with a1=trJEPGa_1 = -\operatorname{tr}J_{EPG}a1=trJEPG, a2a_2a2 the sum of principal minors of JEPGJ_{EPG}JEPG, and a3=detJEPGa_3=\det J_{EPG}a3=detJEPG.
A (generic) Hopf occurs when the Routh--Hurwitz equalities are met:
a1>0,a2>0,a3>0,anda1a2=a3,a_1>0,\quad a_2>0,\quad a_3>0,\quad\text{and}\quad a_1 a_2 = a_3,a1>0,a2>0,a3>0,anda1a2=a3,
with ddERe=h0 \tfrac{d}{d\mu_E}\operatorname{Re}\lambda\big|_{\mu=\mu_h} \neq 0dEdRe=h=0. The first Lyapunov coefficient 1\ell_11 (computed from multilinear terms) sets the criticality: 1<0\ell_1<0\Rightarrow1<0 supercritical (small stable cycles); 1>0\ell_1>0\Rightarrow1>0 subcritical (unstable cycles and hysteresis).
3.B.2 Numerical search and outcome (what we found)
I implemented the 4D model and performed a coarse Hopf scan over E[0,1]\mu_E\in[0,1]E[0,1] for the four leadership profiles (Soeharto, SBY, Jokowi, Prabowo), then attempted 1\ell_11 estimation via finite-difference multilinear forms. With conservative (empirically plausible) gains and lags tied to leadership:
No Hopf crossings were detected in the scanned range; the real parts of the most oscillatory eigenpairs remained negative where imag parts were nonzero.
Interpretation: under the baseline mapping, the feedback loop (PGE,P)(P \to G \to E,P)(PGE,P) is not quite "hot" enough (insufficient gain and/or lag relative to damping c4,dEc_4, d_Ec4,dE) to overturn fold-dominated dynamics.
This result does not rule out Hopf; it diagnoses that the current gain--lag--damping combination sits on the stable side of the Routh--Hurwitz boundary.
3.B.3 How to make Hopf appear (testable parameter windows)
From the linear conditions and inspection of JJJ, three levers move a1a2a_1 a_2a1a2 toward a3a_3a3:
1. Increase effective lag or inertia
Raise \tau (slower policy) or insert an additional first-order lag in the GGG channel. Politically: sluggish, procedural or fragmented response.
2. Raise loop gain
Increase G\phi_GG and G\chi_GG (policy acts strongly on EEE and PPP), and/or kpk_pkp (policy reacts aggressively to PPP). High-gain + delay = overshoot oscillations.
3. Reduce intrinsic damping
Lower c4c_4c4 (less protest damping via cooptation/repression blend) or dEd_EdE (slower economic dissipation).
A practical continuation plan for the manuscript:
Treat :=(GGkp)\zeta:=\tau\cdot (\phi_G\,\chi_G\,k_p):=(GGkp) as a Hopf driver. Continue equilibria in (E,)(\mu_E,\zeta)(E,) and track the Hopf curve H()\mathcal{H}(\Theta)H() where a1a2=a3a_1 a_2=a_3a1a2=a3.
Expect Hopf to emerge for high \zeta (slow + strong policy loop) and diminish for large ktk_tkt (trust-sensitive policy reducing overshoot).
Policy reading: fast, trust-weighted responses (high ktk_tkt, moderate kpk_pkp, moderate G,G\phi_G,\chi_GG,G, small \tau) suppress protest waves; slow, over-reactive responses amplify them.
3.B.4 Normal form and interpretation
