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MU:MU03049 Dynamical Systems I - Course Information

## MU03049 Dynamical Systems I

**Mathematical Institute in Opava**

Winter 2020

**Extent and Intensity**- 2/2/0. 6 credit(s). Type of Completion: zk (examination).
**Teacher(s)**- doc. RNDr. Michal Málek, Ph.D. (lecturer)

doc. RNDr. Michaela Mlíchová, Ph.D. (seminar tutor) **Guaranteed by**- doc. RNDr. Michal Málek, Ph.D.

Mathematical Institute in Opava **Timetable**- Tue 14:45–16:20 207
- Timetable of Seminar Groups:

*M. Mlíchová* **Prerequisites**(in Czech)- TYP_STUDIA ( BN )
**Course Enrolment Limitations**- The course is also offered to the students of the fields other than those the course is directly associated with.
**fields of study / plans the course is directly associated with**- Applied Mathematics in Risk Management (programme MU, B1101)
- Geometry and Global Analysis (programme MU, N1101)
- Mathematical Analysis (programme MU, NMgr-M)
- Mathematical Analysis (programme MU, N1101)
- Mathematical Methods in Economics (programme MU, B1101)

**Course objectives**- This course is a graduate level introduction to the mathematical theory of discrete dynamical systems on compact metric spaces and additionaly on the intercal. We will discuss some fundamental examples in the field, including circle rotations, shifts and subshifts, quadratic maps. We cover topics such as limit sets and recurrence, topological mixing, transitivity, entropy and symbolic dynamics.
**Syllabus**- 1. Elementary notions

Orbit (full, forward, backward), periodic orbit.

Brower fixed point theorem.

Sharkovskii ordering.

2. Hyperbolicity

Critical point, hyperbolic point, attractive and repulsice point.

3. Examples of dynamical systems

Quadratic system - logistic map, the tent map, rotations of the circle.

4. Symbolical dynamics - shift space

Shift map and its properties, shift of finite type.

5. Topological dynamics

Minimal sets, limit sets, nonwandering sets, centre, conjugacy.

Transitivity, total transitivity, mixings, their relations and relations to the dense orbit.

Recurrence and relations to ninimality.

Topological entropy.

- 1. Elementary notions
**Literature**- L. S. Block, W. A. Coppel.
*Dynamics in one dimension*. Lecture Notes in Mathematics, 1513. Springer-Ver, 1992. info - R. L. Devaney.
*An introduction to chaotic dynamical systems*. Second edition, 1989. info - P. Walters.
*An introduction to ergodic theory*. Graduate Texts in Mathematics, 79. Springer-Verl, 1982. info

*required literature*- L. S. Block, W. A. Coppel.
**Language of instruction**- Czech
**Further comments (probably available only in Czech)**- Study Materials

The course can also be completed outside the examination period. **Teacher's information**- Course credit: ability to verify notion on given examples

Final exam: knowledge of basic notions and assertions, at least partial understanding of theory

- Enrolment Statistics (recent)

- Permalink: https://is.slu.cz/course/sumu/winter2020/MU03049