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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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Tracking mortar shell
Miklín, Vojtěch ; Tůma, Jiří (advisor) ; Růžička, Pavel (referee)
This thesis aims to summarize possibilities of using Kalman filter for state estimation of a discrete dynamic system known only from inaccurate measurements. Firstly the notion of statistics, dynamic system and coordinate systems (Cartesian and spherical) are defined. Then the Kalman filter algorithm is described and a physical model for the filter is derived. This thesis is based on chapter 7 of [Tomasi, 1997] (textbook for the CS 205 lecture at Stanford University). Carlo Tomasi applies theory on a simplified example of mortar shell tracking. One of this thesis' objectives was to generalize the method for the case when the observer is not located in the 2D plain of flight of a shell. Another task was to include air resistance and wind in the model equations.
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Nestandardní analýza dynamických systémů
Slavík, Jakub ; Pražák, Dalibor (advisor) ; Růžička, Pavel (referee)
In the presented thesis, we study an application of nonstandard analysis to dynamical systems, in particular to ω-limit set, stability and global attractor. We recall the definition and properties of elementary embedding, in detail ex- plore the introduction of infinitesimals to the real line and study metric spaces using nonstandard methods, in particular continuity and compactness which are closely related to the theory of dynamical systems. Last we attend to dynamical systems and present nonstandard characterizations of some of its properties such as asymptotic compactness and dissipativity and using these characterizations we prove one of the basic results of this theory - existence of a global attractor. 1
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Lyapunov exponents – practical computation
Fischer, Cyril ; Náprstek, Jiří
The Lyapunov exponents serve as numerical characteristics of dynamical systems, which measure possible divergence of nearby trajectories of the solution. In this way they express dependence of the dynamical system on initial conditions. However, the value of Lyapunov exponents consists in their ability to characterise deterministic chaos. The limiting process intrinsic in the definition of Lyapunov exponents, unfortunately, seriously complicates their computation. The short paper presents an overview of difficulties in numerical approaches to enumeration of Lyapunov exponents or at least the largest one and shows a promising method based on QR decomposition of the system Jacobian.
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Analýza atraktorů zobecněných Newtonovských tekutin v 3d oblastech
Žabenský, Josef ; Pražák, Dalibor (advisor) ; Bulíček, Miroslav (referee)
We investigate a system of nonlinear partial differential equations, specifically the so-called Ladyzhenskaya model, in three spatial dimensions. It will be shown that after inclusion of a perturbation of a higher order, the model exhibits a considerably better behavior, in particular it will become quite straightforward to prove differentiability of solutions with respect to the initial condition. Due to this fact we may consequently employ the method of Lyapunov exponents to estimate the fractal dimension of the exponential attractor. First, however, it will be necessary to show existence and uniqueness of solutions, improved regularity and existence of a compact invariant set for the entire system.
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Unconventional Signals Oscillators
Hruboš, Zdeněk ; Galajda, Pavol (referee) ; Štork, Milan (referee) ; Petržela, Jiří (advisor)
Dizertační práce se zabývá elektronicky nastavitelnými oscilátory, studiem nelineárních vlastností spojených s použitými aktivními prvky a posouzením možnosti vzniku chaotického signálu v harmonických oscilátorech. Jednotlivé příklady vzniku podivných atraktorů jsou detailně diskutovány. V doktorské práci je dále prezentováno modelování reálných fyzikálních a biologických systémů vykazujících chaotické chování pomocí analogových elektronických obvodů a moderních aktivních prvků (OTA, MO-OTA, CCII ±, DVCC ±, atd.), včetně experimentálního ověření navržených struktur. Další část práce se zabývá možnostmi v oblasti analogově – digitální syntézy nelineárních dynamických systémů, studiem změny matematických modelů a odpovídajícím řešením. Na závěr je uvedena analýza vlivu a dopadu parazitních vlastností aktivních prvků z hlediska kvalitativních změn v globálním dynamickém chování jednotlivých systémů s možností zániku chaosu v důsledku parazitních vlastností použitých aktivních prvků.
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Nonlinear Physics and Chaos Theory
NÁHLÍK, Tomáš
This thesis deals with nonlinear physics and chaos theory from its begining, through the main people to its application in various fields. This work has also part of fractals and fractal geometry. There are also source codes of various examples.
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Two-mass dynamical system with singular stiffnes matrix
Kozánek, Jan
In this paper the solution of two-mass symmetric dynamical system with singular stiffnes matrix via the state space formulation is given. The coefficient matrix is non-diagonalizable. The terminology problems concerning eigenvectors and latent vectors are discussed, too. Finally the resolvent of this symmetric system was expressed in symmetric form.
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Dvě rovnice popisjící kyvadlový tlumič
Fischer, Cyril ; Náprstek, Jiří
The pendulum damper modelled as a two degree of freedom strongly non-linear auto-parametric system is investigated using two approximate differential systems. Uni-directional harmonic external excitation at the suspension point is considered. Semi-trivial solutions and their stability are analyzed. The thorough analysis of the non-linear system using less simplification than it is used in the previous paper is performed. Both approaches are compared and conclusions are drawn.
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