Dynamics of an Infinite-Dimensional Symmetric Logistic Mapping

Abstract

There are many generalizations of the famous logistic mapping. In the paper, we considered discrete dynamical systems of an infinite number of variables, generated by a symmetric polynomial map that generalizes the logistic mapping for the ring of multisets. Using homomorphisms of the ring of multisets (multinumbers), it is possible to represent the infinite-dimensional symmetric logistic map as a sequence of classical logistic mappings. We observed that the converse statement is also true: for any finite sequence of classical logistic mappings, there exists a logistic map on the ring of multinumbers that agrees with the sequence. We found some nontrivial fixed points of the logistic map on the ring of multinumbers and proved that under some conditions, the initial point of the logistic dynamical system on the ring of multinumbers can be uniquely discovered by any nth iteration. This fact allows us to propose an application of the dynamical system to the construction of collision-free hash functions of data that is important in cryptography. Also, we observed that the logistic map on the ring of multinumbers can be represented as a transformation on a set of rational functions.