This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.
Module version of SS 2019
There are historic module descriptions of this module. A module description is valid until replaced by a newer one.
Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.
|available module versions|
|WS 2021/2||SS 2020||SS 2019||WS 2017/8||SS 2012||WS 2011/2|
MA3081 is a semester module in English language at Master’s level which is offered in summer semester.
This Module is included in the following catalogues within the study programs in physics.
- Catalogue of non-physics elective courses
|Total workload||Contact hours||Credits (ECTS)|
|270 h||90 h||9 CP|
Content, Learning Outcome and Preconditions
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.
Courses, Learning and Teaching Methods and Literature
Courses and Schedule
|VO||4||Dynamical Systems [MA3081]||Kühn, C. Longo, I.||
Mon, 10:15–11:45, MI 00.07.014
Wed, 10:15–11:45, MI 00.07.014
and singular or moved dates
|UE||2||Dynamical Systems (Exercise Session) [MA3081]||Kühn, C. Longo, I.||dates in groups|
Learning and Teaching Methods
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.
Description of exams and course work
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.
The exam may be repeated at the end of the semester.