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,
14:0016:45, BBL 061; location of computer practicums BBL 115)
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. 
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. 
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.
PoincareBendixson's
Theorem. Bendixson's and Dulac's
Criteria. Phase plane analysis of preypredator models. Planar
Hamiltonian and
conservative
ODEs. Newton mechanical 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
onedimensional
ODEs.
AndronovHopf bifrcation in planar ODEs. 
5.5.1 (at the end of Lecture
Notes 11) Computer practicum 16:0017:30 Follow Session XI 
Lectures
10+11 (8 and 15 Nov) 
Fold
and flip
bifurcations of
onedimensional maps.
NeimarkSacker 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 ndimensional ODEs. Fold, perioddoubling, and NeimarkSacker bifurcations of fixed points of ndimensional maps and limit cycles in ndimensional ODEs. 
Computer
practicum 16:0017:30 Follow Session XIII Written home assignment for December 6 (deadline extended to December 13) 
Lecture
13 (29 Nov) 
Onedimensional
dynamics
generated by continuous maps. Feigenbaum's universality. 

Lecture 14 (6 Dec)  Lorenz attractor.  
Lecture 15(13 Dec)  Smale horseshoe. Shilnikov phenomenon. 