Chaos via Furstenberg family couple

In this paper we define (F1,F2)-chaos via Furstenberg family couple F1 and F2. It turns out that the Li–Yorke chaos and distributional chaos can be treated as chaos in Furstenberg families sense. Some sufficient conditions such that a system is the (F1,F2)-chaotic (Theorems 4.2 and 4.4) are given. In addition, we construct an example as an application. It is showed that the second type of distributional chaos cannot imply the first type of distributional chaos even though the scrambled set is uncountable.