Periodicity Analysis of the Logistic Map Over Ring ℤ3n

Periodicity analysis of sequences generated by a deterministic system is long-standing challenge in both theoretical research and engineering applications. To overcome the inevitable degradation Logistic map on finite-precision circuit, its numerical domain commonly converted from real number field to ring or finite field. This paper studies period iterating over $\mathbb{Z}_{3^n}$ perspective associate functional network, where every considered as node, existing mapping relation between any two nodes regarded directed edge. The complete explicit form starting initial value given theoretically verified experimentally. Moreover, conditions control parameter are derived, ensuring corresponding achieve maximum ring. results can be used ground truth for dynamical cryptographical applications various domains.