Abstract: Chaoticity is a vital property of a dynamical system. For a given Furstenberg family \mathcalF, the notion of \mathcalF-chaos has been introduced in a dynamical system recently. In order to extend it to \mathcalF-N-chaos, \mathcalF-scrambled tuples are defined by means of \mathcalF-reciprocating points for a pair (A,B) of sets. Hence, some properties of \mathcalF-N-chaos are considered similarly. A criterion for generically strongly \mathcalF-N-chaotic systems is obtained by heavy use of the theory of Furstenberg families, and it applications in dynamical systems are given with two examples.