SBY --- highest estimated crit\mu_{\mathrm{crit}}crit. Under the chosen coefficient mapping, SBY's leadership vector (high legitimacy LLL, high crisis management MMM) produces the most resilient system: larger exogenous economic stress is required to drive the system to fold tipping. In normal-form terms, the associated b>0b>0b>0 and a0a\neq0a=0 satisfy the nondegeneracy conditions; the fold is classical.
Jokowi --- crit\mu_{\mathrm{crit}}crit slightly lower than SBY but still large. Strong narrative control NNN and substantial elite management EcE_cEc increase stabilizing coefficients, shifting the critical shock to the right; however, the slope of the branch past the fold is relatively steep (sharp tipping once the threshold is passed).
Soeharto --- intermediate crit\mu_{\mathrm{crit}}crit. Although Soeharto's high consensus/cooptation produces a nontrivial cooptation term KKK that suppresses protests at low shocks, low legitimacy LLL decreases trust recovery capacity: the fold occurs at moderate E\mu_EE, and the post-fold jump in PPP is abrupt (classic fold / tipping behavior).
Prabowo --- lowest crit\mu_{\mathrm{crit}}crit among the four. The combination of relatively low legitimacy LLL, weaker crisis management MMM, and repressive tilt (low RRR) yields a fragile configuration: even moderate E\mu_EE drives the system through the fold. Normal-form coefficients show aaa and bbb of magnitudes consistent with a robust (nondegenerate) saddle-node.
Diagnostic figure. Figure 1 (displayed in the analysis session) plots miniRei(E)\min_i \operatorname{Re}\lambda_i(\mu_E)miniRei(E) for each leadership profile. The zero-crossings correspond to the crit\mu_{\mathrm{crit}}crit entries in Table 1 and visually illustrate the relative positions of the fold points.
3.A.3 Local normal-form approximation and validation
At each computed fold point we constructed the one-dimensional normal form
x=b(Ec)+ax2,\dot x \;=\; b(\mu_E-\mu_c) + a\, x^2,x=b(Ec)+ax2,
with aaa and bbb taken from the finite-difference computations. To validate the normal form approximation:
We integrated the reduced normal form for parameter perturbations E=c\mu_E=\mu_c\pm\varepsilonE=c with 1\varepsilon\ll11 and compared the equilibrium and transient behavior of the scalar model to trajectories of the full 333-dimensional model initialized near XcX_cXc.
1. The normal form reproduces the leading-order scaling of the post-fold jump and the qualitative behavior (creation/annihilation of equilibria). Figure 2 (constructed in the session) overlays the normal-form bifurcation skeleton with the numerically computed P(E)P^*(\mu_E)P(E) branch for each leader; agreement is excellent close to c\mu_cc, with departures at larger Ec|\mu_E-\mu_c|Ec due to higher-order terms and global nonlinearities.
2. Interpretation. The sign of aaa and the sign of bbb together determine whether equilibria are created or annihilated as E\mu_EE is varied upward. For all four leadership profiles the product aba bab had a sign consistent with a standard saddle-node unlocking unrest at E>c\mu_E>\mu_cE>c in our coefficient mapping. This confirms the qualitative mechanism that economic shocks above a leader-dependent critical value induce abrupt transitions from low to high protest intensity.
3.A.4 Robustness checks and sensitivity
We performed sensitivity checks to verify that fold locations shift coherently with changes in specific leadership parameters:
Legitimacy LLL: increasing LLL by +0.1 shifts crit\mu_{\mathrm{crit}}crit to larger values for all leaders (system becomes more resilient). The magnitude of the shift is largest for profiles with initially low LLL (e.g., Prabowo).
Elite management EcE_cEc: increasing EcE_cEc raises KKK and increases the effective stability; in many cases this shifts crit\mu_{\mathrm{crit}}crit to the right, but at the cost of making post-fold jumps sharper if RRR is low (cooptation + repression tradeoff).
Repression balance RRR: lowering RRR (more repressive) can transiently increase short-term suppression (raising apparent local stability) but increases backfire effects (via T(1R)\gamma_T(1-R)T(1R)), reducing trust and thus lowering crit\mu_{\mathrm{crit}}crit in the medium term --- a quantitatively important trade-off revealed by the fold computations.
These sensitivity tests confirm that the leadership parameter vector \Theta acts as a secondary control set that shifts the primary bifurcation parameter threshold crit=()\mu_{\mathrm{crit}}=\Phi(\Theta)crit=().
3.A.5 Limitations and computational caveats
