M7960 Dynamical Systems

Faculty of Science
Spring 2022
Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
Taught in person.
Mgr. Petr Liška, Ph.D. (lecturer)
Guaranteed by
doc. RNDr. Michal Veselý, Ph.D.
Department of Mathematics and Statistics - Departments - Faculty of Science
Supplier department: Department of Mathematics and Statistics - Departments - Faculty of Science
Ordinary differential equations: Linear and nonlinear systems of differential equations, existence and uniqueness of solutions, dependence of solutions on initial values and parameters, basics of the stability theory.
Linear algebra: Systems of linear equations, determinants, linear spaces, linear transformation and matrices, canonical form of a matrix.
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
Course objectives
The course is an introduction to the theory of dynamical systems. Attention is paid to continuous dynamical systems, to the theory of autonomous systems of differential equations, and to mathematical modelling.
Learning outcomes
After passing the course, the student will be able: to define and interpret the basic notions used in the fields mentioned above; to formulate relevant mathematical theorems and statements; to use effective techniques utilized in these subject areas; to apply acquired pieces of knowledge for the solution of specific problems; to analyse selected mathematical dynamic deterministic models.
  • 1. Survey of selected resuts from the theory of ordinary differential equations.
  • 2. Autonomous equations - basic notions and properties, elementary types of singular points of two-dimensional systems, classification of singular points of linear and perturbed linear systems, the structure of a limit set in R2, Poincaré-Bendixson theory, Dulac criterion, characteristic directions.
  • 3. General concept of a dynamical system, continuous and discrete dynamical systems.
  • 4. Notion of a mathematical model, classification of models, basic steps of the process of mathematical modelling, formulating a mathematical model, dimensional and mathematical analysis of mathematical models. Selected mathematical models in natural sciences.
    recommended literature
  • KALAS, Josef and Zdeněk POSPÍŠIL. Spojité modely v biologii. 1. vyd. Brno: Masarykova univerzita, 2001. vii, 256. ISBN 802102626X. info
  • BRAUN, Martin. Differential equations and their applications : an introduction to applied mathematics. 2nd ed. New York: Springer-Verlag, 1978. xiii, 518. ISBN 0-387-90266-X. info
  • PERKO, Lawrence. Differential equations and dynamical systems. 2nd ed. New York: Springer-Verlag, 1996. xiv, 519. ISBN 0387947787. info
    not specified
  • VERHULST, Ferdinand. Nonlinear differential equations and dynamical systems. Berlin: Springer Verlag, 1990. 277 s. ISBN 3-540-50628-4. info
  • EDELSTEIN-KESHET, Leah. Mathematical models in biology. Philadelphia: Society for Industrial and Applied Mathematics, 2005. xliii, 586. ISBN 0898715547. info
Teaching methods
lectures and class exercises
Assessment methods
Written and oral examination. For admission to exam students need to submit three homework assignments.
Language of instruction
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: every week.
Teacher's information
The lessons are usually in Czech or in English as needed, and the relevant terminology is always given with English equivalents. The target skills of the study include the ability to use the English language passively and actively in their own expertise and also in potential areas of application of mathematics. Assessment in all cases may be in Czech and English, at the student's choice.
The course is also listed under the following terms Spring 2011 - only for the accreditation, Autumn 2003, Spring 2006, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Spring 2021.
  • Enrolment Statistics (recent)
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