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The Lorenz system: A route from stability to chaos
Arhinful, Daniel Andoh ; Šremr, Jiří (oponent) ; Řehák, Pavel (vedoucí práce)
The theory of deterministic chaos has generated a lot of interest and continues to be one of the much-focused research areas in the field of dynamics today. This is due to its prevalence in essential parts of human lives such as electrical circuits, chemical reactions, the flow of blood through the human system, the weather, etc. This thesis presents a study of the Lorenz equations, a famous example of chaotic systems. In particular, it presents the analysis of the Lorenz equations from stability to chaos and various bifurcation scenarios with numerical and graphical interpretations. It studies concepts of non-linear dynamical systems such as equilibrium points, stability, linearization, bifurcation, Lyapunov function, etc. Finally, it discusses how the Lorenz equations serve as a model for the waterwheel (in detail), and the convection roll for fluid.
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The Lorenz system: A route from stability to chaos
Arhinful, Daniel Andoh ; Šremr, Jiří (oponent) ; Řehák, Pavel (vedoucí práce)
The theory of deterministic chaos has generated a lot of interest and continues to be one of the much-focused research areas in the field of dynamics today. This is due to its prevalence in essential parts of human lives such as electrical circuits, chemical reactions, the flow of blood through the human system, the weather, etc. This thesis presents a study of the Lorenz equations, a famous example of chaotic systems. In particular, it presents the analysis of the Lorenz equations from stability to chaos and various bifurcation scenarios with numerical and graphical interpretations. It studies concepts of non-linear dynamical systems such as equilibrium points, stability, linearization, bifurcation, Lyapunov function, etc. Finally, it discusses how the Lorenz equations serve as a model for the waterwheel (in detail), and the convection roll for fluid.
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Convex hull properties for parabolic systems of partial differential equations
Češík, Antonín ; Schwarzacher, Sebastian (vedoucí práce) ; Bulíček, Miroslav (oponent)
Tématem práce je vlastnost konvexního obalu pro soustavy parciálních dife- renciálních rovnic, jež je přirozeným zobecněním principu maxima pro skalární rovnice. Hlavním výsledkem práce je věta o vlastnosti konvexního obalu pro jis- tou třídu nelineárních parabolických soustav parciálních diferenciálních rovnic. Práce se také zabývá koeficienty lineárních soustav. Tyto výsledky jsou op- timální, jak je ukázáno v protipříkladech k vlastnosti konvexního obalu pro řešení lineárních a parabolických soustav. Celkově se téma dá shrnout tak, že mixování proměnných je to, co rozbije vlastnost konvexního obalu, ne nutně nelinearita rovnice.
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