Topics List---MA 775
This course is a graduate level introduction to the mathematical theory of ordinary differential equations and nonlinear dynamical systems. It is designed for students who want to begin research in dynamical systems, as well as for those who wish to apply dynamical systems techniques in their research. The topics listed below will be illustrated using nonlinear examples from the literature in several areas, including mechanics, reaction-diffusion equations, fluid mechanics, cognitive sciences, and neurophysiology. Problems may also be taken from other areas in the physical, and engineering sciences depending on student interest.
Quick review of ODE basics (existence, uniqueness, smooth dependence on initial data and parameters, Gronwall inequality fixed points, linearization of vector fields, Jordan Normal form of matrices, Poincare-Bendixson Theorem, -- in contrast to previous years, some of this will be done through homework assignments), Poincare maps, stable manifold theory and Wazewski's principle, homoclinic orbits and chaos (including Poincare -- Arnol'd -- Melnikov method and Smale's horseshoe construction), Duffing's equation and the nonlinear pendulum, bifurcation theory (codimension 1 and 2), the center manifold theorem, normal forms, the Lorenz equations, method of averaging, Floquet theory and Hill's equation, relaxation oscillators (van der Pol, Lienard), and one topic of class choice (which could include basic concepts of Hamiltonian dynamical systems or delay differential equations, or any other appropriate subject for which there is sufficient interest).
The two texts I will follow most closely, and which you may purchase in the bookstore, are: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, by J. Guckenheimer and P. Holmes, and Ordinary Differential Equations by Jack Hale. Many of the course topics are treated in the first text, and a rigorous mathematical treatment of many of the topics is given in the second text.
I have also put both of these books on reserve for this course at the Science and Engineering Library (38 Cummington St.). In addition to these two books, I will put other books on reserve for this course. Some of these are: Theory of ordinary differential equations by E. Coddington and N. Levinson, Ordinary differential equations by P. Hartman (two classic o.d.e. books, the latter with an excellent bibliography), Introduction to applied nonlinear dynamical systems and chaos by S. Wiggins (this treats many of the same subjects as Guckenheimer and Holmes and has been very useful for past students for the detailed expositions), The Lorenz equations by C. Sparrow, Applications of centre manifold theory by J. Carr, Theory and application of the Hopf bifurcation by Hassard, et al., (three texts focusing on bifurcation theory), Dynamics and bifurcations by Hale and Kocak, Averaging methods in nonlinear dynamical systems by J. Sanders and F. Verhulst, Introduction to perturbation techniques by Nayfeh, Perturbation methods in applied mathematics by Cole, Nonlinear oscillations by Nayfeh and Mook, Hill's equation by Magnus and Winkler, Introduction to dynamics by Percival and Richards (a nice little paperback on the basics of Hamiltonian systems), Mathematical methods of classical mechanics by V. Arnold (a standard graduate level treatment of classical mechanics), Global stability of dynamical systems by Shub (excellent for the full details of invariant manifolds in Banach spaces), Geometrical methods in the theory of ordinary differential equations by V.I. Arnold, Ordinary differential equations by V.I. Arnold, Ordinary differential equations by Ince (a compendium of many classical equations, see also Kamke), Nonlinear dynamics and chaos by Strogatz.
Finally, as time goes on, I will put a series of approximately thirty research papers -- in which techniques we cover in class are used -- on reserve in the Library during the course of the semester. Ask the librarian at the check-out desk for these. They are filed by the course number.