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
Textbook
S.H. Strogatz, Nonlinear Dynamics and Chaos, Perseus Books, 2000. ISBN 0738204536
(Available at The
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
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 |
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Topics |
Homework Due |
Notes |
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Aug 29 |
1.1-3 |
Introduction, Overview |
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Aug 31 |
2.0-4 |
Geometric viewpoint, Fixed points and stability Linear stability analysis |
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Sep 2 |
2.5-8 |
Existence and uniqueness, Oscillations, Potentials |
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Sep 5 |
|
Labor Day, NO CLASS |
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Sep 7 |
3.0-2 |
Saddle-node bifurcation |
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Sep 9 |
3.2 |
Transcritical bifurcation |
2.2.5, 2.2.8, 2.3.3, 2.4.4, 2.4.8, 2.7.5 |
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Sep 12 |
3.2 |
Transcritical bifurcation cont’d |
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Sep 14 |
3.4-5 |
Supercitical pitchfork bifurcation |
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Sep 16 |
|
Computer Lab, Benedum 1077 |
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Sep 19 |
3.4-5 |
Subcritical pitchfork bifurcation |
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Sep 21 |
3.6 |
Imperfect Bifurcations, Catastrophes |
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Sep 23 |
|
Computer Lab, Benedum 1077 |
3.1.1, 3.2.4, 3.4.2 3.4.14 |
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Sep 26 |
3.6 |
Imperfect Bifurcations, Catastrophes |
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Sep 28 |
3.7 |
Insect outbreak |
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Sep 30 |
|
Computer Lab, Benedum 1077 |
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Oct 3 |
5.0-1 |
Two-dimensional flows |
3.6.2, 3.7.3, 3.7.4 |
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Oct 5 |
5.2 |
Linear systems |
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Oct 7 |
5.2 |
Linear systems |
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Oct 10 |
|
Review |
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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 |
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Oct 14 |
5.2-6.1 |
Linear systems, Stability, Phase plane |
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Oct 17 |
5.2-6.1 |
Linear systems, Stability, Phase plane |
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Oct 19 |
6.2-3 |
Existence, Uniqueness, Topological consequences, Fixed Points and Linearization |
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Oct 21 |
6.4-5 |
Rabbits and sheep, |
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Oct 24 |
6.6 |
Conservative Systems |
5.2.1, 5.2.3, 5.2.4, 5.2.9, 5.3.4 |
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Oct 26 |
6.7 |
Reversible Systems |
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Oct 28 |
|
Computer Lab, Benedum 1077 |
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Oct 31 |
6.6-7 |
Conservative Systems, Reversible Systems |
6.3.1, 6.3.6, 6.3.13, 6.4.4 |
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Nov 2 |
6.8 |
Index theory |
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Nov 4 |
7.0-1 |
Limit cycles |
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Nov 7 |
7.2 |
Gradient systems |
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Nov 9 |
7.2 |
Liapunov function |
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Nov 11 |
7. 3 |
Poincare-Bendixson theorem |
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Nov 14 |
7.4-6 |
Relaxation oscillators |
6.5.7, 6.5.14a), 6.6.1, 6.8.7 |
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Nov 16 |
7.5 |
Relaxation oscillators |
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Nov 18 |
7.6 |
Weakly nonlinear oscillations |
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Nov 21 |
8.1 |
Bifurcations in 2D |
7.2.7, 7.3.3, 7.2.12 |
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Nov 23 |
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NO CLASS - Thanksgiving |
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Nov 25 |
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NO CLASS - Thanksgiving |
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Nov 28 |
8.2 |
Hopf bifurcation |
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Nov 30 |
8.3 |
Chemical reactions |
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Dec 2 |
8.4 |
Global bifurcations of cycles |
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Dec 5 |
9.0-9.2 |
Lorentz equations |
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Dec 7 |
10.0-1 |
One-dimensional maps, Logistic map |
7.6.5, 8.1.6, 8.2.5 (prove Hopf bifurcation analytically) |
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Dec 9 |
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Review |
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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) |
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Dec 16 |
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Projects due 4:00 pm |
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