SubjectsSubjects(version: 908)
Course, academic year 2021/2022
Systems of Ordinary Differential Equations (ODE) - N413007
Title: Soustavy obyčejných diferenciálních rovnic
Guaranteed by: Department of Mathematics (413)
Actual: from 2021
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 [HT]
Capacity: unknown / unknown (unknown)
Min. number of students: unlimited
Language: Czech
Teaching methods: full-time
For type:  
Guarantor: Dubcová Miroslava RNDr. Ph.D.
Is interchangeable with: M413002
Examination dates   Schedule   
Annotation -
Last update: Dubcová Miroslava RNDr. Ph.D. (18.07.2013)
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: Dubcová Miroslava RNDr. Ph.D. (18.07.2013)

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: TAJ413 (28.08.2013)

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. Pearson Prentice Hall, 2004, ISBN 0-13-143140-4

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: Dubcová Miroslava RNDr. Ph.D. (18.07.2013)

Teaching methods -
Last update: TAJ413 (19.07.2013)

Lectures and exercise classes.

Syllabus -
Last update: Dubcová Miroslava RNDr. Ph.D. (18.07.2013)

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. Attractor.

4. Planar systems. Phase portraits of linear systems.

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

6. First integrals and applications.

7. Phase portraits of linear and nonlinear systems in R^3.

8. Stability theory. Poincare mapping.

9. Nonautonomous systems of ODEs.

10. Periodic linear systems. Monodromy matrix. Floquet theory.

11. Systems of ODEs depending on parameters. Bifurcations.

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

13. Discrete dynamical systems, basic notions.

14. Regular and chaotic behavior. Lyapunov exponents.

Registration requirements -
Last update: TAJ413 (16.07.2013)

Mathematics I, Mathematics II

Teaching methods
Activity Credits Hours
Konzultace s vyučujícími 0,5 14
Účast na přednáškách 1 28
Příprava na přednášky, semináře, laboratoře, exkurzi nebo praxi 0,5 14
Příprava na zkoušku a její absolvování 2 56
Účast na seminářích 1 28
5 / 5 140 / 140
Coursework assessment
Form Significance
Regular attendance 10
Examination test 35
Continuous assessment of study performance and course -credit tests 20
Oral examination 35