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Ordinary Differential Equations
Louisiana State University
Spring Semester, 2018
Prof. Stephen Shipman
Place: Lockett 113
Time: 12:00-1:20 Tuesday and Thursday
Office: Room 314 of Lockett Hall
Office Hours: By appointment
Courses in ODEs are often perceived as boring, or at least tedious, because of the expected nature of some of the theorems and the technical nature of some of the proofs. My intention is that this course will deviate from that stereotype, as the field is in fact fascinating, beautiful, and pervasive in science and mathematics. As a foundation for the material, I will use the classic textbook by Vladimir I. Arnol'd, which is universally recognized as a mathematical gem. We will do both linear and nonlinear ODEs. These are genuinely different areas of mathematics, as the questions that are investigated are very different in nature.
The linear theory is sometimes considered part of linear algebra. The overarching concept is the matrix exponential and how its structure reflects canonical forms of linear operators. I will introduce the spectral theory of linear differential operators and its intimate connection with complex analysis. I also want to include some more specialized topics, such as linear systems governed by indefinite quadratic forms.
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.
Obviously, 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.
The primary text book for this course is "Ordinary Differential Equations" by Vladimir I. Arnold. The course will incorporate several topics that are not in that book. Supporting material will come from my notes and other literature.
The prerequisites are real and complex analysis.
There is a bibliography of relevant works below.
Assignments and Evaluation
Enrolled students will be required to do problems weekly from Arnold's book or other sources. 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.
General ideas of flows; Linear equations with constant coefficients
Normal Modes of Linear Equations
Maxwell ODEs for electromagnetics in layered media
Integral equations of Volterra type
Spectrum of ODEs 1
General existence and uniqueness for initial-value problems
Vector fields vs. direction fields
Dimension vs. complexity of flows
Dyamical Systems in Neuroscience. See the book of E. M. Izhikevich, referenced below, chapters 7-9.
See some simulations in my
Hamilton's ODEs. See the book of Evans, referenced below, section 3.3.
Problem Set 1: February 6. Final submission by March 15.
Problem Set 2: February 20. Final submission by March 20.
Problem Set 3: March 13. Final submission by April 3.
Problem Set 4: April 10. Final submission by April 17.
Problem Set 5: April 19. Final submission by April 26.
Final Exam: Monday, May 7, 9:00 AM.
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.
|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%
Vladimir I. Arnold,
Ordinary Differential Equations,
Springer Verlag, 1984 (Russian), 1992 (English translation by Roger Cooke).
B. Malcolm Brown, Michael S. P. Eastham, and Karl Michael Schmidt,
Periodic Differential Operators,
Earl A. Coddington and Norman Levinson,
Theory of Ordinary Differential Equations,
P. Deift and E. Trubowitz,
Inverse Scattering on the Line,
Communications on Pure and Applied Mathematics XXXII (1979) 121-251.
L. C. Evans,
Partial Differential Equations,
Graduate Studies in Mathematics, Volume 19.
G. Freiling and V. Yurko,
Inverse Sturm-Liouville Problems and Their Applications,
Nova Science Publishers, 2001. (See Theorem 5.1.1.)
I. Gohberg, P. Lancaster, and L. Rodman,
Indefinite Linear Algebra and Applications,
Eugene M. Izhikevich,
Dynamical Systems in Neuroscience: The Geometry of excitability and Bursting,
MIT Press, Cambridge, MA, 2007.
Spectral Theory of Ordinary Differential Operators,
Ellis Horwood Ltd., 1981.
F. G. Tricomi,
Interscience Publishers, 1957.
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.
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.