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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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Selected problems in relativistic cosmology
Kerachian, Morteza ; Bičák, Jiří (advisor) ; Balek, Vladimír (referee) ; Vikman, Alexander (referee)
In this work, we studied three selected problems in FRW spacetime. In the first part, we analysed the motion of a test particle in the homogeneous and isotropic universe. We presented a framework in which one can derive the uniformly accelerated trajectory and geodesic motion if a scale factor for a given spacetime is provided as a function of coordinate time. By applying the confomal time transformation, we were able to convert second order differential equations of motion in FRW spacetime to first order differential equations. From this, we managed to obtain a formalism to derive the uniformly accelerated trajectory of a test particle in spatially curved FRW spacetime. The second part of this work is devoted to dynamical cosmology. In particular, we analyse the cases of barotropic fluids and non-minimally coupled scalar field in spatially curved FRW spacetime. First, we set up the dynamical systems for an unspecified EoS of a barotropic fluid case and an unspecified positive potential for a non-minimal coupled scalar field case. For both of these systems, we determined well-defined dynamical variables valid for all curvatures. In the framework of these general setups we discovered several characteristic features of the systems, such as invariant subsets, symmetries, critical points and their...
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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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Dynamical analysis approaches in spatially curved FRW spacetimes
Kerachian, M. ; Acquaviva, G. ; Lukes-Gerakopoulos, Georgios
We summarize two agnostic approaches in the framework of spatially curved Friedmann-Robertson-Walker (FRW) cosmologies discussed in detail in (Kerachian et al., 2020, 2019). The first case concerns the dynamics of a fluid with an unspecified barotropic equation of state (EoS), for which the only assumption made is the non-negativity of the fluid’s energy density. The second case concerns the dynamics of a non-minimally coupled real scalar field with unspecified positive potential. For each of these models, we define a new set of dimensionless variables and a new evolution parameter. In the framework of these agnostic setups, we are able to identify several general features, like symmetries, invariant subsetsand critical points, and provide their cosmological interpretation.
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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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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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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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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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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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