Bifurcation-Based Predictive Modelling of Socio-Political Dynamics

dRdt=TT+UU(t)PPEERR+R(t)\boxed{\begin{aligned} 

\frac{dP}{dt} &= \alpha\,S_E(E;\theta_E,\kappa)\, (1-T)\;+\;\beta_H H \;-\; \big(\gamma_T T + \gamma_{\mathcal{K}}\mathcal{K} + \delta_P\big)\,P \;+\; \xi_P(t)\\[6pt] 

\frac{dT}{dt} &= \eta_U\,U(t)\;-\;\eta_{\mathcal{K}}\,\mathcal{K}(t)\;-\;\eta_P\,P\;-\;\eta_E\,E \;+\; \eta_0 \;-\; \lambda_T\,T \;+\; \xi_T(t)\\[6pt] 

\frac{dE}{dt} &= \phi_0(t)\;+\;\phi_P\,P \;-\; \phi_U\,U(t) \;-\; \mu_E\,E \;+\; \xi_E(t)\\[6pt] 

\frac{dH}{dt} &= \alpha_P\,P \;+\; \alpha_E\,E \;-\; \beta_T\,T \;-\; \gamma_H\,H \;+\; \xi_H(t)\\[6pt] 

\frac{dR}{dt} &= \rho_T\,T \;+\; \rho_U\,U(t)\;-\;\rho_P\,P \;-\;\rho_E\,E \;-\;\lambda_R\,R \;+\; \xi_R(t) \end{aligned}}

Short interpretation of terms

SE(E)(1T)\alpha\,S_E(E)(1-T): baseline mechanism --- economic stress converts into unrest if trust is low; SES_E provides a smooth threshold.
HH\beta_H H: latent black-horse mobilization amplifies protest.
(TT+KK+P)P(\gamma_T T + \gamma_{\mathcal{K}}\mathcal{K} + \delta_P)P: stabilizing decay of protest due to trust, coercion and natural dissipation. Note coercion has two effects: directly reducing PP (through K\gamma_{\mathcal{K}}) but elsewhere it reduces TT (see second equation) and so can be destabilizing indirectly.
dT/dtdT/dt: trust grows with accommodative policy UU, is eroded by coercive policy K\mathcal{K}, by visible protests PP, and by economic stress EE; 0\eta_0 is baseline institutional competence.
dE/dtdE/dt: economic stress evolves by exogenous shocks 0(t)\phi_0(t), is worsened by protests (supply shocks/panic) and reduced by effective relief UU.
dH/dtdH/dt: black-horse potential grows with social turbulence (P, E) and declines with trust and intrinsic decay. When HH crosses a threshold HcH_c, an emergent political actor becomes salient.
dR/dtdR/dt: resilience is boosted by trust and effective policy, eroded by unrest and economic stress.
3) Control / bifurcation parameters

A small set of parameters typically play an outsized role in bifurcation behavior:

K\rho_{\mathcal{K}} and U\rho_U: responsiveness of coercive vs accommodative policy. The ratio

r=KUr \;=\; \frac{\rho_{\mathcal{K}}}{\rho_U}