ORDINARY DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS

Topics

First order equations: Separable equations, linear equations, initial-value problems, explicit and implicit solutions, exact equations and integrating factors, autonomous equations and equilibrium points, stability.

Higher order equations and systems: Higher-order equations and nXn systems, linear equations, Wronskians and Liouville's theorem, higher-order linear equations and systems with constant coefficients, the exponential of a matrix, phase portraits, stable, unstable and center subspaces, multiple roots, Jordan normal form, inhomogeneous systems, method of undetermined coefficients, variation of constants, mechanical oscillations.

Calculus of variations: Functionals and their extrema, Euler's equations, extrema with side conditions, geodesics, Newtonian mechanics, Lagrange's equations, Lagrangian and Hamiltonian functions, Kepler's problem.

Phase space analysis of higher-order systems: Autonomous systems, phase space, equilibrium points, limit cycles, first integrals, conservative systems with one-degree-of-freedom, separatrices, small oscillations, equations on the torus, linearization, Lyapounov stability, asymptotic stability.

Theory of planar systems: Predator-prey models, linearization about and survival of hyperbolic equilibria, index theory, Poincare-Bendixson theorem, Bendixson criterion, small perturbations of conservative systems.

Periodic solutions and Poincare maps: Surface of section, Poincare return map, stability of periodic orbits, linearized Poincare map, Floquet multipliers.

Bifurcations and chaotic dynamics: Saddle-node, transcritical, pitchfork, period doubling, Hopf, unimodal maps and the Feigenbaum cascade, symbolic dynamics, chaos in unimodal maps.

Existence and uniqueness: Contraction mappings, Picard approximations, Lipschitz condition, local existence and uniqueness.

Textbooks

K. T. Aligood, T. D. Sauer, and J. A. Yorke, Chaos, an introduction to dynamical systems, Springer-Verlag.
A. A. Andronov, Qualitative Theory of Second-Order Dynamic Systems, Wiley.
A. A. Andronov, S. E. Khaikin, and A. A. Vitt, Theory of Oscillations, Dover.
V. I. Arnold, Ordinary Differential Equations, MIT press or Springer Verlag.
G. Birkhoff and G. Rota, Ordinary Differential Equations, Wiley.
W. E Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary-Value Problems, Wiley.
F. Brauer and J. A. Nohel, Ordinary Differential Equations, Benjamin.
F. Brauer and J. A. Nohel, Qualitative Theory of Ordinary Differential Equations, Benjamin.
G. F. Carrier and C. E. Pearson, Ordinary Differential Equations, Blaisdell.
E. A. Coddington, An Introduction to Ordinary Differential Equations, Prentice Hall.
R. Courant and D. Hilbert, Methods of Mathematical Physics, Part I, Wiley-Interscience.
R. L Devaney, An Introduction to Chaotic Dynamical Systems, Benjamin-Cummings.
C. Fox, An introduction to the calculus of variations, Dover.
M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press.
E. L. Ince, Ordinary Differential Equations, Dover.
I. Percival and D. Richards, Introduction to Dynamics, Cambridge University Press.
L. S. Pontryagin, Ordinary Differential Equations, Dover.
D. A. Sanchez, Ordinary Differential Equations and Stability Theory: an Introduction, Dover.
G. F. Simmons, Differential Equations with Applications and Historical Notes, McGraw-Hill.
S. H. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering, Addison-Wesley.
R. A. Struble, Nonlinear Differential Equations, McGraw-Hill.
W. Walter, Ordinary Differential Equations, Springer-Verlag.
R. Weinstock, Calculus of Variations, With Applications to Physics and Engineering, Dover.
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer Verlag.

My favorite is the book by Sanchez, which is a very nice standard texbook on ordinary differential equations. It is easy reading, and so I will use parts of it in the course. Unfortunately, it is now out of print, but here are the relevant scanned pages. I have ordered the books by Arnold, Strogatz, Walter, and Weinstock as recommended texts. The book by Weinstock is a standard, accessible, and cheap introduction to the calculus of variations. I will cover some of the material that is covered in that book, so I also recommend buying it. For those of you who want to pursue mathematics in more depth, I recommend buying Arnold's Ordinary Differential Equations. Even though it is somewhat hard to read in detail, this book has a great, very original approach and many excellent illustrations, and is well worth having as a reference. Buy the older MIT edition, it is hardly any different from the newer Springer Verlag edition, and is a lot cheaper. I also recommend at least looking at the book by Strogatz, which is beautifully written, with applications in mind, and has many nice elementary examples from physics, engineering, biology, and chemistry. It is an excellent supplementary text. I will take some of the introductory material from Chapter 1 of the book by Walter. The rest of this book provides an extensive, elegant, and modern treatment of existence and uniqueness issues in ordinary differential equations. I only recommend buying it to those who have a strong interest in pursuing differential equations on the graduate level. Of those that are not in the bookstore, the book by Percival and Richards is a very easy, elementary, but at the same time very instructive introduction to differential equations and variational principles of mechanics. The book by Courant and Hilbert is a classic, although we will only use the chapter on the calculus of variations. The books by Brikhoff and Rota, Brauer and Nohel, Coddington, Hirsch and Smale, Simmons, and Struble are standard textbooks. The book by Carrier and Pearson is a slightly old-fashioned applied book with a very original viewpoint. The book by Ince is a very old classic, with many somewhat exotic topics that you will not find anywhere else. The book by Aligood, Sauer, and Yorke is similar in spirit to the book by Strogatz, but aimed at a more mathematical audience. The books by Devaney and Wiggins deal with the special topics of bifurcations and chaotic dynamics. The book by Boyce and DiPrima is a standard textbook for an elementary course on differential equations. I will assume at least a vague familiarity with chapters 1, 2, 3, 7, 9, and 10 of this book, which constitute our standard sophomore differential equations course, MATH-2400. I strongly recommend this book as a source for reviewing elementary material.

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