Systems of Ordinary Differential Equations (ODE) - M413002
Title: Soustavy obyčejných diferenciálních rovnic
Guaranteed by: Department of Mathematics (413)
Actual: from 2019
Semester: winter
Points: winter s.:5
E-Credits: winter s.:5
Examination process: winter s.:
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited / unlimited (unknown)
Min. number of students: unlimited
Language: Czech
Teaching methods: full-time
For type:  
Note: course can be enrolled in outside the study plan
enabled for web enrollment
Guarantor: Dubcová Miroslava RNDr. Ph.D.
Interchangeability : N413007
Examination dates   
Annotation -
Last update: Pátková Vlasta (09.01.2018)
The course deals with the qualitative theory of differential equations. The theory of differential equations is presented with the emphasis on its geometric and qualitative aspects and is understood as a part of more general theory of dynamical systems.
Aim of the course -
Last update: Pátková Vlasta (09.01.2018)

Students should be able to describe autonomous systems of differential equations qualitatively. Namely, they should be able to determine stability of solutions, to recognize chaotic attractor and to classify bifurcations.

Literature -
Last update: Dubcová Miroslava RNDr. Ph.D. (18.10.2018)

R: A. Klíč, M. Dubcová,L. Buřič: Soustavy obyčejných diferenciálních rovnic, kvalitativní teorie, dynamické systémy, VŠCHT Praha, 2009, ISBN: 978-80-7080-724-8

R: R.C.Robinson: An Introduction to Dynamical Systems: Continuous and Discrete. AMS, 2012 ISBN: 978-0821891353

A: M. W. Hirsch, S. Smale, R. L. Devaney: Differencial Equations, Dynamical Systems & An Introductions to Chaos, Elsevier 2004, ISBN0-12-349703-5

Learning resources -
Last update: Pátková Vlasta (09.01.2018)

Teaching methods -
Last update: Pátková Vlasta (09.01.2018)

Lectures and exercise classes.

Syllabus -
Last update: Dubcová Miroslava RNDr. Ph.D. (16.02.2018)

1. The concept of dynamical systems. Continuous and discrete dynamical systems.

2. Autonomous systems of ODEs. Qualitative approach. Phase flow. The notion of stability.

3. Planar systems. Phase portraits of linear systems.

4. Phase portraits of nonlinear systems. Grobman-Hartman theorem.

5. Closed trajectory. Bendix and Poiancaré criteria

6. First integrals and applications.

7. Population model "Predator - Prey". Hamilton systems in the plane.

8. Newton's equation

9. Phase portraits of linear and nonlinear systems in R3.

10. Ljapun's function. Gradient systems.

11. Systems of ODEs depending on parameters. Bifurcations.

12. Examples: "Brusselator", Lorenz system, dumped oscillator.

13. Discrete dynamical systems, basic notions.

14. Discrete dynamic systems.

Entry requirements -
Last update: Borská Lucie RNDr. Ph.D. (13.05.2019)

Students are expected to have either completed the prerequisite courses Mathematics A and Mathematics B or possess the equivalent knowledge prior to enrolling in the course.

Registration requirements -
Last update: Borská Lucie RNDr. Ph.D. (06.05.2019)

No requirements.

Course completion requirements -
Last update: Dubcová Miroslava RNDr. Ph.D. (16.02.2018)

Assesment, written exam, oral exam

Teaching methods
Activity Credits Hours
Konzultace s vyučujícími 0,5 14
Účast na přednáškách 1 28
Příprava na přednášky, semináře, laboratoře, exkurzi nebo praxi 1,5 42
Příprava na zkoušku a její absolvování 1,5 42
Účast na seminářích 0,5 14
5 / 5 140 / 140