Dynamical Systems

Written examination is on Tuesday January 10, at 14.00 - 17.00.

Please note the following :
-- the examination will take place in Room  AARD GROOT (Gebouw Aardwetenschappen, Budapestlaan 4);
-- you are allowed to bring the lecture notes of the course, your own additional notes as well as text books and to use these during the exam  (but no computers or phones);
-- you can either hand in the essay on the Lorenz system on that day or send it to both  I.A.Kouznetsov@uu.nl and O.Diekmann@uu.nl (with SUNDAY JANUARY 22 as the STRICT DEADLINE).

Who has not filled in this evaluation form yet, please do so and send to:
Mrs. Greta Oliemeulen-Löw
Secretary Regieorgaan Masteropleidingen Wiskunde
Radboud University Nijmegen, FNWI
Heyendaalseweg 135
6525 AJ  NIJMEGEN
The Netherlands
+31 (0)24 – 365 2986
mastermath@math.ru.nl


The first lecture and practicum are on Tuesday, September 6, 2011 (BBL 061, 14:00-16:45).

Instructors: Odo Diekmann & Yuri Kuznetsov

Credits ECTS:  8.0

Language:  English

Preliminaries:  Any standard Bachelor course on ODEs with proofs, e.g. "Differentiaalvergelijkingen" (WISB 231 at UU)

Format: 2 hrs lectures  + 1 hr practicum per week (Tue, 14:00-16:45, BBL 061; location of computer practicums BBL 115) 

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 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    
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    
At the end of the course students will be assigned an examination project. The students should take 7 to 8 days in a period of 3 weeks to produce a written elaboration, which contributes 20% to the final grade. Another 20% come from two home assignments given during the course. The remaining 60 % of the grade are provided by a written examination.

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

    Lecture Notes (by Yu.A. Kuznetsov, O. Diekmann, and W.-J. Beyn), practicum assignments,  and computer session manuals will be made available on-line during the course.

Simulation tools:
    - plotting solutions to 1D and 2D ODEs (DField and PPlane)
    - plotting orbits of 1D maps  (CobwebPlot)

Lecture Notes
 Topics
 Practicum assignments
Lecture 1 (6 Sep 2011) Time, state space and evolution. Definition and examples of dynamical systems. Generators.
Orbits and phase portraits. Equilibria, cycles, homo-/heteroclinic orbits. Invariant sets and their stability.
Equivalence of dynamical systems.		 1.5.1(i), 1.5.5, 1.5.10, 1.5.12, 1.5.13, 1.5.2
Lectures 2+3 (13 and 20 Sep 2011) Linear maps and autonomous ODEs: Dynamics in the eigenspaces. Stable and unstable invariant subspaces.
Spectral projectors.

Hyperbolic linear maps and ODEs. Lyapunov norms in the stable invariant subspaces.
Topological classification of linear maps and linear autonomous ODEs.

2.5.1, 2.5.2, 2.5.5, 2.5.6


2.5.7, 2.5.13, 2.5.14. 2.5.15
Lectures 4+5 (27 Sep and 4 Oct)
 Principle of linearized stability for maps and ODEs. Stability of periodic orbits in ODEs.

Lipschitz maps: Contraction Mapping Principles and Lipschitz Inverse Function Theorem.
Grobman-Hartman Theorems for maps and ODEs.

3.5.1, 3.5.2, 3.5.3, 3.5.4

3.5.5, 3.5.7, 3.5.8
Lectures 6+7 (11 and 18 Oct)
 Limit sets. Poincare-Bendixson's Theorem. Bendixson's and Dulac's Criteria.
Phase plane analysis of prey-predator models.

Planar Hamiltonian and conservative ODEs. Newton mechanical systems with one degree of freedom.
Limit cycles of dissipatively perturbed Hamiltonian systems with one degree of freedom.

4.7.1, 4.7.2, 4.7.4(b),
Written home assignment for October 25

4.7.7, 4.7.8, 4.7.11
Lectures 8+9 (25 Oct and 1 Nov)
 Local bifurcation theory. Fold bifrcation in one-dimensional ODEs.

Andronov-Hopf bifrcation in planar ODEs.

5.5.1 (at the end of Lecture Notes 11)

Computer practicum 16:00-17:30 
Follow Session XI
Lectures 10+11 (8 and 15 Nov)
 Fold and flip bifurcations of one-dimensional maps.

Neimark-Sacker bifurcation of planar maps.

5.5.4, 5.5.5(a)

5.5.6(a,b)
Lecture 12 (22 Nov)
 Center manifold reduction. Fold and Hopf bifurcations in n-dimensional ODEs.
Fold, period-doubling, and Neimark-Sacker bifurcations of fixed points of n-dimensional maps and limit cycles in n-dimensional ODEs.		 Computer practicum 16:00-17:30
Follow Session XIII
Written home assignment for
December 6 (deadline extended to December 13)
Lecture 13 (29 Nov)
 One-dimensional dynamics generated by continuous maps.
Feigenbaum's universality.
Lecture 14 (6 Dec) Lorenz attractor.
Lecture 15(13 Dec) Smale horseshoe. Shilnikov phenomenon.
 Bibliography
Last updated: Thu 22 Dec 2011
kuznet@math.uu.nl