Bifurcation-Based Predictive Modelling of Socio-Political Dynamics

A Hill/sigmoid function to represent how economic stress drives unrest:

SE(E;,)=EE+(>0, 1)S_E(E;\theta,\kappa) \;=\; \frac{E^\kappa}{E^\kappa + \theta^\kappa} \qquad(\theta>0,\ \kappa\ge1)

Saturating policy responses:

U(t)=Umaxtanh(U(wPP+wEE+w0))U(t)=U_{\max}\,\tanh\!\big(\rho_U \cdot (w_P P + w_E E + w_0)\big) K(t)=Kmaxtanh(K(vPPvTT+v0))\mathcal{K}(t)=\mathcal{K}_{\max}\,\tanh\!\big(\rho_{\mathcal{K}}\cdot (v_P P - v_T T + v_0)\big)

These ensure U,K(cap,cap)U,\mathcal{K}\in(-\text{cap},\text{cap}) and capture responsiveness \rho.

2) Core dynamical equations

(Core Equation (Source: Privat Document)

dPdt=SE(E;E,)(1T)+HH(TT+KK+P)P+P(t)

dTdt=UU(t)KK(t)PPEE+0TT+T(t)

dEdt=0(t)+PPUU(t)EE+E(t)

dHdt=PP+EETTHH+H(t)