Abstract: The Arnold mapping with chaotic has achieved good results in the image scrambling and secure communication, however, the Arnold transformation matrix is periodic so that finding the cycle of the transformation matrix is the important basis of scrambling transformation. In order to study the periodicity of the 3-Arnold transform matrix, the new concept of the twin Fibonacci sequence is introduced and four related periodicity theorems are given. And then we prove that the molding cycle of 3-Arnold transform matrix is half of the molding cycle of the twin Fibonacci sequence. Accordingly, a new method to determine the molding cycle of the transformation matrix is formed. At last, a new several-rounds double-scrambling encryption algorithm based on the 3-Arnold mapping is proposed. Simulation results show the proposed algorithm outperforms the 2-Arnold mapping algorithm.