b. Implement ensemble SDE simulator with parallelization (N2k).
c. Calibrate HcH_c and weights ww by backtesting on past emergence episodes.
d. Launch daily runs; produce dashboard with decision rules and alerts.
e. Regularly update model structure after each major event and re-estimate parameters.
This Black-Horse emergence mapping converts the abstract concept H(t)H(t) into a concrete early-warning product: a probability curve, actor attribution, MFPT estimates, and clear decision thresholds. When paired with the full bifurcation model, it supplies leaders and analysts with both when and who questions---enabling targeted, timely, and ethically responsible responses.
C. Phase of Stability vs. Chaotic Transition
This subsection synthesizes model mechanics into a practical taxonomy of system regimes and the observable signals that indicate a shift from stability into chaotic / transition behavior. The goal is to give decision makers clear diagnostic criteria, parameter boundaries, and response prescriptions tied to the model's structure (state vector X=(P,T,E,H,R)X=(P,T,E,H,R), policy levers U,KU,\mathcal{K}, and key bifurcation parameter r=K/Ur=\rho_{\mathcal{K}}/\rho_U).
1. Two principal regimes (qualitative)
Stable (Containment) Regime --- Basin of Attraction A
Characterized by a dominant stable fixed point xAx^\ast_A with: low PP^\ast, moderate--high TT^\ast, declining EE^\ast, and low HH^\ast.
Reinforcing loops (R1--R4) are weak relative to balancing loops (B1--B3).
Small perturbations (noise) decay --- the system returns to xAx^\ast_A.
Chaotic / Transition Regime --- Basin of Attraction B (or Strange Attractor)
