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Autonomous systems of differential equations - classical vs fractional ones
Glozigová, Anna ; Zatočilová, Jitka (referee) ; Nechvátal, Luděk (advisor)
Hlavním zaměřením této práce je hlubší studium a porovnání dvou oblastí diferenciálních rovnic, kde důraz je kladen na neceločíselné řády, neboť během posledních desítek let se tato oblast nejenže stala populární, ale dokonce bylo zjištěno, že standardní přístupy řešení nenaplňují očekávání, tudíž jsou vyžadovány speciální postupy. Práce také obsahuje příklady, experimenty a simulaci pro ověření, případné vyvrácení teoretických výsledků.
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Periodic boundary value problem in mathematical models of nonlinear oscillators
Kyjovský, Adam ; Nechvátal, Luděk (referee) ; Šremr, Jiří (advisor)
This master's thesis deals with qualitative analysis of nonlinear differential equations of second order. For autonomous equations some basic notions of Hamiltonian systems (mainly construction of phase portrait) are presented. For non-autonomous equations the method of lower and upper functions for periodic boundary value problem is used. These notions are then applied to a model of mechanical oscillator, a question of existence of solutions to autonomous and non-autonomous nonlinear differential equations is studied.
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Rotation Number on a Circle
Bíma, Jan ; Vejnar, Benjamin (advisor) ; Pražák, Dalibor (referee)
We apply the dynamical method to obtain structural results concerning certain classes of one-dimensional maps. The notion central to the work is that of a rotation number on a circle; we relate the rotation modulus to periodicity of an orientation-preserving circle homeomorphism and generalize the concept to continuous degree-1 circle maps. We investigate the asymptotic orbit behaviour of circle homeomorphisms with irrational rotation number and develop the Poincaré Classification Theorem which establishes topo- logical (semi-)conjugacy of a circle homeomorphism with an irrational rotation number to a rotation with the same rotation number. 1
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Chimney demolition modeling
Ficker, Tomáš ; Keršner, Zbyněk (referee) ; Frantík, Petr (advisor)
The thesis deals with solving of a discrete dynamic 2D model of a chimney during its demolition. Specific goal of the thesis is to clarify observed phenomenon that is a break of the chimney while falling. The model of the chimney is designed and solved using software FyDiK which is able to solve dynamic effects. For verification of results, two models of chimney have been designed.
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Fractals in Computer Graphics
Šelepa, Jan ; Venera, Jiří (referee) ; Sumec, Stanislav (advisor)
The goal of this work is to give introduction and specification of fractals. The first chapter presents the basics of fractal geometry. The second chapter maps the history of fractals and points out the most important people of fractal science. The third chapter presents fractal classification based on several criteria and gives basic examples. The fourth chapter is a summary of widely used applications for fractal creation. The last chapter describes the application that was made to demonstrate given algorithms and fractals mentioned in this thesis.
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Strategies for computation of Lyapunov exponents estimates from discrete data
Fischer, Cyril ; Náprstek, Jiří
The Lyapunov exponents (LE) provide a simple numerical measure of the sensitive dependence of the dynamical system on initial conditions. The positive LE in dissipative systems is often regarded as an indicator of the occurrence of deterministic chaos. However, the values of LE can also help to assess stability of particular solution branches of dynamical systems. The contribution brings a short review of two methods for estimation of the largest LE from discrete data series. Two methods are analysed and their freely available Matlab implementations are tested using two sets of discrete data: the sampled series of the Lorenz system and the experimental record of the movement of a heavy ball in a spherical cavity. It appears that the most important factor in LE estimation from discrete data series is quality of the available record.
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Stability and control of dynamical systems used in modeling an airplane motion
Novák, Jiří ; Šremr, Jiří (referee) ; Nechvátal, Luděk (advisor)
A modern aircraft (or another machine moving in the air) usually does not rely on its structural stability only. In fact, the motion is conventionally stabilized using a loop feedback control. A dynamical system, which models the aircraft's position and orientation in time, reacts on the state variables, hence, the control signal is dynamically changed. This bachelor's thesis deals with deriving the equations of motion for small perturbations of the state variables as well as with stability and control of such a system of equations. In addition, comparison of both a nonlinear and a linearized model is a part of this work. The programming language Python is used for testing on several examples of a concrete aircraft.
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Nonlinear dynamical systems and chaos
Tesař, Lukáš ; Opluštil, Zdeněk (referee) ; Nechvátal, Luděk (advisor)
The diploma thesis deals with nonlinear dynamical systems with emphasis on typical phenomena like bifurcation or chaotic behavior. The basic theoretical knowledge is applied to analysis of selected (chaotic) models, namely, Lorenz, Rössler and Chen system. The practical part of the work is then focused on a numerical simulation to confirm the correctness of the theoretical results. In particular, an algorithm for calculating the largest Lyapunov exponent is created (under the MATLAB environment). It represents the main tool for indicating chaos in a system.
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Topological entropy
Češík, Antonín ; Vejnar, Benjamin (advisor) ; Pražák, Dalibor (referee)
In this thesis we study topological entropy as an invariant of topological dynamical systems. The first chapter contains basic definitions and examples of topological dynamical systems. In the second chapter we introduce the definition of topological entropy on a compact metric space. We study its properties, in particular the fact that it is invariant under conjugacy. The chapter concludes with calculation of topological entropy for the examples introduced in the first chapter. The last chapter deals with generalizing the notion of topological entropy to noncompact metric spaces. The case of piecewise affine maps on the real line is studied in more detail.
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