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Analysis of Logistic Maps
Adeleke, Joshua Owolabi ; Šremr, Jiří (oponent) ; Řehák, Pavel (vedoucí práce)
A logistic map is related to a discrete logistic equation. Unlike its continuous counterpart, a logistic difference equation exhibits very complicated dynamics including chaotic behavior. This work thus investigated the qualitative behavior of the logistic map by employing some mathematical tools. This dynamics was studied systematically, in such a way that its nature from the pure form to the point when it got complicated to deal with were studied closely. Furthermore, the concept of conjugacy was employed at the point when its analytic computation posed to be complicated, with which its characteristics were further revealed. Notable inferences were made, among which is the description of the chaotic behavior of the logistic map as revealed by its conjugacy with the tent map. Thus, in course of this study, other tool for investigating the chaotic behavior of the logistic map was remarked, which is the symbolic dynamic, with which future study on the logistic map can take up on.
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Analysis of Logistic Maps
Adeleke, Joshua Owolabi ; Šremr, Jiří (oponent) ; Řehák, Pavel (vedoucí práce)
A logistic map is related to a discrete logistic equation. Unlike its continuous counterpart, a logistic difference equation exhibits very complicated dynamics including chaotic behavior. This work thus investigated the qualitative behavior of the logistic map by employing some mathematical tools. This dynamics was studied systematically, in such a way that its nature from the pure form to the point when it got complicated to deal with were studied closely. Furthermore, the concept of conjugacy was employed at the point when its analytic computation posed to be complicated, with which its characteristics were further revealed. Notable inferences were made, among which is the description of the chaotic behavior of the logistic map as revealed by its conjugacy with the tent map. Thus, in course of this study, other tool for investigating the chaotic behavior of the logistic map was remarked, which is the symbolic dynamic, with which future study on the logistic map can take up on.
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Rotation Number on a Circle
Bíma, Jan ; Vejnar, Benjamin (vedoucí práce) ; Pražák, Dalibor (oponent)
Práce se zabývá vybranými partiemi z teorie jednodimenzionálních dynamických sys- témů. Klíčovým pojmem je zde dynamický invariant rotačního čísla na kružnici a jeho vztah k existenci periodických bodů daného orientaci zachovávajícího homeomorfismu kružnice. Pojem rotačního čísla je dále rozšířen pro taková spojitá zobrazení kružnice do sebe, která jsou stupně jedna. Podrobně jsou studovány asymptotické vlastnosti homeo- morfismů kružnice s iracionální hodnotou rotačního čísla, tyto úvahy pak vedou k důkazu Poincarého klasifikační věty postulující (semi-)konjugovanost homeomorfismu kružnice s iracionálním rotačním číslem a rotace se stejným rotačním číslem. 1
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