Article
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Convergence in Discrete Inverse Dynamical Systems
Version 1
: Received: 9 March 2024 / Approved: 12 March 2024 / Online: 17 March 2024 (15:58:52 CET)
How to cite: Diedrich, E. Convergence in Discrete Inverse Dynamical Systems. Preprints 2024, 2024030702. https://doi.org/10.20944/preprints202403.0702.v1 Diedrich, E. Convergence in Discrete Inverse Dynamical Systems. Preprints 2024, 2024030702. https://doi.org/10.20944/preprints202403.0702.v1
Abstract
This article presents a new methodology for modeling and analyzing discrete dynamic systems through the construction of inverse algebraic models, giving rise to the Theory of Inverse Discrete Dynamical Systems. Key concepts such as inverse modeling, structural analysis in inverse algebraic trees, and the establishment of topological equivalences for the analytical transfer of properties between the canonical system and its inverted counterpart are introduced. Central theorems on homeomorphic invariance and topological transport are demonstrated, consolidating the validity of transferring cardinal attributes between both equivalent dynamic representations. As a concrete application, an alternative proof of the historic Collatz Conjecture is presented through the rigorous construction of the associated inverse model and the analytical transfer of properties exhibited in the inverted tree structure. Furthermore, evidence has been presented on Conway's Conjecture in the Game of Life through meticulous inverse modeling in bounded cases, overcoming initial limitations of the theory in combinatorial explosions. Although a complete proof of the conjecture is not achieved, this positively expands the modeling and analysis capacity of the methodology. The impact of properly applying this novel theory to expand understanding and solve open problems in discrete dynamical systems seems vast and profound.
Keywords
Discrete dynamical systems; Inverse modeling; Topological equivalence; Topological transport; Algebraic trees; Combinatorial explosions; Computational complexity; Universal convergence
Subject
Computer Science and Mathematics, Security Systems
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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