Polynomial dynamical systems over finite fields or rings provide a useful tool for studying network dynamics, such as those of gene regulatory networks. In this talk, I will discuss linear dynamical systems over finite commutative rings. The limit cycles of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. The extension of the study to a general finite commutative ring is natural and has applications. To address the difficulties in the commutative ring setting, we developed a computational approach. In an earlier work, we gave an efficient algorithm to determine whether such a system over a finite commutative ring is a fixed-point system or not. In a more recent work, we further analyzed the structure of such a system and provided a method to determine its limit cycles.