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| Main Authors: | , |
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| Format: | Recurso digital |
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Zenodo
2025
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| Online Access: | https://doi.org/10.5281/zenodo.17858988 |
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Table of Contents:
- <p>The Unified Singularity Model (USM) was originally introduced as a structural framework for describing complex behavior as oscillation around a smooth backbone. In its first formulation (USM v1.0), an observable x(t) was decomposed as x(t) = σ(t) + u(t), where σ(t) is a singularity path (backbone) and u(t) is a deviation field encoding oscillations and fluctuations around that backbone. The initial work focused on geometric and stability properties of this decomposition—curvature, amplitude, multiscale structure, and conditions under which u(t) remains coherent and bounded. However, the dynamics of σ(t) and u(t) themselves were left implicit.</p> <p>This paper develops a dynamical extension of the Unified Singularity Model by introducing an explicit forcing structure and prototype evolution equations for σ(t) and u(t). First, we decompose the total effective forcing F(t) into a singularity-level component and a deviation-level component, F(t) = F_sing(t) + F_dev(t), where F_sing(t) acts primarily on the singularity path and F_dev(t) acts primarily on the deviation field. We then propose backbone dynamics of the form D_σ[σ](t) = F_sing(t), with particular attention to first- and second-order ordinary differential equations such as σ''(t) + α_σ σ'(t) = h(σ(t)) + b_sing F_sing(t), in which α_σ is a backbone damping coefficient, h encodes intrinsic drift or curvature-restoring behavior, and b_sing scales the effect of F_sing(t).</p> <p>For the deviation field, we introduce oscillator-like dynamics, u''(t) + 2 λ(t) u'(t) + ω(t)² u(t) = F_dev(t), where λ(t) ≥ 0 is a damping coefficient and ω(t) is an effective frequency that may depend on σ(t) and its derivatives. We analyze the associated deviation energy E_u(t) = ½ u'(t)² + ½ ω(t)² u(t)² and derive an energy balance that clarifies how forcing, damping, and time-varying stiffness inject, dissipate, and redistribute deviation energy. Combining backbone and deviation equations yields a coupled system for (σ(t), u(t), x(t)), providing a structured factorization of x''(t) into backbone drift, deviation dynamics, and the decomposed forcing components F_sing(t) and F_dev(t).</p> <p>As a prototype application, we interpret a financial price series p(t) as x(t) and sketch how σ(t), u(t), F_sing(t), and F_dev(t) might be estimated from empirical data via smoothing, numerical differentiation, and simple parametrizations of drift, damping, and frequency. This example is explicitly phenomenological: it illustrates how the USM dynamical framework can be instantiated in a concrete domain, but does not claim to capture the full complexity of financial markets or to provide trading or investment recommendations.</p> <p>Conceptually, the dynamical Unified Singularity Model completes the original invariance-driven construction: interconnectedness is made precise by shared or related singularity paths σ(t) and shared or related singularity-level forcing F_sing(t), while observable behavior is expressed as a backbone plus a deviation field evolved under structured forcing and damping. The resulting framework is intended as a hypothesis-generating, structurally unified model for backbone–deviation dynamics, providing a clear mechanism that can be extended, tested, and refined across different empirical domains.</p>