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Mathematical models of linear and nonlinear oscillators
Lovas, David ; Opluštil, Zdeněk (referee) ; Čermák, Jan (advisor)
This bachelor thesis deals with mathematical models of linear mechanical oscillators, which represent one of basic applications of ordinary differential equations. There are explained harmonic oscillators, damped oscillators and driven oscillators. The thesis discusses also superposition of oscillators and coupling of oscillators, including their synchronization.
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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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An analysis of differential equations for systems involving bottlenecks
Borkovec, Ondřej ; Opluštil, Zdeněk (referee) ; Kisela, Tomáš (advisor)
This thesis deals with modelling of the flow of products through systems involving bottlenecks using ordinary differential equations. The model is based on hydrodynamics analogy. Further, the conditions for the sustainability of a system, that is the requirements needed not to exceed the maximal capacity, so that the flow of products can flow continuously through the given spot. A model is used to solve examples for vayrying systems.
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Mathematical modelling with differential equations
Béreš, Lukáš ; Šremr, Jiří (referee) ; Opluštil, Zdeněk (advisor)
Diplomová práce je zaměřena na problematiku nelineárních diferenciálních rovnic. Obsahuje věty důležité k určení chování nelineárního systému pouze za pomoci zlinearizovaného systému, což je následně ukázáno na rovnici matematického kyvadla. Dále se práce zabývá problematikou diferenciálních rovnic se zpoždéním. Pomocí těchto rovnic je možné přesněji popsat některé reálné systémy, především systémy, ve kterých se vyskytují časové prodlevy. Zpoždění ale komplikuje řešitelnost těchto rovnic, což je ukázáno na zjednodušené rovnici portálového jeřábu. Následně je zkoumána oscilace lineární rovnice s nekonstantním zpožděním a nalezení podmínek pro koeficienty rovnice zaručující oscilačnost každého řešení.
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Mathematical modelling of flight dynamics
Resl, Ondřej ; Tomášek, Petr (referee) ; Opluštil, Zdeněk (advisor)
This thesis deals with the mathematical models which describe flight dynamics of rocket. It mainly discusses the problem of smooth landing under different conditions, but it also deals with the range of a rocket. Certain models are provided with numerical solutions. The thesis also contains theoretical introduction to given issue.
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Numerical solution of index-2 differenial-algebraic equations
Kroulíková, Tereza ; Opluštil, Zdeněk (referee) ; Zatočilová, Jitka (advisor)
This bachelor´s thesis deals with numerical solution of differential-algebraic equations. At first these equations are described theoretically and their basic properties are presented. Main attention is paid to index and the most used indexes are described in details. Then the thesis concentrates on numerical solution of Hessenberg forms index-2 differential-algebraic equations. Implicit Runge-Kutta methods and backward differentiation formulas are derived. Those are used for solution of index-2 differential-algebraic equations.
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Traffic flow modelling
Ježková, Jitka ; Opluštil, Zdeněk (referee) ; Kisela, Tomáš (advisor)
Tato diplomová práce prezentuje problematiku dopravního toku a jeho modelování. Zabývá se především několika LWR modely, které následně rozebírá a hledá řešení pro počáteční úlohy. Ukazuje se, že ne pro všechny počáteční úlohy lze řešení definovat na celém prostoru, ale jen v určitém okolí počáteční křivky. Proto je dále odvozena metoda výpočtu velikosti tohoto okolí a to nejen zcela obecně, ale i pro dané modely. Teoretický rozbor LWR modelů a řešení počátečních úloh jsou demonstrovány několika příklady, které zřetelně ukazují, jak se dopravní tok simulovaný danými modely chová.
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Continuous mathematical models of population dynamics
Pecka, Luboš ; Opluštil, Zdeněk (referee) ; Franců, Jan (advisor)
The aim of this thesis is to describe the most frequent models describing population dynamics and then to perform some numerical experiments in the MATLAB environment. These simulations should validate our theoretical results. The models are sorted from the basic models to the most complicated and are divided into the models which describe dynamics of one population and models of coexistence of two biological species. The master's thesis icludes also a program for drawing graphs and trajectories of solutions of models described in this thesis including a description of this MATLAB program.
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Macroscopic traffic flow modelling
Pidrová, Kateřina ; Opluštil, Zdeněk (referee) ; Kisela, Tomáš (advisor)
This bachelor thesis is focused on macroscopic traffic flow modelling. First, we present a short introduction into the topic and basic model classification. Then we derive the continuity equation for macroscopic models and outline possible constitutive relations for a flux which determine the key features of the model. The main part of this work is focused on LWR model and its solution by the method of characteristic curves, with emphasize on shockwave creation. The thesis is concluded by an example and experiment comparing of the LWR model with real traffic on highway.
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Asymptotic stability of systems of linear ordinary differential equations in engineering
Mašek, Jakub ; Opluštil, Zdeněk (referee) ; Tomášek, Petr (advisor)
This bachelors thesis is dealing with stability of system of linear ordinary dierential equations and specially lyapunov stability and asymtotic stability.The are established necessary concepts from the theory of stability and form systems of dierential equations at rst. Furthermore there are listed basic methods for determining the stability of linear dierential equations with constant coecients and they are compared. The next part of thesis is dedicated to trajectory in plane with focus on isolated singular points. At the end are two technical applications and they are linked sections and oscillators.
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