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Mathematical modelling of walking robots
Kiša, Daniel ; Opluštil, Zdeněk (referee) ; Tomášek, Petr (advisor)
Tato diplomová práce se zabývá matematickými modely kráčejících robotů. Dva z těchto modelů jsou vybrány a analyzovány. Pasivní model "rimless wheel" , který slouží jako základ pro další, složitější modely, je podrobně analyzován. "Compass gait" model dvounohého robota je v práci analyzován a numericky simulován v programovacím jazyce Python. Metoda pro nalezení podmínek pro pasivní chůzi robota je rovněž implementována.
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Bifurcations in a chaotic dynamical system
Kateregga, George William ; Tomášek, Petr (referee) ; Nechvátal, Luděk (advisor)
Dynamical systems possess an interesting and complex behaviour that have attracted a number of researchers across different fields, such as Biology, Economics and most importantly in Engineering. The complex and unpredictability of nonlinear customary behaviour or the chaotic behaviour, makes it strange to analyse them. This thesis presents the analysis of the system of nonlinear differential equations of the so--called Lu--Chen--Cheng system. The system has similar dynamical behaviour with the famous Lorenz system. The nature of equilibrium points and stability of the system is presented in the thesis. Examples of chaotic dynamical systems are presented in the theory. The thesis shows the dynamical structure of the Lu--Chen--Cheng system depending on the particular values of the system parameters and routes to chaos. This is done by both the qualitative and numerical techniques. The bifurcation diagrams of the Lu--Chen--Cheng system that indicate limit cycles and chaos as one parameter is varied are shown with the help of the largest Lyapunov exponent, which also confirms chaos in the system. It is found out that most of the system's equilibria are unstable especially for positive values of the parameters $a, b$. It is observed that the system is highly sensitive to initial conditions. This study is very important because, it supports the previous findings on chaotic behaviours of different dynamical systems.
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Modelling of postcritical states of slender structures
Mašek, Jan ; Eliáš, Jan (referee) ; Frantík, Petr (advisor)
The aim of the presented thesis is to create a compact publication which deals with properties, solution and examination of behavior of dynamical systems as models of mechanical structures. The opening portion of the theoretical part leads the reader through the subject of description of dynamical systems, offers solution methods and investigates solution stability. As the introduction proceeds, possible forms of structure loading, damping and response are presented. Following chapters discuss extensively the possible approaches to system behavior observation and identification of nonlinear and chaotic phenomena. The attention is also paid to displaying methods and color spaces as these are essential for the examination of complex and sensitive systems. The theoretical part of the thesis ends with an introduction to fractal geometry. As the theoretical background is laid down, the thesis proceeds with an application of the knowledge and shows the approach to numerical simulation and study of models of real structures. First, the reader is introduced to the single pendulum model, as the simplest model to exhibit chaotic behavior. The following double pendulum model shows the obstacles of observing systems with more state variables. The models of free rod and cantilever serve as examples of real structure models with many degrees of freedom. These models show even more that a definite or at least sufficiently relevant monitoring of behavior of such deterministic systems is a challenging task which requires sophisticated approach.
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Attractors in the complex dynamics of turbulent convection
Kašný, Jakub ; Nechvátal, Luděk (referee) ; Macek, Michal (advisor)
This Bachelor's thesis deals with an application of the HAVOK (Hankel Alternative View of Koopman) numerical method, which seeks attractors and predicts intermittent phenomena in dynamical systems, to data from Rayleigh-Bénard convection (RBC), which are measured at Brno Institute of Scientific Instruments in the group of Cryogenics and Superconductivity. This thesis discusses the theory on which the HAVOK is built and further deepens it compared to the article [2]. Furthermore, it enlightens some issues as the best selection of the embedding dimension r, which we selected based on the quality of regression that HAVOK creates, or the use of the Koopman operator and Taken's embedding theorem, that weren't explicitly explained in the article [2]. We discovered three different methods to compute HAVOK regressions based on and using the codes attached to the article. In the thesis, we inspect the matrices of ordinary differential equations, their behaviour when the initial values are changed and their stability for the different regression models and embedding dimensions. The solution with different initial conditions is plotted so that the attractivity can be seen. Part of the thesis contains description of RBC, its equations of motion and characteristic dimensionless numbers that describe the convection. Moreover, the thesis describes how the data are obtained and processed normally and how are processed in new ways based on the HAVOK method.
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Identification of quasiperiodic processes in the vicinity of the resonance
Fischer, Cyril ; Náprstek, Jiří
In nonlinear dynamical systems, strong quasiperiodic beating effects appear due to combination of self-excited and forced vibration. The presence of symmetric or asymmetric beatings indicates an exchange of energy between individual degrees of freedom of the model or by multiple close dominant frequencies. This effect is illustrated by the case of the van der Pol equation in the vicinity of resonance. The approximate analysis of these nonlinear effects uses the harmonic balance method and the multiple scale method.
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Delay differential equations in engineering
Zlámal, Ondřej ; Řehák, Pavel (referee) ; Opluštil, Zdeněk (advisor)
This thesis is about dynamical systems and analysis of their stability. These systems are described using delayed differential equations, whose character is ideal for describing many real life problems. In this thesis it is analysed how size of delay and its rate affects stability of system. Change of stability in system is traced using Hopf bifurcations. Theory of this thesis will be applied on system based on machine tool vibrations and system describing feedback in lasers.
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Attractors in the complex dynamics of turbulent convection
Kašný, Jakub ; Nechvátal, Luděk (referee) ; Macek, Michal (advisor)
This Bachelor's thesis deals with an application of the HAVOK (Hankel Alternative View of Koopman) numerical method, which seeks attractors and predicts intermittent phenomena in dynamical systems, to data from Rayleigh-Bénard convection (RBC), which are measured at Brno Institute of Scientific Instruments in the group of Cryogenics and Superconductivity. This thesis discusses the theory on which the HAVOK is built and further deepens it compared to the article [2]. Furthermore, it enlightens some issues as the best selection of the embedding dimension r, which we selected based on the quality of regression that HAVOK creates, or the use of the Koopman operator and Taken's embedding theorem, that weren't explicitly explained in the article [2]. We discovered three different methods to compute HAVOK regressions based on and using the codes attached to the article. In the thesis, we inspect the matrices of ordinary differential equations, their behaviour when the initial values are changed and their stability for the different regression models and embedding dimensions. The solution with different initial conditions is plotted so that the attractivity can be seen. Part of the thesis contains description of RBC, its equations of motion and characteristic dimensionless numbers that describe the convection. Moreover, the thesis describes how the data are obtained and processed normally and how are processed in new ways based on the HAVOK method.
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A traffic flow with a bottelneck
Kovařík, Adam ; Janovský, Vladimír (advisor) ; Vejchodský, Tomáš (referee)
Title: A traffic flow with a bottelneck Author: Adam Kovařík Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Vladimír Janovský, DrSc. Supervisor's e-mail address: janovsky@karlin.mff.cuni.cz Abstract: In this paper we study a microscopic follow-the-leader traffic model on a circu- lar road with a bottleneck. We assume that all drivers are identical and overtaking is not permitted. We sketch a small part of the rich dynamics of the model including Hopf and Neimark-Sacker bifurcations. We introduce so called POM and quasi-POM solutions and an algorithm how to search them. The main goal of this work is to investigate how the optimal velocity model with a bottleneck deals with so called aggressive behavior of dri- vers. The effect of variable reaction time and a combination of both named factors is also tested. Using numerical simulations we'll find out that aggressiveness and faster reactions have positive effect on traffic flow. In the end we discuss models with two bottlenecks and with one extraordinary driver. Keywords: dynamical systems, ODEs, traffic flow, bottleneck, aggressiveness. 1
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Chaotic system modeling using MATLAB
Lejdar, Lukáš ; Raidl, Aleš (advisor) ; Šindelářová, Kateřina (referee)
In the presented bachelor's thesis we study behavior of dynamical systems. Some interesting attributes of dynamical systems are presented using programs written by the author. For computational part of the programs MATLAB was used and for presentation of output data MATLAB in combination with GNUPLOT were used. Basic terms in chaos theory are explained with examples. In one-dimensional case we focus on the logistic map and we demonstrate a transition to chaos on it. In two-dimensional space we study the Hénon map and in three-dimensional space we take a closer look at some interesting attributes of the famous Lorenz system.
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