Regular bisimulation is the foundation of the non-well-founded axioms and the non-well-founded set theories, and a family of non-well-founded axioms are based on it. This paper defines the regular bisimulations≌*, ≌t and ≡V0, shows the non - well-founded axioms AFA≌*, AFA≌t and AFA≡V0crrosponding to them are respectively equal to the non-well-founded axioms FAFA, SAFA and AFA; FAFA and AFA are at both ends of the family of non-well-founded , SAFA between them , and FAFA, SAFA and AFA are pairwise incompatible; The extensionalities of FAFA, SAFA and AFA increase incrementally , and their ranges decrease successively.