Math 1280: Ordinary Differential Equations II Spring 2017
Instructor: David
Swigon
Office: Thackeray 511, 4126244689, swigon@pitt.edu
Lectures:
MWF 10:0010:50am, Thackeray 525
Office
Hours: MWF 1112:30pm, Thackeray 511,
or by appointment.
Grader: Youngmin
Park, Thackeray 520, yop6@pitt.edu
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 rhythms, and
lasers.
Prerequisites
Singlevariable
calculus (Math 0220, 0230, or equivalent) including
Textbook
S.H. Strogatz, Nonlinear Dynamics and Chaos, 2^{nd} edition, Westview Press (Perseus
Books), 2014.
Grading
Homework
Assignments: 25%
Midterm
exam: 30%
Final
Exam (cumulative): 45%
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, usually on Fridays.
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
not use a computer program (e.g., MATLAB, Maple, Mathematica) in solving the
homework unless I specifically say otherwise. You are allowed to use graphing
calculators both for solving homework and during exams.
Computer
Assignments
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 assignments use MATLAB, a programming environment that is widely used
both in academia and in industry. Its advantage is that it contains built
in subroutines for matrix manipulation, eigenvalue analysis, phase plane
analysis, and numerical solution of differential equations. MATLAB is installed
on the computers in Benedum computing lab, and
student licenses can be purchased for small fee from Computing Services. I will
provide you with handouts outlining the instructions and exercises for each
lab, which you will then complete at your own pace.
You are
free to use any other software, such as XPP/XPPAUT, Mathematica, or Maple.
Week 

Topics 
Homework
Due on Friday 
Jan
4  Jan 6 
1.13 2.04 
Introduction,
Overview Geometric
viewpoint, Fixed points and stability, Population Models, Linear stability
analysis 

Jan
9  Jan 13 
2.58 3.02 
Existence
and uniqueness, Oscillations, Potentials Saddlenode
bifurcation 
2.2.1, 2.2.3, 2.2.8, 2.3.3, 2.4.4, 2.4.5, 2.4.8 
M.L.
King holiday Jan
18  Jan 20 
3.27 
Transcritical bifurcation, pitchfork bifurcation Imperfect
bifurcations 
2.3.2, 2.7.5, 3.1.1, 3.1.4, 3.2.4, 3.2.5 
Jan
23  Jan 27 
3.7 4.03 
Insect
outbreak Flows
on a circle, Oscillators, 
3.4.2, 3.4.9, 3.4.10, 3.4.14, 3.5.8, 3.6.3, 3.7.5 
Jan
30  Feb 3 
4.5 5.03 
Synchronization
Twodimensional
flows, linear systems, stability 
4.1.3, 4.3.3, 4.3.5, 4.3.8, 4.5.1 
Feb
6  Feb 10 
6.03 
Phase
plane, Existence, Uniqueness, Topological consequences, Fixed Points and
Linearization 
5.1.9, 5.1.10 b,d,e, 5.2.4, 5.2.5, 5.2.9, 5.2.12, 5.3.2, 5.3.5 Due
Monday Feb 13 
Feb
13  Feb 17 
6.47 
Competitivecooperative
systems, Conservative systems, Reversible systems 
6.1.5, 6.1.6, 6.3.5, 6.3.6, 6.4.3, 6.4.5, 6.5.2, 6.5.14, 6.6.1 Due
Wednesday Feb 22 
Feb
20 Feb
22 Feb 24 
6.8 
Index
theory Review Midterm Exam 

Feb
27  Mar 3 
7.02 
Limit
cycles; Gradient systems; Liapunov function 

Mar
6  Mar 10 

Spring Break 

Mar
13  Mar 17 
7.3 
PoincareBendixson theorem; 
6.8.7, 6.8.8, 7.1.2, 7.1.3, 7.1.8, 7.2.3, 7.2.6b, 7.2.9a,c, 7.2.12 
Mar
20  Mar 24 
7.56 
Relaxation
oscillators; Weakly nonlinear oscillations 
7.3.1, 7.3.3, 7.3.4, 7.3.5, 7.5.4, 7.5.6 
Mar
27  Mar 31 
8.02 
Bifurcations
in 2D; 
7.5.5,
7.6.3ab, 7.6.5, 7.6.6, 7.6.18 (bonus challenge problem) 
Apr
3  Apr 7 
8.35 
Hopf
bifurcation, Forced pendulum 
8.1.1b, 8.1.6, 8.1.7, 8.1.11, 8.2.1, 8.2.8, 8.3.1 
Apr
10  Apr 14 
8.67,
9.0 
Coupled
Oscillators, Poincare map Lorentz
Equations, 

Apr
17  Apr 19 
9.25,
10.05 
Chaos,
Lorenz map, Logistic map VIDEOS: Chapter 1, Motion and
determinism 
8.4.1, 8.4.2,
8.5.3, 8.6.1, 8.6.4 (bonus
challenge problem) Due
Wednesday Apr 19 
Apr
21 
Review 


Monday, Apr 24 

Final Exam 12:00pm  1:50 pm Room 525 Thackeray 
