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.

本文考虑了由有限域上的向量空间中的映射定义的有限域上的动力学系统（DSFF）。在功能空间上构造了一个关联的线性动力学系统。该系统构成了动力学系统的众所周知的Koopman线性系统框架，因此被称为Koopman线性系统（KLS）。首先表明，可以从KLS的溶液中推断出DSFF溶液的几种结构性质。然后将KLS减小到最小阶（称为RO-KLS），同时仍保留DSFF解结构参数的所有信息。因此，上述非线性DSFF的计算问题可以通过线性代数方法解决。还显示了如何使用RO-KLS计算DSFF的固定点，周期点和链根。更多，\（1-1 \）与RO-KLS的输出轨迹的特殊类别相对应，这表明通过RO-KLS的线性观测器设计可以解决非线性可观测性的问题。