Course detail

Dynamical Systems and Mathematical Modelling

FSI-SA0Acad. year: 2025/2026

The course provides basics of theory of stability, bifurcations and chaos for continuous and discrete dynamic systems.  Applications of the obtained  knowledge in the study of  various problems in technical and scientific branches are stated as well. The study of these problems consists in forming of a differential or difference equation as a corresponding mathematical model, and in analysis of its solution. 

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

prof. RNDr. Jan Čermák, CSc.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Differential and integral calculus of functions in a single and more variables, theory of ordinary differential equations, linear algebra.

Rules for evaluation and completion of the course

Course-unit credit is awarded on the following conditions: Active participation in seminars. Fulfilment of all conditions of the running control of knowledge. 

Examination: The examination tests the knowledge of definitions and theorems (especially the ability of their application to the given problems) and practical skills in solving of examples. The exam is written and oral, the written part (60 minutes) consists of the following topics: Stability of linear and nonlinear ODEs, bifurcation, chaos, ODEs with a time delay, difference equations. 

The final grade reflects the result of the written and oral part of the exam (maximum 100 points).
Grading scheme is as follows: excellent (90-100 points), very good (80-89 points), good (70-79 points), satisfactory (60-69 points), sufficient (50-59 points), failed (0-49 points).


Attendance at lectures is recommended, attendance at seminars is obligatory and checked. Lessons are planned according to the week schedules. Absence from seminars may be compensated for by the agreement with the teacher.

Aims

The aim of the course is to explain basics of the theory of stability, bifurcations and chaos for ordinary differential and difference equations, including time delay equations.  The task of the course is to demonstrate the obtained knowledge in mathematical modelling via dynamic equations, including analysis of  their solutions.


Students will acquire knowledge of basic methods  for analysis of stability, bifurcations and chaos for ordinary differential  and difference equations. They also will master basic procedures  of mathematical modelling by means of studied types of equations , including methods of qualitative analysis of their solutions. 

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Fulford, G., Forrester, P., Jones, A.: Modelling with Differential and Difference Equations, New York, 2001. (EN)
Perko, L.: Differential Equations and Dynamical Systems, Springer-Verlag, 1991.  (EN)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, Berlin, Springer, 1990.  (EN)

Recommended reading

Nahin, P.J.: Chases and Escapes: the mathematics of pursuit and evasion, Princeton University Press, Princetion, 2007. (EN)
 Rachůnková, I,  Fišer, J.: Dynamické systémy 1, UP  Olomouc,  2014 (CS)
Strogatz, S.:  Nonlinear Dynamics and Chaos, With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity), Avalon Publishing,  2014 (EN)

Classification of course in study plans

  • Programme B-MAI-P Bachelor's 2 year of study, summer semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Jan Čermák, CSc.

Syllabus

1. Stability of solutions of ODE systems (basic notions and properties).
2. Linear autonomous systems and their stability, Routh-Hurwitz criterion.
3. Nonlinear autonomní systems, linearization theroem, local  stability of solutions.
4. Global stability of solutions, the Lyapunov method.
5. Limit sets, attractors, periodic orbits.
6. Bifurcations and structural stability  in dimension 1. 
7. Bifurcations and structural stability in higher dimensions.
8. Deterministic chaos, strange attractor
9. ODE with a time delay ( basics of theory).                                              10. Stability of ODE with a time delay.                                                        11. Applications of ODEs with a time deay in control theory (stabilization, destabilization, chaotification).
12. Difference equations  (basics of theory).                                              13. Discrete logistic equation, Sharkovsky theorem.

Exercise

26 hod., compulsory

Teacher / Lecturer

prof. RNDr. Jan Čermák, CSc.

Syllabus

1. Applications of ODEs in mechanics (basic problems).
2. One-body problem, calculations of escape velocities.
3. The first Kepler problem and its solving.
4. Geometric applications of ODEs (constructions of curves with special properties, the Archimedes problem).
5. Applications of ODEs in hydromechanics.
6. Applications of ODEs in hydromechanics (continuation).
7. A basic pursuit strategy (the Bouguer problem).                                   8. Two special pursuit problems.                                                                 9. A basic escape strategy (the  Bailey problem)                                    10. Basic models of systems with a variable mass.
11. ODE models of a single-species and multi-species population (bifurcation analysis).
12. Modeling via ODEs with a time delay.
13. Modeling via difference equations.