Math 1280:      Ordinary Differential Equations II        Spring 2017

Instructor: David Swigon

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

Lectures:  MWF 10:00-10:50am, Thackeray 525

Office Hours: MWF 11-12:30pm, 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

Single-variable calculus (Math 0220, 0230, or equivalent) including Taylor series, multivariable calculus (Math 0240 or equivalent), linear algebra, separable ordinary differential equations and systems of linear differential equations (Math 0290 or 1270)

Textbook

S.H. Strogatz, Nonlinear Dynamics and Chaos, 2nd 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.

Matlab primer (PDF)

Matlab tutorial

Matlab documentation

Week

Reading

Topics

Homework Due on Friday

Jan 4 - Jan 6

1.1-3

2.0-4

Introduction, Overview

Geometric viewpoint, Fixed points and stability, Population Models, Linear stability analysis

 

Jan 9 - Jan 13

2.5-8

3.0-2

Existence and uniqueness, Oscillations, Potentials

Saddle-node bifurcation

2.2.1, 2.2.3, 2.2.8, 2.3.3, 2.4.4, 2.4.5, 2.4.8

Solutions

M.L. King holiday

Jan 18 - Jan 20

3.2-7

Transcritical bifurcation, pitchfork bifurcation

Imperfect bifurcations

2.3.2, 2.7.5, 3.1.1, 3.1.4, 3.2.4, 3.2.5

Solutions

Computer Assignment I

flow.m

Jan 23 - Jan 27

3.7

4.0-3

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

Solutions

Jan 30 - Feb 3

4.5

5.0-3

Synchronization

Two-dimensional flows, linear systems, stability

4.1.3, 4.3.3, 4.3.5, 4.3.8, 4.5.1

Solutions

Computer Assignment II

bifur.m

Feb 6 - Feb 10

6.0-3

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

Solutions

Feb 13 - Feb 17

6.4-7

Competitive-cooperative 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

Solutions

Feb 20

Feb 22

Feb 24

6.8

Index theory

Review

Midterm Exam

Exam Review Sheet

Midterm solutions

 

Feb 27 - Mar 3

7.0-2

Limit cycles; Gradient systems; Liapunov function

 

Mar 6 - Mar 10

 

Spring Break

 

Mar 13 - Mar 17

7.3

Poincare-Bendixson 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

Solutions

Computer Assignment III

Mar 20 - Mar 24

7.5-6

Relaxation oscillators; Weakly nonlinear oscillations

7.3.1, 7.3.3, 7.3.4, 7.3.5, 7.5.4, 7.5.6

Solutions

Mar 27 - Mar 31

8.0-2

Bifurcations in 2D;

7.5.5, 7.6.3ab, 7.6.5, 7.6.6,

7.6.18 (bonus challenge problem)

Solutions

 

Apr 3 - Apr 7

8.3-5

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

Solutions

Apr 10 - Apr 14

8.6-7, 9.0

Coupled Oscillators, Poincare map

Lorentz Equations,

Computer Assignment IV

Apr 17 - Apr 19

9.2-5, 10.0-5

Chaos, Lorenz map, Logistic map

VIDEOS:

Chapter 1, Motion and determinism

Chapter 2, Vector fields

Chapter 3, Mechanics

Chapter 4, Oscillations

Chapter 5, Billiards

Chapter 6, Chaos and the horseshoe

Chapter 7, Strange attractors

Chapter 8, Statistics

Chapter 9, Chaotic or not?

8.4.1, 8.4.2, 8.5.3, 8.6.1,

8.6.4 (bonus challenge problem)

Due Wednesday Apr 19

Solutions

Apr 21

Review

Second-half Review Sheet

 

Monday, Apr 24

 

Final Exam

12:00pm - 1:50 pm

Room 525 Thackeray