Math 2921 *** ORDINARY DIFFERENTIAL EQUATIONS II *** Spring 2011

Instructor: David Swigon

Office: Thackeray 511, 412-624-4689, swigon@pitt.edu

Lectures:  MWF 1:00-1:50pm, Thackeray 524

Office Hours: MWF 2:00-3:00pm 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 non-hyperbolic linear systems, center manifold reductions, bifurcation theory, method of averaging, fast-slow 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, 412-648-7890 or 412-383-7355 (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

Homework 1

Due Jan 14

Solutions

Jan 10 - Jan 14

Wiggins, Chap. 19, 12.1

Perko, Chap. 2.13

Normal forms for vector fields with/without parameters,

Structural stability

Homework 2

Due Jan 21

Solutions

Jan 19 - Jan 21

Wiggins, Chap. 20.1

Perko, Chap 4.2

Bifurcations of non-hyperbolic fixed points

 

Jan 24 - Jan 28

Wiggins, Chap. 20.2-3

Perko, Chap 4.4-5, 7

Poincare-Andronov-Hopf bifurcation

Stability and bifurcations of periodic orbits

Homework 3

Due Jan 31

Solutions

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

Homework 4

Due Feb 16

Solutions

Feb 14 - Feb 18

Wiggins, Chap. 20.6-7

Takens-Bogdanov bifurcation

Hopf-steady state bifurcation

 

Feb 21

 

Review

 

Feb 23

 

Midterm Exam

Solutions

 

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

Conley-Moser conditions

Homework 5

Due Mar 28

Solutions

Mar 21 - Mar 25

Wiggins, Chap. 25

Conley-Moser conditions

 

Mar 28 - Apr 1

Guckenheimer & Holmes, Chap. 4.1-4.4

Method of averaging, Behavior near periodic orbit

 

Apr 4 - Apr 8

Guckenheimer & Holmes, Chap. 4.6

Melnikovís method - homoclinic and subharmonic orbits

Homework 6

Due Apr 22

Solutions

Apr 11 - Apr 15

Guckenheimer & Holmes, Chap. 5.4-5.7

Strange Attractors, Lorenz Attractor

 

Apr 18 - Apr 22

Dynamical systems on manifolds

 

Apr 27

 

FINAL EXAM