#
MU:MU03051 Dynamical Systems II - Course Information

## MU03051 Dynamical Systems II

**Mathematical Institute in Opava**

Summer 2022

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

Mathematical Institute in Opava **Prerequisites**(in Czech)- (
**MU03049**Dynamical Systems I ||**MU03050**Dynamical Systems I ) && 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 continuous dynamical systems on manifolds. We will discuss some fundamental examples in the field and bifurcations.
**Syllabus**- 1. Flow - flow, trajectory, equilibria.

2. Invariant sets - alpha nad omega limit set of the folw, closed orbit, Poincaré - Bendixson Theorem.

3. Bifurcation I. - bifurcation, bifurcation diagram.

4. Examples - pitchfork, transcritical, saddle node and Poincaré - Andronov - Hopf bifurcation.

5. Bifurcation II. - qualitative equivalence of the linear systems, hyperbolic systems, bifurcation of linear systems.

6. Bifurcation III. - Hartman - Grobman and Poincaré - Andronov - Hopf theorems. Examples of nonhyperbolic equilibria, supercritical bifurcation.

7. Centram manifold - central manifolds and their applications.

- 1. Flow - flow, trajectory, equilibria.
**Literature**- D. K. Arrowsmith, C. M. Place.
*An introduction to Dynamical Systems*. Cambridge University Press, 1990. info

*required literature*- D. K. Arrowsmith, C. M. Place.
**Language of instruction**- English
**Further comments (probably available only in Czech)**- 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 (Summer 2022, recent)
- Permalink: https://is.slu.cz/course/sumu/summer2022/MU03051