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Dynamical Systems

Module MA3081

This Module is offered by TUM Department of Mathematics.

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 2020

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 2021/2SS 2020SS 2019WS 2017/8SS 2012WS 2011/2

Basic Information

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 workloadContact hoursCredits (ECTS)
270 h 90 h 9 CP

Content, Learning Outcome and Preconditions


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.

Learning Outcome

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

Courses, Learning and Teaching Methods and Literature

Courses and Schedule

Learning and Teaching Methods

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.




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

Module Exam

Description of exams and course work

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.

Exam Repetition

The exam may be repeated at the end of the semester.

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