Náhodné dynamické systémy generované zobrazeniami intervalu do seba
Jozef Kováč
PhD thesis advisor:
Katarína Janková
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PhD thesis - Full text 
Abstract:
In this thesis, we study random dynamical systems generated by continuous interval maps. The thesis has three main parts. The first part deals with Allee maps, which are used in population dynamics. It is shown that the behavior of the random dynamical systems is very similar to the behavior of the deterministic system if we use strictly increasing Allee maps. However, in the case of unimodal Allee maps, the behavior can dramatically change. The second part deals with distributional chaos and its measure in random dynamical systems. We give some sufficient conditions for a zero measure of chaos and examples of chaotic systems. We demonstrate that the chaoticity of the functions that generate a system does not, in general, affect the chaoticity of the system, i.e. a chaotic system can arise from two nonchaotic functions and vice versa. Finally, we show that distributional chaos for random dynamical system is, in some sense, unstable. In the last part, we study the topological entropy. We show some conditions for upper and lower bounds for the topological entropy and demonstrate examples of systems, in which we are able to calculate the topological entropy. Finally, we deal with the relationship between the topological entropy and distributional chaos.
References
[1] Kováč J., Janková K.: Random Dynamical Dystems Generated by Two Allee Maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 27(8) (2017) pp. 1750117.
[2] Kováč J., Janková K.: Distributional chaos in random dynamical systems, J. Difference Equ. Appl.25(4) (2019), 455–480.