Math 2920 *** ORDINARY DIFFERENTIAL EQUATIONS I *** Fall 2017

Instructor: David Swigon

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

Lectures:  TuTh 4:00-5:15pm, Thackeray 524

Office Hours: TuTh 2:00-3:00pm or by appointment.

Course Web Page: (check frequently for changes and updates) http://www.math.pitt.edu/~swigon/math2920.html

Course Description

The course will develop a basic theory of ordinary differential equations including the existence & uniqueness of solutions of the initial value problem, dependence on initial conditions, theory of linear differential equations and systems, dynamical systems approach to autonomous differential equations local & global behavior, Hartman-Grobman theorem, analysis of planar dynamical systems, etc.

Prerequisites

Linear algebra, advanced calculus. Undergraduate ODE course is welcome but not necessary.

Textbook

Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, AMS 2012, ISBN-13: 978-0821883280
downloadable from http://www.mat.univie.ac.at/~gerald/ftp/book-ode/index.html
(we will not be using the Mathematica codes)

Other recommended books

Jack K. Hale, Ordinary Differential equations, Dover, 2009

Philip Hartman, Ordinary Differential equations, SIAM, 2002

Lawrence Perko, Differential equations and dynamical systems, Springer, 2001

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

Aug 29, Aug 31

Teschl, §1.2-1.5, 2.1

Review: Classification of ODEs,Methods for solving first order equations, Preliminaries

Sep 5, Sep 7

Teschl, §2.2-2.3, 2.7

Initial Value Problems

Existence & uniqueness of solutions, Peano’s theorem

Problems 2.3, 2.4, 2.5, 2.6 (ii) and (iii), 2.8, 2.10

Due Sep 14

Sep 12, Sep 14

Teschl, §2.4, 2.6

Dependence on initial conditions, Extensibility of solutions

Problems 2.14, 2.15, 2.19, 2.20

Due Sep 21

Sep 19, Sep 21

Teschl, §3.1-3.2

Linear Systems

Matrix exponential, autonomous systems and equations

Nineteen Dubious Ways to Compute the Exponential of a Matrix, 25 years later

 

Sep 26, Sep 28

Teschl, §3.3-3.4, 3.6

Wronskian, Variation of parameters, Periodic linear systems

Problems 3.5, 3.16, 3.27, 3.28, 3.40 + Prove Corollary 3.16

Due Oct 5

Oct 3, Oct 5

Teschl, §6.1-6.3

Dynamical Systems

Flow, vector field, integral curves

 

Oct 10

Oct 12

 

Teschl, §6.4-6.5

 

FALL BREAK (Monday schedule)

Orbits, Poincare map, Liapunov and asymptotic stability

Problems 6.1, 6.4, 6.5, 6.7, 6.9

Due Oct 19

Oct 17, Oct 19

Teschl, §6.6

Liapunov’s method, La Salle principle

 

Oct 24

MIDTERM EXAM

Midterm Exam Study Sheet

 

Oct 26

Teschl, §6.6

Liapunov’s method, La Salle principle

 

Oct 31, Nov 2

Teschl, §7.1-7.2

Planar dynamical systems

Lotka-Volterra systems, Lienard’s equation, Van der Pol’s equation

Homework 5

Due Nov 7

Nov 7, Nov 9

Teschl, §7.2-7.3

Jordan curve theorem, Poincare-Bendixson theorem

 

Nov 14, Nov 16

Teschl, §8.1-8.3

Higher-dimensional dynamical systems

Attracting sets, Lorentz equation

Homework 6

Due Nov 21

Nov 21

Teschl, §9.1-9.2

Local Behavior

Stable & unstable manifolds, Stable manifold theorem

 

Nov 28, Nov 30

Teschl, §9.3

Hartman Grobman theorem

Homework 7

Due Dec 11

Dec 5, Dec 7

Teschl, §9.3

topological conjugacy

 

 

FINAL EXAM

Second-Half Semester Study Sheet