Abstract: The equicontinuity has strong stability, it can be ensured within a pre-given error range if the error of its initial state is small enough. This dynamic behavior revealed by equicontinuity has high application value. Therefore, it has important research value in the theory of differential equation stability and topological dynamic system. Equicontinuity and distal are closely related to chaos and so on. In order to study them under the background of fuzzyness, a method of generalization is used to give the definition of the equicontinuity in fuzzy dynamic systems with establishing of fuzzy dynamic system. The following results were obtained: 1)~Equivalent propositions of the equicontinuity in fuzzy dynamic system; 2)~The equicontinuity is preserved under fuzzy uniform topologically conjugate mapping; 3)~The sufficient and necessary condition of two fuzzy dynamical systems to be distal.

Key words: standard fuzzy metric space, fuzzy dynamical system, equicontinuity, fuzzy uniform topologically conjugate, distal