture):
1. Write up and hand in exercises 6.5.1*,
6.5.4, 6.5.6*, 6.5.7, 6.7.1, 6.7.2*, on pages 181-192. (hints: you can use
PPLANE to check some of your answers; in 6.5.6 you can eliminate one parameter
by changing time units.)
2. Reading: sections 7.1-7.5 in the textbook.
Week 10 (due Wednesday Nov 11 just before the
lecture):
1. Write up and hand in exercises 7.1.3*,
7.1.5, 7.2.2, 7.2.12*, 7.2.13*, 7.3.1*, 7.3.3, 7.4.2*, 7.5.1, 7.5.2*, 7.6.2,
7.6.18*.
2. Reading: sections 8.1-8.5 in the textbook.
Week 11 (due Monday Nov 16 just before the lecture):
1. Write up and hand in exercises 8.1.3*,
8.1.6*, 8.1.13, 8.2.1*, 8.2.3, 8.2.8*.
2. Reading: review for midterm exam 2.