Global Orbit Patterns for One Dimensional Dynamical Systems

In this article, we study the behaviour of discrete one-dimensional dynamical systems associated to functions on finite sets. We formalise the global orbit pattern formed by all the periodic orbits (gop) as the ordered set of periods when the initial value thumbs the finite set in the increasing order. We are able to predict, using closed formulas, the number of gop for the set $\mathcal{F}_N$ of all the functions on $X$. We also explore by computer experiments special subsets of $\mathcal{F}_N$ in which interesting patterns of gop are found.Comment: 33 pages, 1 figur

Global Orbit Patterns for Dynamical Systems On Finite Sets

In this paper, the study of the global orbit pattern (gop) formed by all the periodic orbits of discrete dynamical systems on a finite set $X$ allows us to describe precisely the behaviour of such systems. We can predict by means of closed formulas, the number of gop of the set of all the function from $X$ to itself. We also explore, using the brute force of computers, some subsets of locally rigid functions on $X$, for which interesting patterns of periodic orbits are found.Comment: 29 pages, 3 figures. to appear in Proceedings of the International Conference On Modeling of Engineering & Technological Problems (ICMETP), Agra (India), 2009. Published by the American Institute of Physics, U.S.