Math 1360:      Modeling in Applied Math I        CRN 24970

Fall 2005

Instructor: David Swigon

Office: Thackeray 519, 412-624-4689, swigon@pitt.edu

Lectures:  MWF 10:00-10:50am, Benedum 920

Office Hours: MW 2:00-3:00pm, Thackeray 519, or by appointment.

Course Web Page: (check frequently for changes and updates) http://www.math.pitt.edu/~swigon/math1360.html

Course Description

This course is an introduction to mathematical modeling via the mathematics of nonlinear dynamical systems.  It will show, by examples, how important questions about the observed world can be framed in terms of linear and nonlinear differential equations.  By considering 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 rythms, and lasers. 

Prerequisites

Single-variable calculus (Math 0220, 0230, or equivalent) including Taylor series, multivariable calculus (Math 0240 or equivalent), linear algebra and familiarity with separable differential equations is recommended.  Techniques will be reviewed and developed where needed

Textbook

S.H. Strogatz, Nonlinear Dynamics and Chaos, Perseus Books, 2000. ISBN 0738204536

(Available at The Book Center on Fifth Ave.)

Grading Scheme

Homework Assignments: 25%

Project: 25%

Midterm exam: 20%

Final Exam: 30%

Schedule

Each week we will cover approximately one chapter of the book.  The precise schedule and a list of homework problems will be given out in advance and posted on the web.  Homework is due at the beginning of a class one week after it was assigned. 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 use a computer program (e.g., MATLAB, Maple, Mathematica) in solving the homework unless I specifically say otherwise. Each such use should be explicitly documented (e.g., “eigenvalues computed with MATLAB”).

Term projects will be assigned about one month into the semester and will be one week before the final exam.

Programming

I have reserved a computer lab at the Gardner Steel Conference Center (up the street from Thackeray Hall) that we will use about 6 times over the course of the semester for hands-on application of the concepts we will cover in the class.  We will be using MATLAB, a programming environment that is widely used both in academia and in industry.  Its advantage is in that it requires only a little programming overhead because it contains built in subroutines for matrix manipulation, eigenvalue analysis, phase plane analysis, and numerical solution of differential equations. I will provide you with handouts outlining the goals and instructions for each lab.

If you want to use Matlab on your own, it is available at the 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.

Matlab resources

Matlab summary

Matlab tutorial

Matlab documentation

Policy

Attendance is expected.  No late homework will be accepted and there will be no make up exams.

Note

This is a course in applied mathematics and hence emphasis will be placed on problem solving.  No proofs of theorems will be given but the book contains numerous references for those interested.  A very good and rigorous, yet understandable, text on nonlinear dynamics is: Guckenheimer & Holmes, Nonlinear oscillations, dynamical systems and bifurcation of vector fields, Springer, 1983.

Syllabus

Date

Reading

Topics

Homework Due

Notes

Aug 29

1.1-3

Introduction, Overview

 

 

Aug 31

2.0-4

Geometric viewpoint, Fixed points and stability Linear stability analysis

 

 

Sep 2

2.5-8

Existence and uniqueness, Oscillations, Potentials

 

 

Sep 5

 

Labor Day, NO CLASS

 

 

Sep 7

3.0-2

Saddle-node bifurcation

 

 

Sep 9

3.2

Transcritical bifurcation

2.2.5, 2.2.8, 2.3.3, 2.4.4, 2.4.8, 2.7.5

 

Sep 12

3.2

Transcritical bifurcation cont’d

 

 

Sep 14

3.4-5

Supercitical pitchfork bifurcation

 

 

Sep 16

 

Computer Lab, Benedum 1077

 

Lab Handout I

flow.m

Sep 19

3.4-5

Subcritical pitchfork bifurcation

 

 

Sep 21

3.6

Imperfect Bifurcations, Catastrophes

 

 

Sep 23

 

Computer Lab, Benedum 1077

3.1.1, 3.2.4, 3.4.2

3.4.14

Lab Handout II

euler.m

rungekutta45.m

Sep 26

3.6

Imperfect Bifurcations, Catastrophes

 

 

Sep 28

3.7

Insect outbreak

 

 

Sep 30

 

Computer Lab, Benedum 1077

 

pitchfork1.m

pitchfork2.m

Oct 3

5.0-1

Two-dimensional flows

3.6.2, 3.7.3, 3.7.4

 

Oct 5

5.2

Linear systems

 

 

Oct 7

5.2

Linear systems

 

 

Oct 10

 

Review

 

 

Oct 12

 

Midterm Exam

Covers Chapters 2 and 3

Sections 2.0-7, 3.0-2, 3.4, 3.6-7

Practice problems

All examples + exercises 2.2.1-2.2.7,  2.2.8,  2.2.10,  2.3.2-4,  2.4.1-8, 2.7.1-6,  3.1.1-4,  3.2.1-5, 3.4.1-3.4.11, 3.4.14-16, 3.5.2,  3.5.7, 3.6.2-5, 3.7.1-5

Oct 14

5.2-6.1

Linear systems, Stability, Phase plane

 

 

Oct 17

5.2-6.1

Linear systems, Stability, Phase plane

 

 

Oct 19

6.2-3

Existence, Uniqueness, Topological consequences, Fixed Points and Linearization

 

 

Oct 21

6.4-5

Rabbits and sheep,

 

 

Oct 24

6.6

Conservative Systems

5.2.1, 5.2.3, 5.2.4, 5.2.9, 5.3.4

 

Oct 26

6.7

Reversible Systems

 

 

Oct 28

 

Computer Lab, Benedum 1077

 

LabHandoutIII

pplane6.m

dfield6.m

Oct 31

6.6-7

Conservative Systems, Reversible Systems

6.3.1,  6.3.6,  6.3.13,  6.4.4

 

Nov 2

6.8

Index theory

 

 

Nov 4

7.0-1

Limit cycles

 

 

Nov 7

7.2

Gradient systems

 

 

Nov 9

7.2

Liapunov function

 

 

Nov 11

7. 3

Poincare-Bendixson theorem

 

 

Nov 14

7.4-6

Relaxation oscillators

6.5.7, 6.5.14a), 6.6.1, 6.8.7

 

Nov 16

7.5

Relaxation oscillators

 

 

Nov 18

7.6

Weakly nonlinear oscillations

 

 

Nov 21

8.1

Bifurcations in 2D

7.2.7, 7.3.3, 7.2.12

 

Nov 23

 

NO CLASS - Thanksgiving

 

 

Nov 25

 

NO CLASS - Thanksgiving

 

 

Nov 28

8.2

Hopf bifurcation

 

 

Nov 30

8.3

Chemical reactions

 

 

Dec 2

8.4

Global bifurcations of cycles

 

 

Dec 5

9.0-9.2

Lorentz equations

 

 

Dec 7

10.0-1

One-dimensional maps, Logistic map

7.6.5, 8.1.6, 8.2.5 (prove Hopf bifurcation analytically)

 

Dec 9

 

Review

 

 

Dec 12

 

FINAL EXAM

10:00-11:50 am,

BENEDUM 920

Practice problems

All examples + exercises 5.2.3-13, 5.3.1-6, 6.1.1-6, 6.3.1-6, 6.3.8 b), 6.3.9 a)-d), 6.4.1-3, 6.4.5-6, 6.5.1-6, 6.5.19, 6.6.1-3, 7.2.1-3, 7.2.9-10, 7.3.1, 7.3.3, 7.3.5-6, 7.5.4 a)-c), 7.6.4-9, 8.1.7, 8.1.9, 8.1.11, 8.1.13, 8.2.1, 8.2.5-7 (Show that Hopf occurs at mu=0), 8.3.1 a)-d), 8.3.2 a)-c)

Dec 16

 

Projects due 4:00 pm