Stabilization bounds for linear finite dynamical systems

A common problem to all applications of linear finite dynamical systems is analyzing the dynamics without enumerating every possible state transition. Of particular interest is the long term dynamical behaviour. In this paper, we study the number of iterations needed for a system to settle on a fixed set of elements. As our main result, we present two upper bounds on iterations needed, and each one may be readily applied to a fixed point system test or a computation of limit cycles. The bounds are based on submodule properties of iterated images and reduced systems modulo a prime. We also provide examples where our bounds are optimal.