Several Results on the Families of Diffeomorphisms on S¹

In this paper, we give a necessary and sufficient condition for the one-parameter families of diffeomorphisms on S1 to be stable and a necessary condition for the multi-parameter families to be stable; and, moreover, we prove that phase-locking is a generic property of the one- parameter families of diffeomorphisms on S¹. We also get a necessary and sufficient condition of phase-locking for the one-parameter families of integral diffeomorphisms on S¹ which strengthens a result in [2].