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

Abstract

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.