M7960 Dynamical Systems

Faculty of Science
Spring 2015
Extent and Intensity
2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Josef Kalas, CSc. (lecturer)
Guaranteed by
doc. RNDr. Josef Kalas, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Tue 8:00–9:50 MS1,01016
  • Timetable of Seminar Groups:
M7960/01: Tue 10:00–11:50 MS1,01016, J. Kalas
Prerequisites
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. 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.
Syllabus
  • 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.
Literature
  • VERHULST, Ferdinand. Nonlinear differential equations and dynamical systems. Berlin: Springer Verlag, 1990, 277 s. ISBN 3-540-50628-4. info
  • PERKO, Lawrence. Differential equations and dynamical systems. 2nd ed. New York: Springer-Verlag, 1996, xiv, 519. ISBN 0387947787. info
  • 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
Teaching methods
lectures and class exercises
Assessment methods
Examination: written and oral. One written test will be realized during the semester. It is required to obtain at least half of the total amount of points. The exam is composed of a written and an oral part. The written part consists of three exercises. It is necessary to obtain at least 1,5 from possible 3 points.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
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 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024, Spring 2025.
  • Enrolment Statistics (Spring 2015, recent)
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