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@book{60848,
   author = {Mlíchová, Michaela},
   keywords = {Li–Yorke sensitivity; weak mixing system; extension of system; skew-product; minimal set; triangular map; spatio-temporally chaotic; minimal map; almost one-to-one extension},
   language = {eng},
   title = {On Li-Yorke sensitivity and other types of chaos in dynamical systems},
   year = {2019}
}

TY  - 
ID  - 60848
AU  - Mlíchová, Michaela
PY  - 2019
TI  - On Li-Yorke sensitivity and other types of chaos in dynamical systems
KW  - Li–Yorke sensitivity
KW  - weak mixing system
KW  - extension of system
KW  - skew-product
KW  - minimal set
KW  - triangular map
KW  - spatio-temporally chaotic
KW  - minimal map
KW  - almost one-to-one extension
N2  - A dynamical system describes a dependence of the position of a point in some space on the time that is continuous or discrete. The founder of general theory of one and low–dimensional discrete dynamical systems is Alexander Sharkovsky. In 1964 he specified a new total ordering of natural numbers, known today as Sharkovsky’s ordering, which describes the co–existence of periodic orbit for continuous interval maps. In the ’70s it has turned out that this ordering gives direction from more complex to simpler behavior of systems. One of the most significant traits of dynamical systems is just the existence of chaotic behavior. In 1972 during the 139th meeting of the American Association for the Advancement of Science, Lorenz described the Butterfly Effect in his talk entitled “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” The Butterfly Effect shows that very small change in initial conditions can create a significant difference in the results. It is also known as the sensitive dependence on initial conditions. For the first time the notion of chaos was introduced by Li and Yorke in 1975. They showed that not only the existence of periodic point of period 3 implies the existence of periodic points of all periods (which is a special case of Sharkovsky’s Theorem), but also in addition it implies the existence of uncountable set whose points never map to any cycle. There was suggested a defining criterion on the existence of chaos for maps on the interval. This notion can be extended to more general metric spaces and now this kind of chaos is known as Li–Yorke chaos. Various alternative definitions of chaos was introduced later. However, it should be noted that unique and universally accepted definition of chaos does not exists currently and it will probably never exist. A summary of chaos theory is given by Li and Ye. In recent years, the chaotic behavior of dynamical systems is a common concern not only among various branches of mathematics. The chaos is also intensively studied by various branches of science and engineering. Combining Li-Yorke version of chaos with the notion of sensitivity to initial conditions leads to a definition of Li–Yorke sensitivity that is the main area of interest of this thesis. The thesis is structured as follows. After defining some elementary notions and introducing notations in the next section, we explain some kinds of chaos and provide relationship among them. In the last two sections of the first part we recapitulate the main results contained in the papers concerning the thesis. These three papers form the second part of the thesis. We focus on minimal Li-Yorke sensitive systems. In particular, on relations between Li–Yorke sensitivity and spatio-temporal chaos, and extensions and factors of Li–Yorke sensitive systems.
ER  - 

MLÍCHOVÁ, Michaela. \textit{On Li-Yorke sensitivity and other types of chaos in dynamical systems}. 2019.