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Stability of the Zero Solution of Stochastic Differential System
Klimešová, Marie
Stability of stochastic differential equations (SDEs) has become a very popular theme of recent research in mathematics and its applications. The method of Lyapunov functions for the analysis of qualitative behavior of SDEs provide some very powerful instruments in the study of stability properties for concrete stochastic dynamical systems, conditions of existence the stationary solutions of SDEs and related problems.
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Articulated vehicle kinematics
Roman, Matej ; Glos, Jan (oponent) ; Pohl, Lukáš (vedoucí práce)
The goal of this thesis, is to create a kinematic model of an articulated vehicle. The first part deals with finding equations of state of this system and its control driving in reverse. The second part is dedicated to implement the model in the Matlab Simulink environment and to simulate and demonstrate the functionality of control on simple examples, typical for driving in reverse. In the last part a real model is created to test the functionality of control.
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Úloha obchodního cestujícího
Kolář, Adam ; Rozman, Jaroslav (oponent) ; Zbořil, František (vedoucí práce)
Cílem této bakalářské práce je navrhnout prostředí testující problém obchodního cestujícího a porovnat efektivitu jednotlivých přístupů k řešení. V první části jsou diskutovány možnosti genetických algoritmů v závislosti na nastavení křížení, mutací a velikosti populace. V druhé části jsou na stejný problém použity dva druhy neuronových sítí. Za zástupce samoučící varianty byla zvolena Kohonenova neuronová síť. Hopfieldova neuronová síť reprezentuje metodu minimalizace energetické funkce s pevným nastavením koeficientů. U obou neuronových sítí byly popsány možné výhody a nevýhody aplikace. V závěru byly všechny zjištěné poznatky interpretovány ve společném kontextu.
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The Lorenz system: A route from stability to chaos
Arhinful, Daniel Andoh ; Šremr, Jiří (oponent) ; Řehák, Pavel (vedoucí práce)
The theory of deterministic chaos has generated a lot of interest and continues to be one of the much-focused research areas in the field of dynamics today. This is due to its prevalence in essential parts of human lives such as electrical circuits, chemical reactions, the flow of blood through the human system, the weather, etc. This thesis presents a study of the Lorenz equations, a famous example of chaotic systems. In particular, it presents the analysis of the Lorenz equations from stability to chaos and various bifurcation scenarios with numerical and graphical interpretations. It studies concepts of non-linear dynamical systems such as equilibrium points, stability, linearization, bifurcation, Lyapunov function, etc. Finally, it discusses how the Lorenz equations serve as a model for the waterwheel (in detail), and the convection roll for fluid.
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Analyzing stochastic stability of a gyroscope through the stochastic Lyapunov function
Náprstek, Jiří ; Fischer, Cyril
The text delves into the application of first integrals in the construction of Lyapunov functions for analyzing the stability of dynamic systems in stochastic domains. It emphasizes the distinct characteristics of first integrals that warrant the introduction of additional constraints to ensure the essential properties required for a Lyapunov function. These constraints possess physical interpretations associated with system stability. The general approach to testing stochastic stability is illustrated using the example of a 3-degrees-of-freedom system representing a gyroscope.
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Construction of the Lyapunov function reflecting the physical properties of the model
Náprstek, Jiří ; Fischer, Cyril
Practical experience shows that the random excitation component can affect the system response and its dynamic stability not only negatively but also positively. Such mechanisms are usually developed heuristically and are often not sufficiently justified theoretically. The paper presents a possibility of using the properties of first integrals for the construction of a Lyapunov function for the analysis of a dynamic system stability in the stochastic domain. In such case, the Lyapunov function itself contains information on the examined system and, consequently, it is able to provide a more detailed insight into the system stability properties. The procedure is illustrated by a nonlinear SDOF example.
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Application of first integrals in the construction of the Lyapunov function for the random response stability testing
Náprstek, Jiří ; Fischer, Cyril
The paper deals with a possibility of using the properties of first integrals for the construction of Lyapunov function for the analysis of a dynamic system stability in the stochastic domain. It points out certain characteristics of first integrals resulting in the necessity to introduce additional constraints to assure the principal properties of the Lyapunov function. A number of these constraints has their physical interpretation with reference to system stability. The advantage of this method constructing the Lyapunov function consists in the fact that the Lyapunov function itself contains information on the examined system and, consequently, it is not merely a positive definite function without any relation to the actual case concerned. The presented theory finds application in many dynamical systems. The procedure is illustrated by a nonlinear SDOF example.
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The Lorenz system: A route from stability to chaos
Arhinful, Daniel Andoh ; Šremr, Jiří (oponent) ; Řehák, Pavel (vedoucí práce)
The theory of deterministic chaos has generated a lot of interest and continues to be one of the much-focused research areas in the field of dynamics today. This is due to its prevalence in essential parts of human lives such as electrical circuits, chemical reactions, the flow of blood through the human system, the weather, etc. This thesis presents a study of the Lorenz equations, a famous example of chaotic systems. In particular, it presents the analysis of the Lorenz equations from stability to chaos and various bifurcation scenarios with numerical and graphical interpretations. It studies concepts of non-linear dynamical systems such as equilibrium points, stability, linearization, bifurcation, Lyapunov function, etc. Finally, it discusses how the Lorenz equations serve as a model for the waterwheel (in detail), and the convection roll for fluid.
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