The analysis used finite differences for second derivatives; publication-grade symbolic derivatives and numerical continuation (AUTO / MATCONT) are recommended to refine bifurcation curves and produce rigorous continuation branches.
Results are conditional on the chosen coefficient mappings from leadership parameters to model coefficients. Empirical calibration (likelihood / Bayesian fitting) would reduce subjectivity.
The reduced T,E,PT,E,PT,E,P model captures the essential fold mechanism but omits explicit dynamical coupling with HHH (black-horse potential). Section 4 provides full simulations including HHH.
3.A.6 Concluding remarks for the fold analysis
The fold analysis demonstrates that leadership quality, encoded in the seven-dimensional \Theta, systematically modulates the critical economic shock amplitude crit\mu_{\mathrm{crit}}crit required for abrupt social unrest. The four case studies illustrate a clear ordering of resilience (SBY Jokowi Soeharto Prabowo) under the adopted mappings. The extracted normal-form coefficients validate that the transitions are standard saddle-node bifurcations and provide interpretable sensitivity measures for policy: increasing legitimacy and crisis management robustness yields the largest shifts in crit\mu_{\mathrm{crit}}crit, raising the regime's resilience to economic perturbations.
B Hopf normal-form calculations and limit-cycle analysis (augmented model)
3.B.1 Why augment the model?
In the reduced (T,E,P)(T,E,P)(T,E,P) system the numerics showed fold (saddle-node) tipping as the dominant route to unrest; a coarse Hopf search returned no complex-pair crossings. To allow oscillatory protest waves (small-amplitude cycles born at Hopf), we introduce an explicit policy/response lag that closes a negative-feedback loop with delay-like dynamics---well known to generate Hopf when gain and lag exceed damping.
We add a slow state GGG (aggregate policy response), yielding a 4D ODE that preserves our leadership linkages:
T=TS(E)(L+NN+MM)(1T)(T+T(1R))P,E=EdEE+PPTTGG,P=PP(1PPmax)+ES(E)TTc4PGG,G=G+kpPktT.\begin{aligned} \dot T &= \alpha_T\,S(E)\,\big(L+\sigma_N N+\sigma_M M\big)\,(1-T)\;-\;(\beta_T+\gamma_T (1\!-\!R))\,P,\\ \dot E &= \mu_E - d_E E + \phi_P P - \phi_T T \;\;-\; \phi_G G,\\ \dot P &= \rho_P P\Big(1-\tfrac{P}{P_{\max}}\Big) + \eta_E S(E) - \delta_T T - c_4 P \;\;-\; \chi_G G,\\ \dot G &= \frac{-G + k_p P - k_t T}{\tau}. \end{aligned}TEPG=TS(E)(L+NN+MM)(1T)(T+T(1R))P,=EdEE+PPTTGG,=PP(1PmaxP)+ES(E)TTc4PGG,=G+kpPktT.
Here S(E)=EE+S(E)=\dfrac{E^\kappa}{E^\kappa+\theta^\kappa}S(E)=E+E is the same stress activation as before. The leadership parameters =(C,L,M,N,ES,EM,R)\Theta=(C,L,M,N,ES,EM,R)=(C,L,M,N,ES,EM,R) modulate coefficients as previously defined; new couplings obey:
Gain: kpk_p\uparrowkp with M,EMM, EMM,EM; ktk_t\uparrowkt with M,LM, LM,L.
Lag: \tau\uparrow when CC\downarrowC and MM\downarrowM (slower, more inertial state reaction).
Policy impacts: G,G\phi_G,\chi_G\uparrowG,G with LLL and ESESES (legitimate policies help both economy and de-escalation).
Linear Hopf conditions (cubic sub-block)
Linearizing at an equilibrium X=(T,E,P,G)X^*=(T^*,E^*,P^*,G^*)X=(T,E,P,G) gives Jacobian JJJ. Because GGG only couples with (T,E,P)(T,E,P)(T,E,P), the oscillatory onset is governed by a cubic characteristic polynomial on the (E,P,G)(E,P,G)(E,P,G) feedback loop (trust acts as a stabilizing leak and coupling):
3+a12+a2+a3=0,\lambda^3 + a_1 \lambda^2 + a_2 \lambda + a_3 =0,3+a12+a2+a3=0,
