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