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2026
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| Online Access: | https://doi.org/10.5281/zenodo.19426875 |
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| author | von Mallinckrodt, Bernd |
| author_facet | von Mallinckrodt, Bernd |
| contents | <p>This paper introduces Structural Compression (Φ) as a geometrically grounded early warning signal for fold-type (saddle-node) critical transitions in complex dynamical systems.</p> <p> </p> <p>Classical early warning signals based on critical slowing down (CSD), such as rising variance and lag-1 autocorrelation, are widely used but remain mechanism-agnostic and primarily amplitude-based. They do not provide a structural account of how a system progressively loses resilience prior to a transition.</p> <p> </p> <p>To address this limitation, Structural Compression is defined as the normalized spectral entropy of a rolling covariance matrix, capturing the effective dimensionality of multivariate state-space dynamics. As a system approaches a fold bifurcation, variance concentrates along a critical mode, resulting in a measurable contraction of degrees of freedom reflected in decreasing Φ.</p> <p> </p> <p>The framework introduces the Compression–Response Transition Index (CRTI), defined as</p> <p>T = R̂ / Φ,</p> <p>where R̂ denotes adaptive response capacity derived from lag-1 autocorrelation of the leading principal component. A formal validity boundary is established via the Structural–Dynamic Separability (SDS) condition, requiring low correlation between Φ and R̂ for meaningful composite interpretation.</p> <p> </p> <p>Simulation results indicate that Φ declines and T rises consistently prior to fold-type bifurcations, while these patterns are not reproduced under Hopf bifurcation or noise-induced transition conditions. These findings support a mechanism-specific interpretation of early warning signals based on geometric structure rather than amplitude alone.</p> <p> </p> <p>This paper provides the primary theoretical formulation of the CRTI framework. All related extensions, methodological refinements, and domain applications build upon the definitions and boundary conditions established here.</p> <p> </p> <p>Limitations: The framework applies specifically to fold-type bifurcations and is not applicable to Hopf or noise-induced transitions. Results are based on simulation experiments and require empirical validation across independent datasets. No operational thresholds are defined.</p> <p> </p> <p>Preprint DOI: https://doi.org/10.5281/zenodo.19410081</p> <p> </p> <p> </p> <p>early warning signals; structural compression; spectral entropy; critical transitions; CRTI; fold bifurcation; covariance dynamics; multivariate systems; resilience indicators; dynamical systems</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19426875 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Structural Compression as a Mechanism-Specific Early Warning Signal Beyond Critical Slowing Down (CRTI Framework) von Mallinckrodt, Bernd <p>This paper introduces Structural Compression (Φ) as a geometrically grounded early warning signal for fold-type (saddle-node) critical transitions in complex dynamical systems.</p> <p> </p> <p>Classical early warning signals based on critical slowing down (CSD), such as rising variance and lag-1 autocorrelation, are widely used but remain mechanism-agnostic and primarily amplitude-based. They do not provide a structural account of how a system progressively loses resilience prior to a transition.</p> <p> </p> <p>To address this limitation, Structural Compression is defined as the normalized spectral entropy of a rolling covariance matrix, capturing the effective dimensionality of multivariate state-space dynamics. As a system approaches a fold bifurcation, variance concentrates along a critical mode, resulting in a measurable contraction of degrees of freedom reflected in decreasing Φ.</p> <p> </p> <p>The framework introduces the Compression–Response Transition Index (CRTI), defined as</p> <p>T = R̂ / Φ,</p> <p>where R̂ denotes adaptive response capacity derived from lag-1 autocorrelation of the leading principal component. A formal validity boundary is established via the Structural–Dynamic Separability (SDS) condition, requiring low correlation between Φ and R̂ for meaningful composite interpretation.</p> <p> </p> <p>Simulation results indicate that Φ declines and T rises consistently prior to fold-type bifurcations, while these patterns are not reproduced under Hopf bifurcation or noise-induced transition conditions. These findings support a mechanism-specific interpretation of early warning signals based on geometric structure rather than amplitude alone.</p> <p> </p> <p>This paper provides the primary theoretical formulation of the CRTI framework. All related extensions, methodological refinements, and domain applications build upon the definitions and boundary conditions established here.</p> <p> </p> <p>Limitations: The framework applies specifically to fold-type bifurcations and is not applicable to Hopf or noise-induced transitions. Results are based on simulation experiments and require empirical validation across independent datasets. No operational thresholds are defined.</p> <p> </p> <p>Preprint DOI: https://doi.org/10.5281/zenodo.19410081</p> <p> </p> <p> </p> <p>early warning signals; structural compression; spectral entropy; critical transitions; CRTI; fold bifurcation; covariance dynamics; multivariate systems; resilience indicators; dynamical systems</p> |
| title | Structural Compression as a Mechanism-Specific Early Warning Signal Beyond Critical Slowing Down (CRTI Framework) |
| url | https://doi.org/10.5281/zenodo.19426875 |