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Dynamical Systems generated by ODE and Maps
| Credits | 8 credit points |
| Instructors | Kuznetsov, Yu.A. (Universiteit Utrecht), Diekmann, O. (Universiteit Utrecht) |
| I.A.Kouznetsov@uu.nl, O.Diekmann@math.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. This course is a part of the MRI Master Class "Numerical bifurcation analysis of dynamical systems" but can be followed independently. |
| 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 a parameter is 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 practicum per week. The course material includes pencil and paper exercises as well as exercises that require the use of symbolic manipulation tools, such as MAPLE, as well as simple simulation programs. Training in the use of these tools is an integrated part of the course. |
| Examination | Written exam on Tuesday January 26 from 14.00 - 17.00, room |
| 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. |