‎in the following text for arbitrary $x$ with at least two elements‎, ‎nonempty countable set $gamma$‎ ‎we make a comparative study on the collection of generalized shift dynamical systems like $(x^gamma,sigma_varphi)$ where $varphi:gammatogamma$ is an arbitrary self-map‎. ‎we pay attention to sub-systems and combinations of generalized shifts with counterexamples regarding devaney‎, ‎exact dev...

Since Li and Yorke first gave the definition of chaos by using strict mathematical language in 1975 [1], the research on chaos has greatly influenced modern science, not just natural sciences but also several social sciences, such as economics, sociology, and philosophy. The theory of chaos convinced scientists that a simple definite system can produce complicated features and a complex system ...

Distributional chaos was introduced in Schweizer and Smítal (Trans Am Math Soc 344:737–754, 1994) for continuous maps of the interval, as chaotic behavior based on development distances between orbits points system. In Balibrea et al. (Chaos Solitons Fractals 23(5):1581–1583, 2005), this phenomenon generalized to compact metric space distinguished into three different forms, DC1, DC2 DC3. Loran...

We show the existence of a distal dynamical system which possess an uncountable distributionally scrambled set of type 3. This distal system has no Li-Yorke pairs, hence the distributional chaos of type 3 does not imply the Li-Yorke chaos. Moreover, the system is conjugated to a system without any distributionally scrambled pair. Therefore the distributional chaos of type 3 is not preserved by ...