Math 404 – Intermediate (Applied)
Differential Equations (Winter 2008)
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Time and Location: |
TuTh 8:30-10, 1372 East Hall. |
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Instructor: |
Richard Kollár, East Hall
1859, phone: (734) 647 -
5368. |
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E-mail: |
kollar....umich.edu (the preferred way
of communication). |
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Office Hours: |
Tue 10-12, Wed 2-3, or by appointment. |
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Textbook (required): |
Nonlinear Dynamics and Chaos by Steven H. Strogatz, Westview Press, 1994, 2000 |
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Additional Material: |
Online Video Material (S. H.
Strogatz): http://dspace.library.cornell.edu/handle/1813/97 |
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Recommended Literature: |
Dynamics: The
Geometry of Behavior,
R.H.Abraham, C.D. Shaw, Aerial Press (2nd print or electronic edition) |
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Advanced Mathematical Methods for
Scientists and Engineers by Carl
M. Bender and Steven A. Orszag, McGraw-Hill, 1978, |
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Introduction to
Applied Nonlinear Dynamical Systems and Chaos, S. Wiggins, Springer |
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Nonlinear
Oscillations, Dynamical systems, and Bifurcations of Vector Fields by J. Guckenheimer and P.
Holmes, Springer. |
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Nonlinear Systems, P.G. Drazin, Cambridge
Texts in Applied Mathematics, |
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Syllabus: |
This course is an introduction to the
modern qualitative theory of ordinary differential equations with emphasis on
geometric concepts and techniques and visualization. Much of the motivation
for this approach comes from various applications to Physics, Biology,
Chemistry and Engineering. Examples of applications of differential equations
are a significant part of the course. There are relatively few proofs. |
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Topic Covered: |
The course covers the first eight
chapters of the textbook: one and two-dimensional dynamical systems,
geometric representation of solutions, fixed points, periodic orbits and
limit cycles, linear stability analysis, bifurcations, oscillations in
physical and biological systems, and elementary numerical methods (in
MATLAB). If time permits, the next topic will be chaotic dynamical systems
(Chapter 9). |
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Grading Policy: |
Grades will be based on homework and
exams with the following weights:
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Homework Assignments: |
Homework assignments will be regularly
posted on the homework page. |
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Lesson plan: |
The topics covered will be regularly
updated: |
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Jan 3 |
Thu |
1.1 – 1.3 Introduction to dynamical
systems. Why are they interesting? |
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Jan 8 |
Tue |
ODE/Calculus/Algebra review, Dynamical
systems movies. |
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Jan 10 |
Thu |
2.1 – 2.6 Flows on the line: Fixed
Points, Stability, Population Growth, Existence and Uniqueness |
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Jan 15 |
Tue |
2.7 – 2.8 Potential Flows, Numerical Methods
(MATLAB), 3.1 Saddle-node Bifurcation |
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Jan 17 |
Thu |
3.2 – 3.4 Bifurcations, Due
Homework #1 |
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Jan 22 |
Tue |
3.5 – 3.6 Dimensional Analysis,
Scaling |
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Jan 24 |
Thu |
3.7 Catastrophes, Due Homework #2 |
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Jan 29 |
Tue |
3.7 Insect Outbreak, Slow and Fast
Time Scales |
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Jan 31 |
Thu |
4.1 – 4.5 Flows on the Circle, Due
Homework #3 |
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Feb 5 |
Tue |
5.1 – 5.2 2D Linear Systems |
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Feb 7 |
Thu |
5.2 – 5.3 2D Linear Systems, Love
Affairs Model, Due Homework #4 |
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Feb 12 |
Tue |
6.1 – 6.3 2D Nonlinear Systems,
Existence, Uniqueness, Linearization |
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Feb 14 |
Thu |
6.4 – 6.6 Lotka-Volterra Model,
Conservative Systems, Due Homework #5 |
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Feb 19 |
Tue |
Midterm Exam (in-class) |
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Feb 21 |
Thu |
6.6 – 6.7 Reversible System, Pendulum |
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Feb 26 |
Tue |
No class – Spring Break |
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Feb 28 |
Thu |
No class – Spring Break |
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Mar 4 |
Tue |
6.8 Index Theory |
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Mar 6 |
Thu |
7.1 – 7.2 Limit Cycles, Movies of
Mechanical and Chemical Oscillations, Due Homework #6 |
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Mar 11 |
Tue |
7.2 – 7.3 Lyapunov Function, Dulac’s
Criterion |
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Mar 13 |
Thu |
7.3 – 7.4 Poincare-Bendixson
Theorem, Due Homework #7 |
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Mar 18 |
Tue |
7.5 Relaxation Oscillations |
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Mar 20 |
Thu |
7.6 Weakly Nonlinear Oscillators,
Regular Perturbation Theory, Due Homework #8 |
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Mar 25 |
Tue |
8.1 – 8.2 2D Bifurcations |
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Mar 27 |
Thu |
8.2 Hopf Bifurcations, Due Homework
#9 |
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Apr 1 |
Tue |
8.3 Chemical Oscillations |
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Apr 3 |
Thu |
8.4, 8.6 Global Bifurcations of
Cycles, Coupled Oscillators, Quasi-periodicity, Due Homework #10 |
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Apr 8 |
Tue |
9.1 Chaos, Lorenz Equations, Chaotic
Waterwheel |
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Apr 10 |
Thu |
9.2 – 9.3 Properties of Lorenz
Equations, Strange Attractors, Due Homework #11 |
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Apr 15 |
Tue |
9.3 Strange Attractors |
Page Updated: 04/22/2008