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Indefinite Descriptive Proximities Inherent in Dynamical Systems. An Axiomatic Approach

James Francis Peters, Tane Vergili, Fatih Ucan, Divagar Vakeesan

TL;DR

This work develops an axiomatic framework for indefinite descriptive proximities in dynamical systems, introducing the indefinite descriptive distance $d^{\lim\Phi}$ and the induced indefinite descriptive Hausdorff topology $(\mathcal{K}_{\Phi},d_H^{\Phi},\tau_H^{\Phi})$. It proves foundational results that every descriptive proximity space on a dynamical system is indefinite (Theorem 1) and that every dynamical system possesses an indefinite descriptive Hausdorff topology (Theorem 3), while also establishing that the waveform energy $E_{m(t)}$ varies with each clock tick (Theorem 4). The paper then develops a practical application using a relaxed proximity $\delta_{\Phi_o}$ and Hilbert envelope lobes to detect and quantify stable, low-energy-dissipation portions of system waveforms. Overall, the work connects descriptive proximity theory with dynamical-system energy analysis to identify stable regions in complex, self-similar or chaotic dynamics, providing a rigorous axiomatic toolkit for probing energy dissipation and stability in dynamical scenes.

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.

Indefinite Descriptive Proximities Inherent in Dynamical Systems. An Axiomatic Approach

TL;DR

This work develops an axiomatic framework for indefinite descriptive proximities in dynamical systems, introducing the indefinite descriptive distance and the induced indefinite descriptive Hausdorff topology . It proves foundational results that every descriptive proximity space on a dynamical system is indefinite (Theorem 1) and that every dynamical system possesses an indefinite descriptive Hausdorff topology (Theorem 3), while also establishing that the waveform energy varies with each clock tick (Theorem 4). The paper then develops a practical application using a relaxed proximity and Hilbert envelope lobes to detect and quantify stable, low-energy-dissipation portions of system waveforms. Overall, the work connects descriptive proximity theory with dynamical-system energy analysis to identify stable regions in complex, self-similar or chaotic dynamics, providing a rigorous axiomatic toolkit for probing energy dissipation and stability in dynamical scenes.

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.
Paper Structure (7 sections, 13 theorems, 9 equations, 6 figures, 4 tables)

This paper contains 7 sections, 13 theorems, 9 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Chaotic dynamical system
  4. Descriptive Case
  5. Descriptive Case
  6. Descriptive Dynamical System
  7. Application: Relaxed Descriptive Proximities

Key Result

Corollary 1

There is a descriptive Hausdorff topology $\tau_H^{\Phi}$ on every collection of compact complete descriptions of a descriptive proximity space $(X,\delta_{\Phi})$.

Figures (6)

  • Figure 1: Torus near sets
  • Figure 2: Self-similar biker motion waveform
  • Figure 3: Relaxed Proximities Between Hilbert Lobes
  • Figure 4: Source of Hilbert energy lobes in Table \ref{['table:runner19']}
  • Figure 5: Source of Hilbert energy lobes in Table \ref{['table:runner24']}
  • ...and 1 more figures

Theorems & Definitions (52)

  • Definition 1
  • Example 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 2
  • Definition 5
  • Corollary 1
  • Remark 3
  • ...and 42 more