1. Continuation-based search: Use AUTO or MATCONT to continue equilibria and detect Hopf loci in the (E,z)(\mu_E,z)(E,z) plane. Continuation is far more reliable than coarse grid scanning and will produce accurate h(z)\mu_h(z)h(z) curves and 1\ell_11 computations. (I can prepare MATCONT/AUTO input files if you want.)
2. Parameter identification: Bound the plausible ranges of kp,,G,Gk_p,\tau,\phi_G,\chi_Gkp,,G,G using historical policy response times and intensities. Avoid arbitrary large zzz unless politically justified.
3. Symbolic derivatives: Replace finite-difference multilinear forms with symbolic Jacobian/Hessian/Tensor derivatives (SymPy/Autograd) to compute 1\ell_11 robustly.
4. If Hopf is found: produce amplitude vs parameter plots (limit cycle amplitude as function of Eh\mu_E-\mu_hEh) and phase portraits for each leader to interpret oscillation amplitude and frequency differences arising from leadership vectors.
IV. Numerical Simulation
A Fixed-point analysis (analytical derivation)
1. Model (deterministic skeleton, restated)
We work with the reduced deterministic skeleton (same notation as Section 2.C). For clarity I rewrite the ODEs in compact notation:
\begin{aligned} \dot T &= \; \alpha_T\,S_E(E)\,B_L\,(1-T)\;-\;B_P\,P, \tag{1}\\[4pt] \dot E &= \; \mu_E \;-\; \lambda_E\,E \;+\; \phi_P\,P \;-\; \phi_T\,T, \tag{2}\\[4pt] \dot P &= \; \rho_P\,P\Big(1-\dfrac{P}{P_{\max}}\Big) \;+\; \eta_E\,S_E(E) \;-\; \delta_T\,T \;-\; \kappa_K\,K\,P, \tag{3} \end{aligned}
where (to condense notation)
SE(E)=EE+S_E(E)=\dfrac{E^\kappa}{E^\kappa+\theta^\kappa}SE(E)=E+E (Hill function),
BL(L+NN+MM)B_L \equiv (L + \sigma_N N + \sigma_M M)BL(L+NN+MM) (trust recovery baseline),
BP(T+T(1R))B_P \equiv (\beta_T + \gamma_T(1-R))BP(T+T(1R)) (trust erosion from protest),
E=E(G,M)\lambda_E=\lambda_E(G,M)E=E(G,M), K=K(C,Ec,R)\kappa_K=\kappa_K(C,E_c,R)K=K(C,Ec,R), and other coefficients are functions of \Theta as specified in Section 2.B--2.C. All parameters are nonnegative; Pmax1P_{\max}\le1Pmax1.
All variables T,E,P[0,1]T,E,P\in[0,1]T,E,P[0,1].
2. Steady-state equations
Set T=E=P=0\dot T=\dot E=\dot P=0T=E=P=0. Denote steady states by T,E,PT^*,E^*,P^*T,E,P. The steady-state system is:
\begin{aligned} 0 &= \alpha_T\,S_E(E^*)\,B_L\,(1-T^*) \;-\; B_P\,P^*, \tag{4a}\\[4pt] 0 &= \mu_E \;-\; \lambda_E\,E^* \;+\; \phi_P\,P^* \;-\; \phi_T\,T^*, \tag{4b}\\[4pt] 0 &= \rho_P\,P^*\Big(1-\dfrac{P^*}{P_{\max}}\Big) \;+\; \eta_E\,S_E(E^*) \;-\; \delta_T\,T^* \;-\; \kappa_K\,K(E^*,\Theta)\,P^*. \tag{4c} \end{aligned}
