Once a Hopf point (X,h)(X^*,\mu_h)(X,h) is located, the center-manifold reduction yields
z=(+i0)z+1zz2+O(z4),\dot z = (\sigma + i \omega_0) z + \ell_1 z |z|^2 + \mathcal{O}(|z|^4),z=(+i0)z+1zz2+O(z4),
with =(Eh)\sigma=\alpha(\mu_E-\mu_h)=(Eh). The sign of 1\ell_11 determines whether cycles are born stable (supercritical, 1<0\ell_1<01<0, small periodic protest waves) or born unstable (subcritical, 1>0\ell_1>01>0, dangerous hysteresis lobes where finite shocks kick the system onto large excursions). In our baseline scans, the system remained on the pre-Hopf side; if you want, we can increase \zeta systematically and produce (h,0,1)(\mu_h,\omega_0,\ell_1)(h,0,1) tables plus limit-cycle amplitude curves.
3.B.5 Methods --- targeted Hopf search in the augmented model
To test whether oscillatory protest dynamics (small-amplitude limit cycles) could appear in our framework we augmented the reduced model (T,E,P)(T,E,P)(T,E,P) with an explicit policy/response variable GGG. The augmented deterministic system is
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)\,(L+\sigma_N N + \sigma_M M)\,(1-T)\;-\;(\beta_T+\gamma_T(1-R))\,P,\\[4pt] \dot E &= \mu_E - d_E E + \phi_P P - \phi_T T - \phi_G G,\\[4pt] \dot P &= \rho_P P\Big(1-\frac{P}{P_{\max}}\Big) + \eta_E S(E) - \delta_T T - c_4 P - \chi_G G,\\[4pt] \dot G &= \dfrac{-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,
where S(E)S(E)S(E) is the same saturating stress activation used previously. Leadership parameters \Theta modulate the coefficients as in Section 2C; additional GGG-loop coefficients (kp,kt,,G,Gk_p,k_t,\tau,\phi_G,\chi_Gkp,kt,,G,G) are set by plausible mappings from the leadership vector (e.g., stronger crisis management and elite coordination increase kpk_pkp; lower consensus increases \tau).
Because Hopf bifurcations require both loop gain and effective delay/inertia, we carried out a targeted numerical experiment that systematically raised a scalar multiplier zzz (the "gain lag" multiplier) that scales the GGG-loop gain and lifts the effective lag:
For each leader (Soeharto, SBY, Jokowi, Prabowo) we scanned z{1.0,1.5,2.0,2.5,3.0,3.5,4.0,5.0}z \in \{1.0,1.5,2.0,2.5,3.0,3.5,4.0,5.0\}z{1.0,1.5,2.0,2.5,3.0,3.5,4.0,5.0} and E[0,1]\mu_E\in[0,1]E[0,1] on a coarse grid.
For each (z,E)(z,\mu_E)(z,E) we computed the steady state numerically (forward integration to convergence), evaluated the Jacobian, and tested for an eigenvalue pair with near-zero real part and nonzero imaginary part (the Hopf linear condition).
When a candidate crossing was found we refined E\mu_EE locally and attempted to compute the first Lyapunov coefficient 1\ell_11 via finite-difference multilinear forms to assess criticality (super/subcritical).
All numerical derivatives used conservative finite-difference increments; runs were intentionally coarse to explore the parameter space quickly. The code and parameter mapping used are archived in the analysis notebook.
3.B.6 Results --- targeted search outcomes
Summary (numeric run). The targeted sweep (z up to 5) did not detect Hopf bifurcations for any of the four leadership profiles under the coefficient mappings employed. The run summary CSV is available as:
