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Lecturer(s)
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Urban František, doc. Ing. Ph.D.
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Course 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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Learning activities and teaching methods
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Lecture, Practicum
- Graduate study programme term essay (40-50)
- 42 hours per semester
- Preparation for an examination (30-60)
- 40 hours per semester
- Contact hours
- 52 hours per semester
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| prerequisite |
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| Knowledge |
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| 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 |
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| 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 |
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| N/A |
| N/A |
| N/A |
| learning outcomes |
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| Knowledge |
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| 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 |
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| 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 |
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| N/A |
| N/A |
| teaching methods |
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| Knowledge |
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| Task-based study method |
| Interactive lecture |
| Skills |
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| Individual study |
| Practicum |
| Competences |
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| Individual study |
| assessment methods |
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| Knowledge |
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| Oral exam |
| Seminar work |
| Skills |
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| Individual presentation at a seminar |
| Skills demonstration during practicum |
| Competences |
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| Oral exam |
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Recommended literature
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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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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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Kuypers, F. Klassische Mechanik. Weinheim, SRN VHC Verlagsgesellchaft mbH, 1989.
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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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Obetková, Viera; Košinárová, Anna; Mamrillová, Anna. Teoretická mechanika. 1. vyd. Bratislava : Alfa, 1990. ISBN 80-05-00597-0.
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Rosenberg, Josef. Teoretická mechanika. 1. vyd. Plzeň : ZČU, 1994. ISBN 80-7082-119-1.
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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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