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System of autonomous differential equations
Benáčková, Jana ; Tomášek, Petr (referee) ; Opluštil, Zdeněk (advisor)
In his work dealing with applications, systems theory of autonomous differential equations in biology to the analysis model of coexistence of two populations. Mathematical models are described in general non-linear autonomous system of differential equations. I introduced the classification of types of singular points that are important for the following solutions to specific models. In the last part is an overview of the most famous models of the two populations (predator × prey) and specific models for the communities of invertebrate animals and mammals.
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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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Stability analysis of systems of ordinary differential equations
Trejtnar, Miloš ; Opluštil, Zdeněk (referee) ; Tomášek, Petr (advisor)
This thesis deals with a stability analysis of the first order systems of ordinary differential equations. There are introduced some stability approaches in the thesis and they are discussed in the several examples. The attention is focused to the case of linear autonomous systems, where the classification of the singular points is realized. The thesis is closed by the application of the stability theory in mathematical model of electric current conduction in a primary and secondary coil of a transformer.
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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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Delay differential equations in engineering
Zlámal, Ondřej ; Řehák, Pavel (referee) ; Opluštil, Zdeněk (advisor)
This thesis is about dynamical systems and analysis of their stability. These systems are described using delayed differential equations, whose character is ideal for describing many real life problems. In this thesis it is analysed how size of delay and its rate affects stability of system. Change of stability in system is traced using Hopf bifurcations. Theory of this thesis will be applied on system based on machine tool vibrations and system describing feedback in lasers.
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Regular variation and its applications
Ženatá, Kamila ; Opluštil, Zdeněk (referee) ; Řehák, Pavel (advisor)
This barchelor thesis deals with concept of regular vaiation and its applications in various areas of mathematics. The thesis provides an overview of the basic properties of regularly varying functions, related concepts and specific applications of the findings in differential equations and infinite series.
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Epidemic models
Machalová, Monika ; Šremr, Jiří (referee) ; Opluštil, Zdeněk (advisor)
Epidemiological models and their analysis are an essential tool for studying, understanding, and controlling the spread of epidemics. It is a discipline in which we are trying to answer questions such as how long an epidemic will last, what its flow will be, or how much it will affect the population. There are a large number of models and approaches for describing these epidemics and their behavior. In this thesis, we focus on models described by a system of differential equations, and we analyze these models in terms of stability.
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Application of boundary value problems for ordinary differential equations in engineering
Zapoměl, Jakub ; Šremr, Jiří (referee) ; Opluštil, Zdeněk (advisor)
This bachelor thesis deals with the determination of the shape of the deflection line for boundary value problems in strength of materials. There are several methods for solving boundary value problems. This thesis focuses on the Green's function method. It provides a basic review of the properties of ordinary differential equations, an introduction to the Green's function method and the actual application of the findings to beam bending models. The concrete models are solved using an interactive program developed in Matlab software.
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