Three multi-variant versions of distributional chaos in non-autonomous discrete systems

This paper investigates three multi-variant versions of distributional chaos in non-autonomous discrete systems (NADSs). These three types of multi-variant distributional chaos are named n-DC1, n-DC2, and n-DC3, respectively. A method is proposed to construct the three types of chaos (n-DC1, n-DC2, and n-DC3) for any integer n≥2. Furthermore, it is shown that n-DCi can imply 2-DCi, but the inverse is not valid, where i∈{1,2,3}. It is further proved that n-DC1 and n-DC2 can be preserved under iteration, while n-DC3 needs to be preserved under certain conditions. It is also shown that the chaotic dynamic behaviour of the uniformly convergent NADSs can be characterized by their sub-orbits. Moreover, it is shown that n-DC1 and n-DC2 can be preserved under topological conjugacy, but n-DC3 cannot. In three distinct cases, the equivalent conditions are provided for the limit system induced by a uniformly convergent NADS to exhibit n-DC1, n-DC2, or n-DC3.