Autonomous Operator and Differential Systems of order 1 over Integral Domains

Ronald Orozco López

Published 2018-12-30Version 1

In this paper we introduce the autonomous operator which is a nonlinear map that sends sequences to sequences in an integral domain $R$, in particular sequences over the ring of exponential generating functions. Properties of this operator are studied: its relationship with the derivative of the ring of sequences, its characterization by groups of $k$-homogeneity, condictions of injectivity and surjectivity and its restriction to a linear operator. Autonomous differential dynamical systems over integral domain are defined. It is also shown that the flow of such differential equations is the exponential generating function of the sequence obtained via the action of autonomous operator acting over the Hurwitz expansion of the vectorial field of such equation. It is shown that the flow is a torsion-free cyclic $R$-module. The $R^{\times}$-modules associates to the groups of $k$-homogeneity are constructed. Some images of the autonomous operator are given and upper bounds for the flow are finded. We finish showing some examples of autonomous differential equations and we find its $R^{\times}$-modules when $R$ is one of the rings: $\Z$, $\C$, $\Z[i]$, $\Z[\omega]$ and $\Z[2]$.