Terminological cycles have been a very hard problem in description logics for a long time, and their essential problems, i.e. semantics and reasoning problems, have not been solved reasonably. Current research progress and the existing problems of terminological cycles in description logics are analyzed in this paper. Based on the work of Baader, a new direction in terminological cycles is put forward. Aiming at more expressive description logic, the semantics and reasoning mechanism of terminological cycles are studied. The number restrictions are added to description logic εL, and the descripion logic εLN is presented. The semantics (fixpoint semantics and descriptive semantics) of εLN are given. To meet the requirement of εLN, the description graphs (syntax description graph and semantics description graph) are redefined. The satisfiability and subsumption reasoning algorithms of terminological cycles in description logic εLN w.r.t. fixpoint semantics are presented with simulation between description graphs. It is proved that the satisfiability and subsumption reasoning algorithms of terminological cycles in εLN are polynomial.