Dynamical Systems

Stability, Lyapunov functions. Poincare map. Stable/Unstable manifolds. Hartman-Grobman Theorem. Structural stability. Hyperbolic sets, Anosov diffeomorphisms. Local Bifurcations of
flows and maps. Normal forms. Center manifolds. Global bifurcations, Shilnikov condition.
Melnikov’s method. Chaos, Smale horseshoe, symbolic dynamics, strange attractors, transitivity,
Lyapunov exponents. Unimodal maps, Sharkovsky’s Theorem, circle maps. Invariant measures,
Krylov-Bogulubov, Poincare recurrence. Ergodicity, (pointwise) Ergodic Theorem, mixing.
Entropy. Examples from theoretical and applied contexts.

After successful completion of the module the students are able to understand and apply the mathematical theory of dynamical systems stressing the analysis of geometric and topological properties of solutions of nonlinear ordinary differential equations as well as iterated maps in finite-dimensional Euclidean phase spaces. Furthermore the students are able to analyse elements for the theory of systems on differentiable manifolds and in infinite-dimensional phase spaces.

MA2005 Ordinary Differential Equations, MA3080 Introduction to Nonlinear Dynamics

lecture, exercise course, self-study assignments
The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should motivate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress.

blackboard

Guckenheimer, John, and Holmes, Philip: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields; Springer-Verlag, New York 1983.

The module examination is based on a written exam (90 minutes).
In this the students demonstrate that they comprehensively understand the theoretical basics of dynamical systems and can apply them in limited time. On the basis of specific examples the students exhibit their skills in analyzing geometric and topologic properties of solutions of nonlinear ordinary differential equations and of iterative maps in finite-dimensional Euclidean phase spaces.