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Main menu for Browse IS/STAG
Course info
KME / DYCH
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Course description
Department/Unit / Abbreviation
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KME
/
DYCH
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Academic Year
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2023/2024
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Academic Year
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2023/2024
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Title
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Nonlinear dynamics and chaos
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Form of course completion
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Exam
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Form of course completion
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Exam
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Accredited / Credits
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Yes,
5
Cred.
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Type of completion
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Oral
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Type of completion
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Oral
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Time requirements
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Lecture
3
[Hours/Week]
Tutorial
2
[Hours/Week]
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Course credit prior to examination
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Yes
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Course credit prior to examination
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Yes
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Automatic acceptance of credit before examination
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Yes in the case of a previous evaluation 4 nebo nic.
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Included in study average
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YES
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Language of instruction
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Czech, English
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Occ/max
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|
|
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Automatic acceptance of credit before examination
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Yes in the case of a previous evaluation 4 nebo nic.
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Summer semester
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0 / -
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1 / -
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0 / -
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Included in study average
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YES
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Winter semester
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0 / -
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0 / -
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0 / -
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Repeated registration
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NO
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Repeated registration
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NO
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Timetable
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Yes
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Semester taught
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Summer semester
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Semester taught
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Summer semester
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Minimum (B + C) students
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10
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Optional course |
Yes
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Optional course
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Yes
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Language of instruction
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Czech, English
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Internship duration
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0
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No. of hours of on-premise lessons |
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Evaluation scale |
1|2|3|4 |
Periodicity |
každý rok
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Evaluation scale for credit before examination |
S|N |
Periodicita upřesnění |
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Fundamental theoretical course |
No
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Fundamental course |
No
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Fundamental theoretical course |
No
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Evaluation scale |
1|2|3|4 |
Evaluation scale for credit before examination |
S|N |
Substituted course
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None
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Preclusive courses
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N/A
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Prerequisite courses
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N/A
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Informally recommended courses
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N/A
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Courses depending on this Course
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N/A
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Histogram of students' grades over the years:
Graphic PNG
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XLS
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Course objectives:
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The student will be introduced with
- basic principles and theorems of the theoretical mechanics of discrete dynamical systems in the configurational space (Lagrangian approach) and in the phase space (hamiltonian approach).
- the principles and theorems of the mathematical theory of dynamical systems
- the examples of dynamical systems related to the oscillators and biological systems
- methods of approximation of the nonlinear dynamical systems solution
- forms of transition to the deterministic chaos
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Requirements on student
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Requirements on students:
Development and delivery of the semester work at the appropriate level
Examination Requirements:
An active substance taught knowledge and application of theoretical knowledge to solve specific problems of discrete mechanical systems.
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Content
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1.Discrete dynamical system. Generalized coordinates, constrains, configurational and phase space. Principle of the virtual work, equilibrium stability.
2.Hamiltonial principle. Lagrangian equations of second order, dissipation. Balance laws, Noether theorem, Liouville theorem, Poisson brackets
3. Canonical equations and transformations. Legendre transformation, Hamiltonian equations (Hamilton-Jacobi theory).
4.Basic terms of the nonlinear dynamical systems theory, continuous and discrete dynamical systems
5.Fixed points and attractors in autonomous systems - ecological systems
6.Limit cycles in autonomous systems - bifurcation types, bifurcation in chemical oscillator, quasiperiodic solution
7.Periodic and chaotic attractors of excited oscillators - Poincare's mapping,
Van der Pol oscillator, Birkhoff-Shaw chaotic attractor
8.Stability and bifurcation of iterative mappings. Chaos of iterative mappings, logical mapping, Smale horseshoe
9. Multiple scale method
10.Types of chaos transition, period doubling, intermitance, quasiperiodic way, crisis
11.Applications, Lorenz system, Rossler band
12.Chaos in the hamoltonian systems
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Activities
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Fields of study
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Studentům je k dispozici kurz v Google Classroom se všemi podstatnými informacemi a materiály.
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Guarantors and lecturers
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Literature
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Basic:
Horák, Jiří; Krlín, Ladislav; Raidl, Aleš. Deterministický chaos a jeho fyzikální aplikace. Vyd. 1. Praha : Academia, 2003. ISBN 80-200-0910-8.
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Basic:
Rosenberg, Josef. Teoretická mechanika. 1. vyd. Plzeň : ZČU, 1994. ISBN 80-7082-119-1.
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Recommended:
Nayfeh, Ali Hasan; Balachandran, Balakumar. Applied nonlinear dynamics : analytical, computational, and experimental methods. New York : John Wiley & Sons, 1995. ISBN 0-471-59348-6.
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Recommended:
Kuypers, F. Klassische Mechanik. Weinheim, SRN VHC Verlagsgesellchaft mbH, 1989.
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Recommended:
Thompson, J. M. T.; Stewart, H. B. Nonlinear dynamics and chaos. 2nd ed. Chichester : John Wiley & Sons, 2002. ISBN 0-471-87645-3.
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Recommended:
Obetková, Viera; Košinárová, Anna; Mamrillová, Anna. Teoretická mechanika. 1. vyd. Bratislava : Alfa, 1990. ISBN 80-05-00597-0.
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Recommended:
Brdička, Miroslav; Hladík, Arnošt. Teoretická mechanika : Celost. vysokošk. učebnice pro stud. matematicko-fyz. a pedagog. fakult, stud. oboru učitelství všeobecně vzdělávacích předmětů. 1. vyd. Praha : Academia, 1987.
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On-line library catalogues
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Time requirements
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All forms of study
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Activities
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Time requirements for activity [h]
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Graduate study programme term essay (40-50)
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42
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Contact hours
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52
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Preparation for an examination (30-60)
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40
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Total
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134
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Prerequisites
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Knowledge - students are expected to possess the following knowledge before the course commences to finish it successfully: |
orient yourself in differential equations |
orient yourself in differential and integral calculus |
orient yourself in the classical mechanics of material points and bodies |
orient yourself in numerical mathematics |
Skills - students are expected to possess the following skills before the course commences to finish it successfully: |
describe and solve specific problems of differential and integral calculus in application to mechanical systems |
describe and solve basic types of first and second order differential equations with applications in physics |
describe and solve the balance of a system of material points and bodies (static and dynamic problems) |
Competences - students are expected to possess the following competences before the course commences to finish it successfully: |
N/A |
N/A |
N/A |
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Learning outcomes
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Knowledge - knowledge resulting from the course: |
describe approximation methods for solving nonlinear problems (method of multiple scales and reduction to a central variety) |
describe the division of dynamical systems |
describe problems in Newtonian and Hamiltonian mechanics |
enumerate and explain the basic concepts and theorems of the theory of nonlinear dynamical systems |
explain the basics of deterministic chaos theory |
Skills - skills resulting from the course: |
characterize the properties of the obtained solution (stability, chaos, etc.) |
find approximations of the solution using the method of multiple scales or reduction to the central variety |
solve problems of the dynamics of linear and non-linear systems |
determine bifurcations of codimension 1 |
Competences - competences resulting from the course: |
N/A |
N/A |
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Assessment methods
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Knowledge - knowledge achieved by taking this course are verified by the following means: |
Oral exam |
Seminar work |
Skills - skills achieved by taking this course are verified by the following means: |
Individual presentation at a seminar |
Skills demonstration during practicum |
Competences - competence achieved by taking this course are verified by the following means: |
Oral exam |
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Teaching methods
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Knowledge - the following training methods are used to achieve the required knowledge: |
Interactive lecture |
Task-based study method |
Skills - the following training methods are used to achieve the required skills: |
Individual study |
Practicum |
Competences - the following training methods are used to achieve the required competences: |
Individual study |
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