Math 2921
*** ORDINARY DIFFERENTIAL EQUATIONS II *** Spring 2011
Instructor: David
Swigon
Office: Thackeray 511, 4126244689, swigon@pitt.edu
Lectures:
MWF 1:001:50pm, Thackeray 524
Office
Hours: MWF 2:003:00pm or by
appointment.
Course
Web Page: (check frequently for changes and updates) http://www.math.pitt.edu/~swigon/math2921.html
Course
Description
Math 2921
is a continuation of MATH 2920 as an introduction to the area of nonlinear
dynamical systems with an exposition of advanced techniques useful for
applications. The topics covered will include perturbations of nonhyperbolic
linear systems, center manifold reductions, bifurcation theory, method of
averaging, fastslow decomposition, Melnikov's method, and an introduction to
chaos.
Prerequisites
Introductory
graduate course in ordinary differential equations (MATH 2920 or equivalent).
Textbook
Stephen
Wiggins, Introduction to Applied
Nonlinear Dynamical Systems and Chaos, Springer, 2003.
Lawrence
Perko, Differential equations and
dynamical systems, Springer, 2001
John
Guckenheimer & Philip Holmes,
Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector
Fields, Springer 2002.
Further
reading
Yuri
Kuznetsov, Elements of Applied
Bifurcation Theory, Springer, 2004
Jack K.
Hale, Ordinary Differential equations,
Dover, 2009
Philip
Hartman, Ordinary Differential equations,
SIAM, 2002
M. W.
Hirsch, S. Smale, and R. L. Devaney, Differential
Equations, Dynamical Systems, and an Introduction to Chaos,
Elsevier/Academic Press, 2004
Grading
Scheme
Homework: 30%
Midterm
Exam: 30%
Final
Exam (cumulative): 40%
Schedule
Tentative
syllabus is given below. Homework assignments will be posted online and are due
one week after they are assigned.
Disability Resource Services
If
you have a disability for which you are or may be requesting an accommodation,
you are encouraged to contact both your instructor and Disability Resources and
Services, 140 William Pitt Union, 4126487890 or 4123837355 (TTY) as early
as possible in the term. DRS will verify your disability and determine
reasonable accommodations for this course.
Academic Integrity
Cheating/plagiarism will not be tolerated. Students
suspected of violating the University of Pittsburgh Policy on Academic
Integrity will incur a minimum sanction of a zero score for the quiz, exam or
paper in question. Additional sanctions may be imposed, depending on the
severity of the infraction. On homework,
you may work with other students or use library resources, but each student
must write up his or her solutions independently. Copying solutions from other
students will be considered cheating, and handled accordingly.
Syllabus
Week 
Reading 
Topics 
HW 
Jan
5  Jan 7 
Wiggins,
Chap. 18 Perko,
Chap 2.12 
Center
manifolds, Computation and properties of center manifolds 
Due
Jan 14 
Jan
10  Jan 14 
Wiggins,
Chap. 19, 12.1 Perko,
Chap. 2.13 
Normal
forms for vector fields with/without parameters, Structural
stability 
Due Jan 21 
Jan
19  Jan 21 
Wiggins,
Chap. 20.1 Perko,
Chap 4.2 
Bifurcations
of nonhyperbolic fixed points 

Jan
24  Jan 28 
Wiggins,
Chap. 20.23 Perko,
Chap 4.45, 7 
PoincareAndronovHopf
bifurcation Stability
and bifurcations of periodic orbits 
Due Jan 31 
Jan
31  Feb 4 
Wiggins,
Chap. 33, 10.3 Perko,
Chap. 4.8 
Homoclinic
bifurcation, Heteroclinic bifurcation SNIC


Feb
7  Feb 11 
Wiggins,
Chap. 20.4 
Stability
of bifurcations Codimension
of a bifurcation 
Due Feb 16 
Feb
14  Feb 18 
Wiggins,
Chap. 20.67 
TakensBogdanov
bifurcation Hopfsteady
state bifurcation 

Feb
21 

Review 

Feb 23 

Midterm Exam 

Feb
25 
Wiggins,
Chap. 29 
Logistic
map, Liapunov exponent 

Feb
28  Mar 4 
Wiggins,
Chap. 23, 24 
Smale
horseshoe, symbolic dynamics 

Mar 7  Mar 11 

SPRING BREAK 

Mar
14  Mar 18 
Wiggins,
Chap. 25 
ConleyMoser
conditions 
Due Mar 28 
Mar
21  Mar 25 
Wiggins,
Chap. 25 
ConleyMoser
conditions 

Mar
28  Apr 1 
Guckenheimer
& Holmes, Chap. 4.14.4 
Method
of averaging, Behavior near periodic orbit 

Apr
4  Apr 8 
Guckenheimer
& Holmes, Chap. 4.6 
Melnikov’s
method  homoclinic and subharmonic orbits 
Due Apr 22 
Apr
11  Apr 15 
Guckenheimer
& Holmes, Chap. 5.45.7 
Strange
Attractors, Lorenz Attractor 

Apr
18  Apr 22 
Dynamical
systems on manifolds 


Apr 27 

FINAL EXAM 
