||Ordinary Differential Equations
|| MATH 6410 - 1
|Place & Time:
|| M, W, F, 2:00 - 2:50 in JWB 308
||10:40-11:30 M, W, F, in JWB 224 (tent.)
||Luis Barreira & Claudia Valls, Ordinary Differential Equations: Qualitative Theory A.M.S., 2012. ISBN 978-0-8218-8749-3.
We shall basically follow
the text. But much of the material is standard and widely available. Therefore, students might be able to get by without owning the text, although the majority of the problems will come from the text. I'll provide references and put copies in the math library. I will also cover additional topics that were suggested in consultation with Professors Adler, Dobson and Keener. Come to class for details and references. Here is a partial list of alternative sources that cover the material.
I have tried several different texts for this course: Amann, Perko, Chicone, Liu and Cronin. Perhaps not so unexpectedly, the texts the students liked the best, Perko and Liu, are not the ones I liked the best, Amann and Chicone. None of the texts perfectly suits the material we teach at the University of Utah. Grant's notes as far as they go are modeled on the course he took as a graduate student here at Utah. All the texts cover about two thirds of the course, and the rest has to be supplemented. The authors all have their hobby horses, and they discuss their favorite special topics beyond what would be appropriate for a beginning course.
This course is designed to be a balance of application and theory that is optimized for the needs of students at Utah, be they interested is applied mathematics, mathematical biology, numerical analysis, probability,
differential equations or geometric analysis. As mathematicians, it is our prerogative and, indeed duty, to understand why theorems work, so that we may modify or code them as we encounter them in the future.
However, only a minimal amount of doing proofs will be required. Besides the general understanding why any of the theorems hold,
I will only require that students know completely the proofs of two things: local existence and linearized stability at a rest point.
The choice of topics in Math 6410 varies slightly from instructor to instructor, although the variance is far less than you might think hearing old graduate student gossip. One only needs to look at previous syllabi, or the Differential Equations Preliminary Exams derived from these courses over the last decade to see that they are very consistent. Schmitt and Thompson's notes designed for this course spins toward nonlinear functional analysis. A course based on Jordan & Smith's, Verhulst's and Glendenning's texts spins towards perturbation methods.
Texts Suitable for an Undergraduate Course in ODE's.
- Fred Bauer & John Nohel, Dover 1989; reprint of W. A. Benjamin, 1969.
- Roger Grimshaw, Nonlinear Ordinary Differential Equations, CRC Press, 1993.
- Richard Miller & Anthony Michel, Ordinary Differential Equations, Dover, 2007; reprint of Academic Press 1982.
- R. Clark Robinson, An Introduction to Dynamical Systems: Continuous and Discrete Pearson Prentice Hall, 2004.
- Steven Strogatz, Nonlinear Dynamics and Chaos, Perseus Books, 1994.
- Michael Taylor, Introduction to Differential Equations, American Mathematical Society Pure and Applied Undergraduate Texts 14, 2010.
Texts Suitable for a Graduate Course for Students who have not Studied Measure Theory.
- Ravi Agarwal & Donal O'Regan, An Introduction to Ordinary Differential Equations, Springer Universitext, 2008.
- Vladimir Arnold, Ordinary Differential Equations, 3rd. ed., Springer Universitext, 1992.
- Luis Barriera and Claudia Valls, Ordinary Differential Equations: Qualitative Theory, American Mathematical Society Graduate Studies in Mathematics 137, 2010.
- David Betounes, Differential Equations: Theory and Applications, 2nd. ed., Springer 2010.
- Jane Cronin, Ordinary Differential Equations: Introduction and Qualitative Theory, 3rd. ed., CRC Press, 2008.
- Paul Glendenning, Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations, Cambridge, 1994.
- Christopher Grant, Theory of Ordinary Differential Equations(chapter index), pdf(158 pages), on-line lecture notes with Solutions.
- M. Hirsch, S. Smale & R. Devaney, Differential Equations, Dynamical Systems & an Introduction to Chaos 2nd. ed., Elsevier, 2004.
- James Hetao Liu, A First Course in the Qualitative Theory of Differential Equations, Prentice Hall 2003.
- D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations-- An Introduction for Scientists and Engineers, 4th ed., Oxford, 2007.
- Lawrence Perko, Differential Equations and Dynamical Systems, Springer Texts in
Applied Mathematics 7, 1991.
- Stephen Salaff and Shing-Tung Yau, Ordinary Differential Equations, 2nd ed., International Press, 1998.
- Gerald Teschl, Ordinary Differential Equations and Dynamical Systems American Mathematical Society Graduate Studies in Mathematics 140, 2012.
- Ferdinand Verhulst, Nonlinear Differential Equations and Dynamical Systems, 2nd. ed., Springer Universitext, 2006.
Texts Suitable for a Graduate Course for Students who have Studied Measure Theory.
- Herbert Amann, Ordinary Differential Equations--An Introduction to Nonlinear Analysis, Walter de Gruyter, 1990.
- Carmen Chicone, Differential Equations with Applications, Springer Texts in
Applied Mathematics 34, 1999.
- Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations, Krieger Publishing Company, 1984; reprint of McGraw Hill, 1955.
- Jack Hale, Ordinary Differential Equations, 2nd. ed., Dover 2009; reprint of Krieger 1980; re-reprint of John Wiley & Sons, 1969.
- Philip Hartman, Ordinary Differential Equations, 2nd ed., SIAM 2002; reprint of Birkhaüser, 1982; re-reprint of John Wiley & Sons, 1964.
- Fritz John, Ordinary Differential Equations, Courant Institute of Mathematics Lecture Notes, 1965.
- Klaus Schmitt and Russell Thompson, Nolinear Analysis and Differential Equations: An Introduction, University of Utah Lecture Notes 2009.pdf.(152 pages.)
- Wolfgang Walter, Ordinary Differential Equations, Springer Graduate Texts in Mathematics 182, 1991.
Specialist's Books on Specific Topics. Unsuitable for Math 6410.
- Richard Bellman, Stability Theory of Differential Equations, Dover 1969; reprint of McGraw-Hill, 1953.
- John Guckenheimer & Philip Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences 42, Springer, 1983.
- Yuri Kuznetsov, Elements of Bifurcation Theory, 3rd. ed., Applied Mathematical Sciences 112, Springer, 2010.
- Stephen Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd ed., Springer Texts in
Applied Mathematics 2, 2000.
Last updated: 7 - 29 - 13