Dynamical Systems Generated by Ordinary Differential Equations and Maps

Credits	8 credit points
Instructors	Diekmann, O. (Universiteit Utrecht), Kuznetsov, Yu.A. (Universiteit Utrecht)
E-mail	O.Diekmann@uu.nl, I.A.Kouznetsov@uu.nl
Aim	The aim of this course is to introduce basic ideas, concepts, examples, results, techniques and methods for studying the orbit structure of smooth dynamical systems on finite dimensional spaces generated by ordinary differential equations (ODEs) or iterated maps.
Description	Subjects that will be treated in detail are : -- linearization near steady states : the Principle of Linearized Stability and local topological equivalence (the Grobman-Hartman Theorem) -- phase plane analysis: Poincare'-Bendixson theory, planar Hamiltonian systems from mechanics and their perturbations, predator-prey systems -- bifurcation theory (how does the orbit structure change when parameters are varied) for ODE and for maps -- stability of periodic solutions of ODE : Poincare' maps and Floquet multipliers -- combined Center Manifold and Normal Form reduction -- the horseshoe map and symbolic dynamics (and chaotic behaviour)
Organization	2 hrs lectures + 1 hr computer practicum per week. The course material includes pencil and paper exercises as well as exercises that require the use of sophisticated computer tools, such as CONTENT. Training in the use of this tool is an integrated part of the course.
Examination	Every week an assignment will be given that has to be handed in. The grades for these assignments contribute 40 % to the final grade. At the end of the course students will be assigned a final project . The students should take 7 to 8 days in a period of 3 weeks to produce a written elaboration, which contributes 40 % to the final grade. At the end of the course we organise a session in which each student gives a presentation about his/her project. This contributes the remaining 20 % to the final grade.
Literature	Literature: - Yu.A. Kuznetsov. Elements of Applied Bifurcation Theory. 3rd ed. Springer-Verlag, New York, 2004. - F. Verhulst. Nonlinear Differential Equations and Dynamical Systems. Springer, Universitext, 1996
Prerequisites	Any standard Bachelor course on ODEs with proofs. Lecture notes and computer session manuals will be made available on-line during the course.

Last changed: 18-01-2012 10:21