The discrete dynamic system on a double circulant with different functions at the vertices

The structure of the functional graph is studied for a discrete dynamic system consisting of two circulants $G_{n,k}$ with different orientations and functionings and with the corresponding vertices being conjugate. The recurrent relation for the number of fixed points is obtained, and the asymptotic behaviour of this number is described. In the case $k = 2$ the theorems characterizing structural properties, fixed points, pendant vertices and cycles of length 2 of the functional graphs are proved. In particular, the explicit formulas for the number of fixed points and pendant vertices are found.