Chronos Spatial Stability: Dispersion Relations for a Coupled Time Field and Density System
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| Formato: | Recurso digital |
| Lenguaje: | inglés |
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2025
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| _version_ | 1866901661504503808 |
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| author | Hall, Matthew |
| author_facet | Hall, Matthew |
| contents | <p>This paper extends the Chronos time-field framework by incorporating spatial dependence and Fourier-mode analysis into the coupled time field Θ(t, x) and density field ρ(t, x). Starting from a reaction–diffusion-type equation for ρ and a diffusion–relaxation equation for Θ, the system is linearized around a homogeneous equilibrium, yielding a 2×2 mode matrix M(k) for each wavenumber k.</p> <p>The dispersion relation λ±(k) derived from M(k) determines whether spatial perturbations grow or decay. In the symmetric parameter regime (Dρ = DΘ, κ = σ), the determinant simplifies to (κ + Dk²)² − CG, which is minimized at k = 0. This implies that global stability across all spatial modes is governed by the same Chronos threshold χ < 1 obtained in the homogeneous model. The Chronos constant χ thus controls stability universally, independent of wavelength.</p> <p>A verification protocol is provided to ensure that any researcher or automated reasoning system can independently confirm the linearization, mode matrix, dispersion relation, and Chronos threshold.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17808122 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Chronos Spatial Stability: Dispersion Relations for a Coupled Time Field and Density System Hall, Matthew Chronos constant dispersion relation stability analysis Fourier modes reaction–diffusion systems time field linearization eigenvalue analysis CHaSSE mathematical physics time-field dynamics <p>This paper extends the Chronos time-field framework by incorporating spatial dependence and Fourier-mode analysis into the coupled time field Θ(t, x) and density field ρ(t, x). Starting from a reaction–diffusion-type equation for ρ and a diffusion–relaxation equation for Θ, the system is linearized around a homogeneous equilibrium, yielding a 2×2 mode matrix M(k) for each wavenumber k.</p> <p>The dispersion relation λ±(k) derived from M(k) determines whether spatial perturbations grow or decay. In the symmetric parameter regime (Dρ = DΘ, κ = σ), the determinant simplifies to (κ + Dk²)² − CG, which is minimized at k = 0. This implies that global stability across all spatial modes is governed by the same Chronos threshold χ < 1 obtained in the homogeneous model. The Chronos constant χ thus controls stability universally, independent of wavelength.</p> <p>A verification protocol is provided to ensure that any researcher or automated reasoning system can independently confirm the linearization, mode matrix, dispersion relation, and Chronos threshold.</p> |
| title | Chronos Spatial Stability: Dispersion Relations for a Coupled Time Field and Density System |
| topic | Chronos constant dispersion relation stability analysis Fourier modes reaction–diffusion systems time field linearization eigenvalue analysis CHaSSE mathematical physics time-field dynamics |
| url | https://doi.org/10.5281/zenodo.17808122 |