A new version of distributional chaos and the relations between distributional chaos in a sequence and other concepts of chaos

TL;DR

This paper explores the relationships between various forms of distributional chaos, introduces a new chaos concept called DC 2', and establishes conditions under which different chaos types are equivalent or independent.

Contribution

It provides a sufficient condition for the equivalence of distributional chaos and distributional chaos in a sequence, and introduces the new chaos concept DC 2' with its properties.

Findings

Distributional chaos is equivalent to chaos in a sequence under certain conditions.

Distributional chaos in a sequence does not imply other chaos types like w-chaos, R-T chaos, or DC 3.

The new chaos concept DC 2' is invariant under iteration for any positive integer N.

Abstract

In this paper we consider relations between distributional chaos in a sequence with distributional chaos, w-chaos, R-T chaos, DC 3, respectively). We give a sufficient condition and prove that the distributional chaos is equivalent to the distributional chaos in a sequence under this condition. Besides, we get that distributional chaos in a sequence and w-chaos(R-T chaos, DC 3, respectively)do not imply each other. Finally, we give a new definition of chaos, named DC 2', which is similar to DC 2, and show that for any integer N > 0, f is DC 2' if and only if f^N is also DC 2'.