C. Sensitivity analysis: how legitimacy, narrative, and repression shift _crit
To assess how specific leadership dimensions affect systemic resilience, we performed a parameter sensitivity analysis centered on the critical economic stress threshold crit\mu_{\text{crit}}crit. This threshold marks the onset of instability, where equilibrium stability is lost via a saddle--node or Hopf bifurcation. By perturbing selected leadership parameters around their baseline values, we can quantify both the direction and magnitude of their impact on crit\mu_{\text{crit}}crit.
1. Legitimacy
Legitimacy acts primarily through the stabilizing coefficient T\delta_TT, which links trust erosion to the underlying political mandate. Increasing legitimacy raises T\delta_TT, thereby dampening the erosion of trust in the presence of protests. Numerical sweeps show that higher legitimacy systematically shifts crit\mu_{\text{crit}}crit upward, meaning that a system with strong legitimacy can absorb greater economic shocks before reaching instability. Conversely, low legitimacy sharply lowers crit\mu_{\text{crit}}crit, making unrest more likely even under modest stress.
2. Narrative control
Narrative influences the trust-building pathway BLB_LBL and the damping of protest growth. Strong narrative capacity amplifies the conversion of economic stress into resilience by maintaining coherence in public perception. Sensitivity curves demonstrate that improving narrative control increases the slope of the stability region in the (E,P)(E,P)(E,P) plane, thereby delaying bifurcation. However, unlike legitimacy, narrative has a nonlinear impact: once narrative control surpasses a moderate threshold, marginal improvements yield diminishing returns. Weak narrative, on the other hand, can collapse stability abruptly, lowering crit\mu_{\text{crit}}crit to values near zero.
3. Repression vs. consensus
The repression--consensus balance modifies both the direct protest damping term (c4c_4c4) and the backfire channel feeding protest intensity. Sensitivity experiments reveal a U-shaped effect:
At moderate levels, repression stabilizes the system by containing protests, thus pushing crit\mu_{\text{crit}}crit upward.
At excessive levels, repression generates backlash, reducing consensus and indirectly amplifying protest dynamics. This lowers crit\mu_{\text{crit}}crit, making the system fragile to even small shocks.
Conversely, a tilt toward consensus-building (low repression) broadens the stability basin, but only if supported by sufficient legitimacy. Otherwise, low repression combined with low legitimacy leads to rapid erosion of trust.
4. Comparative influence
When normalized, legitimacy exerts the strongest linear effect on crit\mu_{\text{crit}}crit, followed by repression (nonlinear and regime-dependent), and finally narrative (saturating impact). Taken together, these three parameters provide interpretable levers of systemic resilience. Our formalism thus operationalizes the intuitive claim that leaders with higher legitimacy, credible narratives, and calibrated use of repression extend the stress threshold before unrest becomes inevitable.
D Simulation of black horse emergence (HHH)
