Ordinary Differential Equations

Math 7320
Louisiana State University
Spring Semester, 2021

Prof. Stephen Shipman

Place: Lockett 243
Time: 12:00-1:20 Tuesday and Thursday

Office: Room 314 of Lockett Hall
Telephone: 225/578-1674
Email: shipman@lsu.edu
Office Hours: By Zoom: Monday and Thursday at 9:00 am, or by appointment. Please access the Zoom link from Moodle.

Course Description

This course will include the theories of linear and nonlinear ordinary differential equations. These are genuinely different areas of mathematics. Linear ODEs fall within the area of spectral theory, while nonlinear ODEs are the core of the study of dynamical systems.

The overarching concepts for the nonlinear theory are flows of vector fields and dynamical systems. Upon that basis, one studies diverse phenomena such as bifurcations, separation of time scales, bursting (such as in neurobiology), hysteresis, stability (this is the connection between linear and nonlinear), control systems, chaos, and strange attractors.

The overarching concepts in solving linear ODEs are (1) for initial-value problems: the matrix exponential and how it reflects the canonical forms of matrices and (2) for boundary-value problems: the solution operator, or Green function. I will introduce the spectral theory of linear differential operators and its intimate connection with complex analysis.

Certainly, a course cannot come close to doing justice to all of these topics, and neither can one person. My goal is twofold: (1) to present the foundational rigorous theory of ODEs, and (2) to introduce a broad variety of topics in ODEs that highlight what makes the field interesting.

Textbook

Prerequisite

Literature

There is a bibliography of relevant works below.

Assignments and Evaluation

Routine problems will be worth 70% of the grade. Assigned problem sets are listed below.

A set of problems at the end of the semester will take the place of the final exam. These will be worth 30% of the grade.

Students may discuss problems with each other and other people (including me, of course) and consult other literature; in fact students are encouraged to search the literature and discuss ideas. However, all work that is turned in must ultimately be that of the submitter alone. If a student receives aid on an assigned problem from discussions with people or other sources, he or she must begin from scratch in writing the solution so that the result is the product of his or her own understanding alone.

Lecture Notes

Mathematica file of ODE examples

Problem Sets

Final Exam

No collaboration or communication with any human being is allowed regarding any of the problems on the final exam, except that you may ask me questions for clarification. You must cite all references that you utilize in devising your solution.

Grading scale

A+: at least 95%	A: at least 90%	A-: at least 88%
B+: at least 85%	B: at least 80%	B-: at least 78%
C+: at least 75%	C: at least 70%	C-: at least 68%
D+: at least 65%	D: at least 60%	D-: at least 50%
F: less than 50%

Bibliography

1. Vladimir I. Arnold, Ordinary Differential Equations, Springer Verlag, 1984 (Russian), 1992 (English translation by Roger Cooke).
2. B. Malcolm Brown, Michael S. P. Eastham, and Karl Michael Schmidt, Periodic Differential Operators, Birkhäuser 2013.
3. Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955.
4. P. Deift and E. Trubowitz, Inverse Scattering on the Line, Communications on Pure and Applied Mathematics XXXII (1979) 121-251.
5. G. Freiling and V. Yurko, Inverse Sturm-Liouville Problems and Their Applications, Nova Science Publishers, 2001. (See Theorem 5.1.1.)
6. I. Gohberg, P. Lancaster, and L. Rodman, Indefinite Linear Algebra and Applications, Birkhäuser 2005.
7. Eugene M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of excitability and Bursting, MIT Press, Cambridge, MA, 2007.
8. Erich Müller-Pfeiffer, Spectral Theory of Ordinary Differential Operators, Ellis Horwood Ltd., 1981.
9. J. Pöschel and E. Trubowitz, Inverse Spectral Theory Academic Press, 1987.
10. F. G. Tricomi, Integral Equations, Interscience Publishers, 1957.
11. Stephen P. Shipman and Aaron T. Welters, Resonant electromagnetic scattering in anisotropic layered media, Journal of Mathematical Physics, Vol. 54, Issue 10 (2013) 103511:1-40.
12. Stephen P. Shipman and Aaron T. Welters, Pathological scattering by a defect in a slow-light periodic layered medium, J. Math. Phys. 57 (2016) 022902.
13. V. A. Yabukovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients I and II, Halsted Press 1975.