Orbits of families of discrete dynamical systems evolving in the natural numbers

In this paper, we give a class of one-dimensional discrete dynamical systems with state space $N +$⁠. This class of systems is defined by two parameters: one of them sets the number of nearest neighbors that determine the rule of evolution, and the other parameter determines a segment of natural numbers $Λ={1,2,…,b}$⁠. In particular, we investigate the behavior of a class of one-dimensional maps where an integer moves to an other integer given by the sum of the nearest neighbors minus a multiple of $b∈ N +$⁠. We find the coexistence of fixed points and periodic cycles. Two single parameter families of maps are introduced and their dynamics in the segment of natural sequence $Λ$⁠. Furthermore, an order of the numbers of the set $Λ−b$ is given by these families. Last, we present a connection of the $N +$ generated by the orbits of a particular case.