Koopman operator approach for computing structure of solutions and observability of nonlinear dynamical systems over finite fields

This paper considers dynamical systems over finite fields (DSFF) defined by a map in a vector space over a finite field. An associated linear dynamical system is constructed over the space of functions. This system constitutes the well known Koopman linear system framework of dynamical systems, hence called the Koopman linear system (KLS). It is first shown that several structural properties of solutions of the DSFF can be inferred from the solutions of the KLS. The KLS is then reduced to the smallest order (called RO-KLS) while still retaining all the information of the parameters of structure of solutions of the DSFF. Hence, the above computational problems of nonlinear DSFF are solvable by linear algebraic methods. It is also shown how fixed points, periodic points and roots of chains of the DSFF can be computed using the RO-KLS. Further, for DSFF with outputs, the output trajectories of the DSFF are in 1 - 1 correspondence with special class of output trajectories of RO-KLS and it is shown that the problem of nonlinear observability can be solved by a linear observer design for the RO-KLS.