Indefinite Descriptive Proximities Inherent in Dynamical Systems. An Axiomatic Approach

TL;DR

This paper develops an axiomatic framework for indefinite proximities in dynamical systems, leading to a new type of Hausdorff topology and insights into system stability and energy dissipation.

Contribution

It introduces a novel axiomatic approach to indefinite proximities, establishing a new descriptive topology for dynamical systems and analyzing energy variation over time.

Findings

Every descriptive proximity space on a dynamical system is indefinite.

Every dynamical system admits an indefinite descriptive Hausdorff topology.

The energy of a dynamical system varies with each clock tick.

Abstract

This paper introduces indefinite proximities inherent in the collection of physical objects found in a dynamical system. Axiomatically, these indefinite proximities lead to a new form of Hausdorff topology, which is indefinite descriptively. The main results in this paper are (1) Every descriptive proximity space on a dynamical system is indefinite (Theorem 1), (2) Every dynamical system has an indefinite descriptive Hausdorff topology (Theorem 3), and (3) The energy of a dynamical system varies with every clock tick (Theorem 4). An application of these results is given in terms of the detection of those portions of a dynamical system that are stable and that have low energy dissipation.