Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Recurso digital |
| Lingua: | |
| Pubblicazione: |
Zenodo
2026
|
| Accesso online: | https://doi.org/10.5281/zenodo.19059331 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866901984271925248 |
|---|---|
| author | von Mallinckrodt, Bernd |
| author_facet | von Mallinckrodt, Bernd |
| contents | <p>This paper provides a complete phase space analysis of the CRTI dynamical system — dR/dt = ρ·R·(1−R) − α·R·Φ, dΦ/dt = β·Φ·(1−Φ) − γ·R·Φ — where R is adaptive reorganization capacity and Φ is structural compression. The analysis reframes the CRTI framework around phase space topology rather than equilibrium claims, and establishes the central result that systemic collapse is governed by basin structure, not by scalar thresholds.<br>All four fixed points of the system are identified and classified by full Jacobian analysis: P₀ = (0,0) is an unstable node; P₁ = (1,0) is a stable node (adaptive attractor) when β < γ; P₂ = (0,1) is a stable node (compression attractor) when ρ < α; and the interior point P* = (R*, Φ*), when it exists, is always a saddle point. This corrects a stability classification in the preceding paper (Mallinckrodt, 2026b), which incorrectly identified P* as a stable node due to an incomplete determinant computation. The full expression det(J*) = RΦ(ρβ − αγ) is negative under the conditions required for P* to exist in the interior of [0,1]², establishing the saddle classification rigorously.<br>In the bistable parameter regime (λ = ρ/α < 1 and μ = β/γ < 1), the stable manifold W^s(P*) constitutes the separatrix dividing the phase space into two basins of attraction: the recovery basin converging to P₁ and the collapse basin converging to P₂. Collapse is defined dynamically as a trajectory in the collapse basin, not as a point event or scalar threshold crossing. The system is formally identified as belonging to the class of two-species competitive Lotka-Volterra systems with logistic self-limitation, for which all four qualitative phase portraits are established results in the mathematical ecology literature.<br>The Mallinckrodt Cycle is reformulated as a piecewise dynamical system: the continuous ODE governs inter-collapse dynamics, and a discrete reset map re-injects the system into the recovery basin. Reorganization viability requires the post-collapse state to lie in the recovery basin — a geometric condition on the separatrix, not a scalar condition on T. Systemic temperature T(t) = R(t)/Φ(t) is reinterpreted as a scalar projection of a two-dimensional phase portrait: it does not uniquely determine basin membership and cannot serve as a standalone collapse indicator.<br><br></p> <p>CRTI · phase space · separatrix · basin of attraction · saddle point · bistability · collapse pathways · adaptive capacity · structural compression · systemic temperature · competitive Lotka-Volterra · fixed point analysis · Jacobian stability · determinant correction · Mallinckrodt Cycle · Mallinckrodt Framework · nonlinear dynamics · complexity science · resilience theory · piecewise dynamical system</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19059331 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Phase Space Structure, Separatrices, and Collapse Pathways in the CRTI Framework von Mallinckrodt, Bernd <p>This paper provides a complete phase space analysis of the CRTI dynamical system — dR/dt = ρ·R·(1−R) − α·R·Φ, dΦ/dt = β·Φ·(1−Φ) − γ·R·Φ — where R is adaptive reorganization capacity and Φ is structural compression. The analysis reframes the CRTI framework around phase space topology rather than equilibrium claims, and establishes the central result that systemic collapse is governed by basin structure, not by scalar thresholds.<br>All four fixed points of the system are identified and classified by full Jacobian analysis: P₀ = (0,0) is an unstable node; P₁ = (1,0) is a stable node (adaptive attractor) when β < γ; P₂ = (0,1) is a stable node (compression attractor) when ρ < α; and the interior point P* = (R*, Φ*), when it exists, is always a saddle point. This corrects a stability classification in the preceding paper (Mallinckrodt, 2026b), which incorrectly identified P* as a stable node due to an incomplete determinant computation. The full expression det(J*) = RΦ(ρβ − αγ) is negative under the conditions required for P* to exist in the interior of [0,1]², establishing the saddle classification rigorously.<br>In the bistable parameter regime (λ = ρ/α < 1 and μ = β/γ < 1), the stable manifold W^s(P*) constitutes the separatrix dividing the phase space into two basins of attraction: the recovery basin converging to P₁ and the collapse basin converging to P₂. Collapse is defined dynamically as a trajectory in the collapse basin, not as a point event or scalar threshold crossing. The system is formally identified as belonging to the class of two-species competitive Lotka-Volterra systems with logistic self-limitation, for which all four qualitative phase portraits are established results in the mathematical ecology literature.<br>The Mallinckrodt Cycle is reformulated as a piecewise dynamical system: the continuous ODE governs inter-collapse dynamics, and a discrete reset map re-injects the system into the recovery basin. Reorganization viability requires the post-collapse state to lie in the recovery basin — a geometric condition on the separatrix, not a scalar condition on T. Systemic temperature T(t) = R(t)/Φ(t) is reinterpreted as a scalar projection of a two-dimensional phase portrait: it does not uniquely determine basin membership and cannot serve as a standalone collapse indicator.<br><br></p> <p>CRTI · phase space · separatrix · basin of attraction · saddle point · bistability · collapse pathways · adaptive capacity · structural compression · systemic temperature · competitive Lotka-Volterra · fixed point analysis · Jacobian stability · determinant correction · Mallinckrodt Cycle · Mallinckrodt Framework · nonlinear dynamics · complexity science · resilience theory · piecewise dynamical system</p> |
| title | Phase Space Structure, Separatrices, and Collapse Pathways in the CRTI Framework |
| url | https://doi.org/10.5281/zenodo.19059331 |