![]() ![]() Let the upper and lower (distance) distribution functions, F*y and Fxy, are defined for any as the lim sup and lim inf as of the average number of times that the distance between the trajectories of x and y is less than t during the first n iterations. This paper gives one such condition that under the shadowing property, one can imply distributional chaos from Devaney chaos, building a connection between the global and local chaotic behaviors of such systems. So, a natural question is that under what kinds of conditions can one relate these two types of chaos. Neither Li-Yorke chaos nor distributional chaos can imply Devaney chaos, in general. The notion of Devaney chaos is a global property of the dynamical system, while distributional chaos is a local one. The other is called distributional chaos, which is related to the complexity of the trajectory behavior of points in the space. One is Devaney chaos, which means that the system is topologically transitive, sensitive to initial conditions, and the set of periodic points in the space is dense. In this paper, we consider the relationship between two important notions of chaos. Two examples are finally provided for illustration.Ĭhaos theory is a hot topic in area of topological dynamics, and different definitions of chaos have been introduced. Some of these results not only extend the existing related results for autonomous discrete systems to non-autonomous discrete systems, but also relax the assumptions of the counterparts in the literature. Consequently, estimations of the topological entropy for the induced set-valued system are obtained, and several criteria of Li-Yorke chaos and distributional chaos in a sequence are established. Based on these results, the paper furthermore establishes the topological (equi-)semiconjugacy and (equi-)conjugacy between induced set-valued systems and subshifts of finite type. Second, the relationships of several related dynamical behaviors between the non-autonomous discrete system and its induced set-valued system are investigated. Consequently, estimations of topological entropy and several criteria of Li-Yorke chaos and distributional chaos in a sequence are derived. Further, several sufficient conditions for it to be topologically equi-semiconjugate or equi-conjugate to a subshift of finite type are obtained. First, some necessary and sufficient conditions are given for a non-autonomous discrete system to be topologically semiconjugate or conjugate to a subshift of finite type. This paper establishes topological (equi-)semiconjugacy and (equi-)conjugacy between induced non-autonomous set-valued systems and subshifts of finite type. Two examples are finally provided for illustration. Consequently, estimations of the topological entropy for the induced set-valued system are obtained, and several criteria of Li–Yorke chaos and distributional chaos in a sequence are established. Consequently, estimations of topological entropy and several criteria of Li–Yorke chaos and distributional chaos in a sequence are derived. This paper establishes topological (equi-)semiconjugacy and (equi-) conjugacy between induced non-autonomous set-valued systems and subshifts of finite type. ![]()
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