Dynamical Systems

COURSE OBJECTIVE
At the end of this course students ...
... know the existence an uniqueness theorem for initial value problems;
... are able to solve constant coefficient linear (matrix) differential equations and know the elementary linear algebra needed for that;
... can analyze one- and two-dimensional dynamics and are able to draw phase space / phase plane pictures;
... know elementary bifurcation theory and in particular the saddle-node bifurcations;
... can recognize and analyze gradient, conservative and Hamiltonian dynamical systems;
... know and understand the stable and unstable manifold theorem and the principle of linearization;
... know what limit sets are;
... are able to apply the Poincaré-Bendixson theorem;
... understand the concept of compactification and know the Poincaré sphere.

COURSE CONTENT
This course entails the theory of ordinary differential equations from the modern point of view of dynamical systems.
Subjects are:
1. Existence an uniqueness of initial value problems;
2. Linear systems and elementary linear algebra;
3. One-dimensional dynamics and two-dimensional phase plane pictures;
4. Elementary bifurcation theory and saddle-node bifurcations;
5. Gradient dynamics;
6. Conservative systems and Hamiltonian dynamics;
7. Stable and unstable manifolds and linearization;
8. Limiting behavior;
9. The Poincaré-Bendixson theorem;
10. Compactification and the Poincaré sphere.

TEACHING METHODS
Regular instruction class in combination with tutorial classes.

TYPE OF ASSESSMENT
Hand-in exercises, a midterm and a final exam. The hand-ins count for 10% each. The first midterm counts for 30% and the second midterm counts for 50%.

The resit exam counts for 100% (hand-ins don't count anymore).

RECOMMENDED BACKGROUND KNOWLEDGE
First year courses Calculus and Analysis.