Academic year 2023/2024 |
Supervisor: | prof. RNDr. Jan Čermák, CSc. | |||
Supervising institute: | ÚM | |||
Teaching language: | Czech | |||
Aims of the course unit: | ||||
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. |
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Learning outcomes and competences: | ||||
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. |
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Prerequisites: | ||||
Differential and integral calculus of functions in a single and more variables, theory of ordinary differential equations, linear algebra. |
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Course contents: | ||||
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. |
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Teaching methods and criteria: | ||||
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on mathematical modelling via differential equations, and on practical topics presented in lectures.
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Assesment methods and criteria linked to learning outcomes: | ||||
Course-unit credit is awarded on the following conditions: Active participation in seminars. Fulfilment of all conditions of the running control of knowledge. |
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Controlled participation in lessons: | ||||
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. |
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Type of course unit: | ||||
Lecture | 13 × 2 hrs. | optionally | ||
Exercise | 13 × 2 hrs. | compulsory | ||
Course curriculum: | ||||
Lecture | 1. Stability of solutions of ODE systems (basic notions and properties). |
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Exercise | 1. Applications of ODEs in mechanics (basic problems). |
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Literature - fundamental: | ||||
1. Perko, L.: Differential Equations and Dynamical Systems, Springer-Verlag, 1991. | ||||
2. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, Berlin, Springer, 1990. |
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3. Fulford, G., Forrester, P., Jones, A.: Modelling with Differential and Difference Equations, New York, 2001. | ||||
Literature - recommended: | ||||
1. Strogatz, S.: Nonlinear Dynamics and Chaos, With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity), Avalon Publishing, 2014 | ||||
2. Nahin, P.J.: Chases and Escapes: the mathematics of pursuit and evasion, Princeton University Press, Princetion, 2007. | ||||
3. Rachůnková, I, Fišer, J.: Dynamické systémy 1, UP Olomouc, 2014 |
The study programmes with the given course: | |||||||||
Programme | Study form | Branch | Spec. | Final classification | Course-unit credits | Obligation | Level | Year | Semester |
B-MAI-P | full-time study | --- no specialisation | -- | Cr,Ex | 5 | Compulsory-optional | 1 | 2 | S |
Faculty of Mechanical Engineering
Brno University of Technology
Technická 2896/2
616 69 Brno
Czech Republic
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