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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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On a model of corruption in a democratic society
Splítek, Martin ; Janovský, Vladimír (advisor) ; Mlčoch, Lubomír (referee)
The aim of this work is to study the behavior of serious social pheno- menon - corruption, and we do this through a mathematical model of corruption in a democratic society, published in [1]. The model is a dynamical system of three differential equations, specified by three variables and ten parameters. The model is studied by means of numerical analysis, namely, the method of nume- rical integration of ordinary differential equations and the method of numerical continuation. We used toolbox Matcont [2], which works in the environment of program MATLAB [3]. The result is commented parametric study of the pheno- menon of corruption. Keywords: ordinary diferential equations, dynamic systems, bifurcation ana- lysis 1
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Rovnice geodetiky v prostoročasech s helikální symetrií
Tomášik, Miroslav ; Scholtz, Martin (advisor) ; Žofka, Martin (referee)
In this bachelor thesis we investigate geodesics in helically symmetric spacetimes in the framework of linearized Einstein's gravity. Work is an extension of paper by Bičák, Scholtz and Bohata [2], which is under preparation. First we introduce standard numerical methods for solving systems of ordinary differential equations. Next we present helically symmetric solution of linearized Einstein's equations and numerical code solving the geodesic equation on given background. We discuss conditions of existence of helically symmetric solution and finally we present selected results obtained by numerical simulations. We give present few particular examples of geodesics, selected phase portraits obtained by the method of the Lyapunovov exponents and visualize the causal structure of helically symmetric spacetime.
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Mathematical models of ecosystems
Scholle, David ; Janovský, Vladimír (advisor) ; Kofroň, Josef (referee)
This work is about models of population growth in different situations. At first, we will examine amount of spiders and their prey in the region of Langa Astigiana, based on models of dynamical systems. We will also consider the usage of spraying of near vineyards and effect of this on the ecosystem. The aim of this work is also to check the possibility of periodical cycles, and thus also of the Hopf Bifurcation, appearing. Next part talks about the model of a beehive and examines the influence of insecticides on the population of bee drones and worker bees. The aim of the last chapter is to examine the effectivity and possible impact of human intervention in the region of Šumava forest. The model will check the necessity of such action against parasites. The software used for these tasks will be mainly the continuation toolbox MatCont, which is a part of the program MatLab.
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Numerical Optimization Methods for the Falsification of Hybrid Dynamical Systems
Kuřátko, Jan ; Ratschan, Stefan (advisor) ; Bergamaschi, Luca (referee) ; Lukšan, Ladislav (referee)
Title: Numerical Optimization Methods for the Falsification of Hybrid Dynamical Systems Author: Jan Kuřátko Department: Department of Numerical Mathematics Supervisor: Stefan Ratschan, Institute of Computer Science, The Czech Academy of Sciences Abstract: This thesis consists of three published papers that contribute to the finding of error trajectories of hybrid dynamical systems. A hybrid dynamical system is a dynamical system that has both discrete and continuous state. For example, one can use it as a model for a thermostat in a room: Such a thermostat may have two discrete states, one where the heating is off, and another one, where the heating is on. Its continuous state is the temperature in the room. For such a model one may be interested in finding an error trajectory, that is, an evolution of the system that reaches an unsafe state that is to be avoided. Industry is in need of methods for automatized testing and verification of safety conditions in order to identify flaws in the design of systems. The thesis contains several contributions to finding error trajectories that are based on numerical optimization. Keywords: optimization, dynamical systems, saddle-point matrix
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Dynamical systems in cosmology
Knob, Lukáš ; Acquaviva, Giovanni (advisor) ; Loukes Gerakopoulos, Georgios (referee)
The main aim of this thesis is the analysis of different cosmological models from the standpoint of dynamical systems theory. We consider mostly spatially curved FLRW metric with different source terms, some of them possible candidates for dark matter and dark energy, particularly linear barotropic fluids, Chaplygin gas and canonical scalar field with exponential and general form of potential. We rewrite the cosmological equations as the system of the first order differential equations in dimensionless variables and study globally their phase space and the stability of the critical points. We also present few interesting features of models with interactions between two cosmic fluid constituents and mention dynamical properties of orthogonal Bianchi I models. 1
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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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Investigation of chaotic dynamical systems using the methods of Riemannian geometry
Matejov, Dávid ; Scholtz, Martin (advisor) ; Ledvinka, Tomáš (referee)
In this thesis we investigate a chaos in dynamical systems described by the Hamilton function using a new geometric method. At first, necessary definitions and terms are introduced, like dynamical systems, Lyapunov ex- ponents and Poncaré sections. Subsequently we deal with the new geometric method (Horowitz et al., 2007), which predicts the stability of a system un- der consideration. An extensive part is devoted to the general relativity and spinor formalism, the Newman-Penrose formalism and the theory of optical scalars. In the computational part of the thesis, we investigate the pp-waves and chaotic behaviour of geodesics in this class of space-times. All calcula- tions are done in the Python programming language, so we the chapter on numerical calculations is included. 1
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Inherent instability of financial markets
Hladík, Jan ; Buben, Radek (advisor) ; Znoj, Milan (referee)
The main aim of this presented diploma thesis is to help build a systematic understanding of the political and social foundations of global financial markets, their operations and impacts on the global power affairs. The thesis highlights the dynamic complexity of the post financial crisis state of the World with its itra- and inter-social features. It instrumentaly uses critique of a free market agenda and neo-classical economy which contrasts the Efficient Markets Hypothesis with Hyman Minsky's Financial Instability Hypothesis (FIH), taking into account the dynamic complexity of financial markets. This approach offers analytical tools that can account for crisis through processes endogenous to contemporary financial capitalism. I shall argue that a financially complex system is, according to the FIH, inherently flawed and unstable. After a theoretical and historical review, the thesis discusses various aspects of the process of austerity regime and its social consequences. This provides an opportunity for analyses of the ongoing existence of interstate competition, of militarised foreign policy, and of other international, at times violent conflicts. In an effort to make sense of some of these phenomena, I instrumentaly use the study of geoeconomics that builds on some fundamental assumptions...
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