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Rovnice geodetiky v prostoročasech s helikální symetrií
Tomášik, Miroslav ; Scholtz, Martin (advisor) ; Žofka, Martin (referee)
In this bachelor thesis we investigate geodesics in helically symmetric spacetimes in the framework of linearized Einstein's gravity. Work is an extension of paper by Bičák, Scholtz and Bohata [2], which is under preparation. First we introduce standard numerical methods for solving systems of ordinary differential equations. Next we present helically symmetric solution of linearized Einstein's equations and numerical code solving the geodesic equation on given background. We discuss conditions of existence of helically symmetric solution and finally we present selected results obtained by numerical simulations. We give present few particular examples of geodesics, selected phase portraits obtained by the method of the Lyapunovov exponents and visualize the causal structure of helically symmetric spacetime.
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Modelling of postcritical states of slender structures
Mašek, Jan ; Eliáš, Jan (referee) ; Frantík, Petr (advisor)
The aim of the presented thesis is to create a compact publication which deals with properties, solution and examination of behavior of dynamical systems as models of mechanical structures. The opening portion of the theoretical part leads the reader through the subject of description of dynamical systems, offers solution methods and investigates solution stability. As the introduction proceeds, possible forms of structure loading, damping and response are presented. Following chapters discuss extensively the possible approaches to system behavior observation and identification of nonlinear and chaotic phenomena. The attention is also paid to displaying methods and color spaces as these are essential for the examination of complex and sensitive systems. The theoretical part of the thesis ends with an introduction to fractal geometry. As the theoretical background is laid down, the thesis proceeds with an application of the knowledge and shows the approach to numerical simulation and study of models of real structures. First, the reader is introduced to the single pendulum model, as the simplest model to exhibit chaotic behavior. The following double pendulum model shows the obstacles of observing systems with more state variables. The models of free rod and cantilever serve as examples of real structure models with many degrees of freedom. These models show even more that a definite or at least sufficiently relevant monitoring of behavior of such deterministic systems is a challenging task which requires sophisticated approach.
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Application of chaotic dynamics in natural and technical sciences
Kratochvíl, C. ; Švéda, P. ; Hortel, Milan ; Škuderová, Alena ; Houfek, M.
During the entire 20th century there was a gradual transformation of scientific research, which has produced in science and technology especially in the extraordinary interest in complex dynamic systems. They are non-linear systems, operating environments and the irreversible complexity in their name means they have complex structures, relations and interactions (often of different physical nature). It turned out that an inherent attribute of these complex systems and chaos (deterministic and stochastic). The article will focus on some aspects of the manifestations of chaos, its spread, as well as identification, suppression and control. We will mention also other important phenomena - the possible emergence of a new order out of chaos.
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ANALYTIC FORMULAE FOR THE CONSTRUCTION OF SYMMETRIC DYNAMICAL SYSTEMS
Kozánek, Jan
In this paper the formulae for complex modal matrix and symmetric regular stiffness matrix and symmetric matrix of viscous damping from real, positive definite mass matrix and from diagonal spectral matrix were deduced. Because of the non-unicity of this problem, the solution is based on the fact, that the helping matrix has the same ratio of the eigenvalues as the inversion of given mass matrix. Finally, the resolvent of this symmetric system was expressed in simple additive form.
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ANALYTIC FORMULAE FOR THE CONSTRUCTION OF NON-DIAGONALIZABLE DYNAMICAL SYSTEMS
Kozánek, Jan
In this paper the formulae for regular stiffness matrix and matrix of viscous damping and for complex left modal matrix from regular mass matrix, nondiagonal jordan matrix with one complex eigenvalue of multiplicity 2 and from complex right modal matrix were deduced. Finally, the resolvent of this system was expressed in additive form. The corresponding formulae for dynamical systems with commutative matrix of viscous damping and with real right modal matrix were given, too.
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ANALYTIC FORMULAE FOR THE CONSTRUCTION OF DIAGONALIZABLE DYNAMICAL SYSTEMS
Kozánek, Jan
In this paper the formulae for regular stiffness matrix and matrix of viscous damping and for complex modal matrix of left eigenvectors from regular mass matrix, diagonal spectral matrix and from complex modal matrix of right eiegenvectors was deduced. Finally, the resolvent of this system was expressed in simple additive form. The corresponding formulae for dynamical systems with commutative matrix of viscous damping and with corresponding real modal matrix of right eigenvectors was given, too.
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Výzkum turbulance pomocí metod pro analýzu dynamických systémů
Uruba, Václav
Turbulence represent itself very important and common behaviour of real fluids. It is usually treated using special methods based on probabilistic approach. In the presented study methods originally developed for chaotic dynamical systems are applied to turbulent fluids. Problem of the continuous system discretization to develop phase space is analyzed in details.
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