  This paper develops methods for analyzing periodic orbits of states of linear
feedback shift registers with periodic coefficients and estimating their
lengths. These shift registers are among the simplest nonlinear feedback shift
registers (FSRs) whose orbit lengths can be determined by feasible computation.
In general such a problem for nonlinear FSRs involves infeasible computation.
The dynamical systems whose model includes such FSRs are termed as Periodic
Finite State systems (PFSS). This paper advances theory of such dynamical
systems. Due to the finite field valued coefficients, the theory of such
systems turns out to be radically different from that of linear continuous or
discrete time periodic systems with real coefficients well known in literature.
A special finite field version of the Floquet theory of such periodic systems
is developed and the structure of trajectories of the PFSS is analyzed through
that of a shift invariant linear system after Floquet transformation. The
concept of extension of a dynamical system is proposed for such systems
whenever the equivalent shift invariant system can be obtained over an
extension field.
