A multi-dimensional-time dynamical system

TL;DR

This paper introduces a novel multi-dimensional-time dynamical system framework generated by multiple functions, exploring fixed points, periodic orbits, and applications like banking deposit income, with several open problems discussed.

Contribution

It defines the concept of MDTDS using a free group as the time space, analyzes fixed points and periodic orbits, and applies the framework to financial and circle systems.

Findings

Fixed points are intersections of individual fixed points.

Periodic orbits are constructed via subgroup structures.

Applications include modeling deposit income and circle dynamics.

Abstract

In this paper we give a concept of multi-dimensional-time dynamical system (MDTDS). Such dynamical system is generated by a finite family of functions $\{f_i\}$. The multi-dimensional-time space is taken as a free group. Using the subgroups of the free group we define periodic orbits of MDTDS and construct such orbits. We study fixed points of MDTDS and show that the set of the fixed points is the intersection of the set of fixed points of each $f_i$. The $\omega$-limit set of the MDTDS is defined and some properties of the set is studied. Moreover we construct a MDTDS for income of a deposit from several banks and construct a MDTDS on circle. For these MDTDSs we describe the set of fixed and periodic points. We discuss several open problems.