Math 1280: Ordinary Differential Equations II Spring 2007
Instructor: David Swigon
Office: Thackeray 519, 412-624-4689, swigon@pitt.edu
Lectures:
MWF 10:00-10:50am, Thackeray 627
Office
Hours: MW 1-3pm, Thackeray 519, or by
appointment.
Course
Web Page: (check frequently for changes and updates) http://www.math.pitt.edu/~swigon/math1280.html
Course
Description
This course
is continuation of the Fall semester course in
ordinary differential equations, Math 1270.
The focus will be on the mathematics of nonlinear dynamical systems. By using concrete problems from
physics, biology, chemistry, and engineering, the course will illustrate such
concepts as equilibrium and stability, bifurcation, limit cycles, and
chaos. You will also learn important analytical techniques such as
linearization, phase plane analysis, and dimensional analysis. Advanced
topics, treated in an elementary way, will be hysteresis, coupled oscillators, Hopf bifurcations, and strange attractors. Applications
will include mechanical vibrations, dynamics of interacting populations,
biological rythms, and lasers.
Prerequisites
Single-variable
calculus (Math 0220, 0230, or equivalent) including
Textbook
S.H. Strogatz, Nonlinear Dynamics and
Chaos, Perseus Books, 2000. ISBN 0738204536
(Available
at The
Grading
Scheme
Homework
& Computer Lab Assignments: 25%
Project: 15%
Midterm
exam: 25%
Final
Exam: 35%
Schedule
Each week
we will cover approximately one chapter of the book. Homework is due at
the beginning of a class one week after it was assigned. You may work with
other students on homework but do not submit solutions that are identical
copies of each other, as such will be discarded. You may use a computer
program (e.g., MATLAB, Maple, Mathematica)
in solving the homework unless I specifically say otherwise. Each such use
should be explicitly documented (e.g., eigenvalues computed with MATLAB).
Projects
Term projects
will be assigned about one month into the semester and will be one week before
the final exam. The students will form teams of 2-4 members and work on a
problem of their choice (within the guidelines to be given later with the
consent of the instructor). The final
report is due before the last lecture, where the results will be presented by
one or more of team members.
Computer
Assignments
Computer assignments will require you to use a computing equipment.
You can use your own computer or any of the campus computing labs in
Alumni Hall (on Linux machines) and Benedum Hall (on
UNIX workstations). You can also remotely login to the UNIX timesharing server
unixs.cis.pitt.edu.
Instructions
for the exercises will use MATLAB, a programming environment that is widely
used both in academia and in industry. Its advantage is in that it
requires only a little programming overhead because it contains built in
subroutines for matrix manipulation, eigenvalue analysis, phase plane analysis,
and numerical solution of differential equations. I will provide you with
handouts outlining the goals and instructions for each lab.
You are
free to use any other software, such as XPP/XPPAUT, Mathematica,
or Maple.
Syllabus
|
Week |
|
Topics |
Homework |
Notes |
|
Jan
3 Jan 5 |
1.1-3 2.0-4 |
Introduction,
Overview Geometric
viewpoint, Fixed points and stability Linear stability analysis |
2.2.2,
2.2.4, 2.2.8, 2.4.4, 2.4.5, 2.4.8 |
|
|
Jan
8 Jan 12 |
2.5-8 3.0-2 |
Existence
and uniqueness, Oscillations, Potentials Saddle-node
bifurcation |
Fri,
Jan 12, 10-11am Benedum
1077 |
Make
sure you know your login & password to pitt
account |
|
M.L.
King holiday Jan
17 Jan 19 |
3.2-5 |
Transcritical bifurcation, pitchfork bifurcations |
3.1.4,
3.1.4, 3.2.4, 3.2.6 |
|
|
Jan
22 Jan 26 |
3.6-7 |
Imperfect
bifurcations; catastrophes |
3.4.2,
3.4.6, 3.4.11, 3.4.14, 3.5.7 |
|
|
Jan
29 Feb 2 |
4.0-3 |
Flows
on a circle, Oscillators |
Fri,
Feb 2, 10-11am Benedum
1077 |
|
|
Feb
5 Feb 9 |
4.4-6 |
Synchronization |
3.6.2,
3.6.3, 3.7.4, 4.3.3, 4.3.4 |
|
|
Feb
12 Feb 16 |
5.0-3 |
Two-dimensional
flows, linear systems, stability |
5.1.10
a,c,d |
|
|
Feb
19 Feb 23 |
6.1-3 |
Phase
plane, Existence, Uniqueness, Topological consequences, Fixed Points and
Linearization |
|
|
|
Feb
26 Feb 28 |
|
Review |
|
|
|
Mar
2 |
|
Midterm Exam |
|
|
|
Mar
5 Mar 9 |
|
Spring Break |
|
|
|
Mar
12 Mar 16 |
6.4-5 |
Competitive-cooperative
systems, Conservative systems, |
6.1.4, 6.1.6, 6.3.6, 6.3.10, 6.4.2, 6.5.2 |
|
|
Mar
19 Mar 23 |
6.6-8 |
Reversible
systems, Index theory |
6.5.14, 6.5.19, 6.6.1, 6.6.7 (use
for Matlab version R12) (use
for Matlab version R14, save as pplane6.m) |
Fri,
Mar 23, 10-11am Benedum
1077 |
|
Mar
26 Mar 30 |
7.0-2 |
Limit
cycles; Gradient systems; Liapunov function |
6.7.3,
7.1.8, 7.2.3, 7.2.9ab |
Fri,
Mar 30, 10-11am Benedum
1077 |
|
Apr
2 Apr 6 |
7.3-6 |
Poincare-Bendixson theorem; Relaxation oscillators; Weakly
nonlinear oscillations |
7.3.1,
6.8.7, 7.3.3, 7.3.7, 7.5.4, 7.6.5 |
|
|
Apr
9 Apr 13 |
8.1-3 |
Bifurcations
in 2D; Hopf bifurcation |
|
|
|
Apr
16 Apr 20 |
8.4 |
Global
bifurcations |
|
|
|
Apr
20 |
|
Project presentations |
|
|
|
Wednesday Apr 25 2pm 4pm |
|
Final Exam |
|
|