Course detail
Dynamical Systems and Mathematical Modelling
FSI-SA0 Acad. year: 2023/2024 Summer semester
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
Supervisor
Department
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.
The study programmes with the given course
Programme B-MAI-P: Mathematical Engineering, Bachelor's, compulsory-optional
Type of course unit
Lecture
26 hours, optionally
Teacher / Lecturer
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 hours, compulsory
Teacher / Lecturer
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.