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2026
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| Acceso en línea: | https://doi.org/10.5281/zenodo.19568424 |
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| _version_ | 1866901705726099456 |
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| author | von Mallinckrodt, Bernd |
| author_facet | von Mallinckrodt, Bernd |
| contents | <p>Structural Compression Precedes Variance Amplification Near Fold Bifurcations</p> <p> </p> <p>This work analyzes the behavior of multivariate stochastic systems approaching fold (saddle-node) bifurcations through the geometry of their stationary covariance structure. While classical early warning signals focus on variance amplification and critical slowing down, these indicators primarily capture late-stage dynamics close to the critical point.</p> <p> </p> <p>We introduce a structural perspective based on the spectral entropy of the covariance matrix and its normalized effective rank (Φ_norm). Using a linearized Itô system and the Lyapunov equation, we show that covariance eigenvalue concentration leads to a progressive reduction of effective dimensionality as the dominant mode diverges. This process—referred to as structural compression—is observable across the full pre-bifurcation regime.</p> <p> </p> <p>Analytical results demonstrate that Φ_norm decreases monotonically toward zero under standard assumptions (spectral gap, non-degenerate noise, near-normal dynamics), with asymptotic scaling proportional to |α₁|·|log|α₁||. Numerical simulations of a 4-dimensional Ornstein–Uhlenbeck system confirm that structural compression unfolds gradually, while total variance remains nearly constant until the final stage before the bifurcation.</p> <p> </p> <p>The key empirical finding is not an earlier threshold crossing in a statistical sense, but a redistribution of informative signal across the control parameter: Φ_norm utilizes its dynamic range continuously throughout the approach, whereas variance concentrates its change near criticality.</p> <p> </p> <p>We interpret Φ_norm as a quasi-order parameter describing the geometric collapse of covariance structure. A complementary recovery proxy (AR(1) coefficient of the leading principal component) provides an auxiliary dynamical channel, enabling a joint structural–temporal diagnostic framework.</p> <p> </p> <p>The results establish structural compression as a robust and interpretable mechanism underlying early warning behavior in multivariate systems, with potential applications in ecology, climate dynamics, financial systems, and other complex adaptive domains.</p> <p> </p> <p> </p> <p> <span class="Apple-converted-space"> </span></p> <p>Core scientific</p> <p> </p> <p> </p> <ul> <li>early warning signals</li> <li>critical transitions</li> <li>fold bifurcation</li> <li>saddle-node bifurcation</li> <li>critical slowing down</li> </ul> <p> </p> <p> </p> <p> </p> <p> <span class="Apple-converted-space"> </span></p> <p>Method / novelty</p> <p> </p> <p> </p> <ul> <li>structural compression</li> <li>spectral entropy</li> <li>effective rank</li> <li>covariance analysis</li> <li>multivariate systems</li> </ul> <p> </p> <p> </p> <p> </p> <p> <span class="Apple-converted-space"> </span></p> <p>Mathematical / physical</p> <p> </p> <p> </p> <ul> <li>Lyapunov equation</li> <li>Ornstein-Uhlenbeck process</li> <li>stochastic dynamics</li> <li>eigenvalue spectrum</li> <li>dimensionality reduction</li> </ul> <p> </p> <p> </p> <p> </p> <p> <span class="Apple-converted-space"> </span></p> <p>Application fields</p> <p> </p> <p> </p> <ul> <li>complex systems</li> <li>system stability</li> <li>tipping points</li> <li>resilience</li> </ul> <p> </p> <p> </p> <p>early warning signals, critical transitions, fold bifurcation, structural compression, spectral entropy, effective rank, covariance analysis, Lyapunov equation, stochastic systems, Ornstein-Uhlenbeck process, tipping points, system stability</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19568424 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Structural Compression Precedes Variance Amplification Near Fold Bifurcations: A Covariance-Based Analysis von Mallinckrodt, Bernd <p>Structural Compression Precedes Variance Amplification Near Fold Bifurcations</p> <p> </p> <p>This work analyzes the behavior of multivariate stochastic systems approaching fold (saddle-node) bifurcations through the geometry of their stationary covariance structure. While classical early warning signals focus on variance amplification and critical slowing down, these indicators primarily capture late-stage dynamics close to the critical point.</p> <p> </p> <p>We introduce a structural perspective based on the spectral entropy of the covariance matrix and its normalized effective rank (Φ_norm). Using a linearized Itô system and the Lyapunov equation, we show that covariance eigenvalue concentration leads to a progressive reduction of effective dimensionality as the dominant mode diverges. This process—referred to as structural compression—is observable across the full pre-bifurcation regime.</p> <p> </p> <p>Analytical results demonstrate that Φ_norm decreases monotonically toward zero under standard assumptions (spectral gap, non-degenerate noise, near-normal dynamics), with asymptotic scaling proportional to |α₁|·|log|α₁||. Numerical simulations of a 4-dimensional Ornstein–Uhlenbeck system confirm that structural compression unfolds gradually, while total variance remains nearly constant until the final stage before the bifurcation.</p> <p> </p> <p>The key empirical finding is not an earlier threshold crossing in a statistical sense, but a redistribution of informative signal across the control parameter: Φ_norm utilizes its dynamic range continuously throughout the approach, whereas variance concentrates its change near criticality.</p> <p> </p> <p>We interpret Φ_norm as a quasi-order parameter describing the geometric collapse of covariance structure. A complementary recovery proxy (AR(1) coefficient of the leading principal component) provides an auxiliary dynamical channel, enabling a joint structural–temporal diagnostic framework.</p> <p> </p> <p>The results establish structural compression as a robust and interpretable mechanism underlying early warning behavior in multivariate systems, with potential applications in ecology, climate dynamics, financial systems, and other complex adaptive domains.</p> <p> </p> <p> </p> <p> <span class="Apple-converted-space"> </span></p> <p>Core scientific</p> <p> </p> <p> </p> <ul> <li>early warning signals</li> <li>critical transitions</li> <li>fold bifurcation</li> <li>saddle-node bifurcation</li> <li>critical slowing down</li> </ul> <p> </p> <p> </p> <p> </p> <p> <span class="Apple-converted-space"> </span></p> <p>Method / novelty</p> <p> </p> <p> </p> <ul> <li>structural compression</li> <li>spectral entropy</li> <li>effective rank</li> <li>covariance analysis</li> <li>multivariate systems</li> </ul> <p> </p> <p> </p> <p> </p> <p> <span class="Apple-converted-space"> </span></p> <p>Mathematical / physical</p> <p> </p> <p> </p> <ul> <li>Lyapunov equation</li> <li>Ornstein-Uhlenbeck process</li> <li>stochastic dynamics</li> <li>eigenvalue spectrum</li> <li>dimensionality reduction</li> </ul> <p> </p> <p> </p> <p> </p> <p> <span class="Apple-converted-space"> </span></p> <p>Application fields</p> <p> </p> <p> </p> <ul> <li>complex systems</li> <li>system stability</li> <li>tipping points</li> <li>resilience</li> </ul> <p> </p> <p> </p> <p>early warning signals, critical transitions, fold bifurcation, structural compression, spectral entropy, effective rank, covariance analysis, Lyapunov equation, stochastic systems, Ornstein-Uhlenbeck process, tipping points, system stability</p> |
| title | Structural Compression Precedes Variance Amplification Near Fold Bifurcations: A Covariance-Based Analysis |
| url | https://doi.org/10.5281/zenodo.19568424 |