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Filippov dynamical systems with applications
Šimonová, Dorota ; Janovský, Vladimír (advisor) ; Ratschan, Stefan (referee)
The thesis is motivated by problems of contact mechanics with friction. At the beginning we describe a class of piecewise smooth systems with discontinuous vector field called Filippov systems. We also show how to solve them. The rest of this thesis is focused on applications, especially dry friction model and finite element model of Coulomb friction with one contact point. We propose a technique for simulation of the second mentioned model which combines sovling methods for Filippov systems and impact oscillators. Powered by TCPDF (www.tcpdf.org)
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Numerická optimalizace
Márová, Kateřina ; Janovský, Vladimír (advisor) ; Lukšan, Ladislav (referee)
This thesis addresses the topic of unconstrained optimization. It describes seven derivative-free optimization methods for objective functions of multiple variables. Three groups of methods are distinguished. The Alternating Variable method and the method of Hooke and Jeeves represent the pattern search methods. Then there are two simplex algorithms: one by Spendley, Hext and Himsworth and the amoeba algorithm of Nelder and Mead. The family of methods with adaptive sets of search directions consists of Rosenbrock's method, the method of Davies, Swann and Campey, and Powell's method. All algorithms are implemented in MATLAB and tested on three functions of two variables. Their progression is illustrated by multiple figures and their comparative analysis is given. Powered by TCPDF (www.tcpdf.org)
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On a model of corruption in a democratic society
Splítek, Martin ; Janovský, Vladimír (advisor) ; Mlčoch, Lubomír (referee)
The aim of this work is to study the behavior of serious social pheno- menon - corruption, and we do this through a mathematical model of corruption in a democratic society, published in [1]. The model is a dynamical system of three differential equations, specified by three variables and ten parameters. The model is studied by means of numerical analysis, namely, the method of nume- rical integration of ordinary differential equations and the method of numerical continuation. We used toolbox Matcont [2], which works in the environment of program MATLAB [3]. The result is commented parametric study of the pheno- menon of corruption. Keywords: ordinary diferential equations, dynamic systems, bifurcation ana- lysis 1
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Steady states of dynamical systems
Šerý, David ; Janovský, Vladimír (advisor) ; Vlasák, Miloslav (referee)
In the thesis we analyse qualitative properties of dynamical systems near equilibria. We mainly deal with planar equations. The key notion is the stability of steady state. The stability analysis is closely connected to linearisation, which in many cases doesn't suffice. In that case Lyapunov function may help. We define stable and unstable manifold, basin of attraction, topological equivalence of equations and demonstrate their significance in qualitative analysis. The theory will be illustrated on examples. In the third chapter we briefly mention numerical continuation of steady states with respect to a parameter. 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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Numerical solution of equations describing the dynamics of flocking
Živčáková, Andrea ; Kučera, Václav (advisor) ; Janovský, Vladimír (referee)
This work is devoted to the numerical solution of equations describing the dynamics of flocks of birds. Specifically, we pay attention to the Euler equations for compressible flow with a right-hand side correction. This model is based on the work Fornasier et al. (2010). Due to the complexity of the model, we focus only on the one-dimensional case. For the numerical solution we use a semi-implicit discontinuous Galerkin method. Discretization of the right-hand side is chosen so that we preserve the structure of the semi-implicit scheme for the Euler equations presented in the work Feistauer, Kučera (2007). The proposed numerical scheme was implemented and numerical experiments showing the robustness of the scheme were carried out. Powered by TCPDF (www.tcpdf.org)
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Numerical solution of ordinary differential equations
Monhartová, Petra ; Feistauer, Miloslav (advisor) ; Janovský, Vladimír (referee)
In the present work we study numerical methods for the nu- merical solution of initial value problems for ordinary differential equations. With the aid of the Taylor formula we derive several one-step methods. We compare numerical solution computed with explicit and implicit Eu- ler methods. Moreove, we are concerned with second-order and fourth-order Runge-Kutta methods. We find how accurately the numerical methods obta- ined with the aid of these methods approximate the exact solution. Further we estimate the error of these method by the half-step method. 1
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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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Application of the Laplace transoform and the homotopy perturbation method for the Burgers equation
Chaloupka, Tomáš ; Felcman, Jiří (advisor) ; Janovský, Vladimír (referee)
We use the homotopy perturbation method for solving different types of functional equations. The method is formulated in Introduction. Several types of functional equations are solved in Chapter one. In Chapter two, we define the Laplace transformation and combine it with the homotopy perturbation method in order to solve some differential equations. Last chapter is focused on attempts to find the solution of the Burgers equation with different initial conditions. For these conditions, we try to prove the existence of the solution or to find a suitable approximation of the solution. We compared the method with the method of characteristics. We investigate the behaviour of the homotopy perturbation method where method of characteristics doesn't exclude the existence of the classic solution. We discuss the practical application of the homotopy perturbation method to the Burgers equation.
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A quadrature formula of Clenshaw-Curtis type for the Gegenbauer weight-function
Labant, Ján ; Kofroň, Josef (advisor) ; Janovský, Vladimír (referee)
In this thesis we study especially quadrature formulae based on the Cheby- shev expansion, known as the Clenshaw-Curtis quadrature. The first part is focused on the Chebyshev polynomials, their definitions and properties. This knowledge will be used to derivate the Clenshaw-Curtis quadrature. Consider- able part of this work is dedicated to comparison of this and the well-known Gauss quadrature both theoretically and practicaly. In the further work we will extend the Clenshaw-Curtis quadrature by the Gegenbauer weight function which gives us new methods for numerical integration. These methods allow us to find a solution of some known problems what will be pointed out also on some nu- merical experimets. 1
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