Leadership Parameters and Bifurcation of Political Unrest: a Mathematical Formalism with Cases Study

\dot T &= \; \underbrace{\alpha_T \, S_E(E;\theta_E,\kappa_E)\,(L + \sigma_N N + \sigma_M M)\,(1-T)}_{\text{trust recovery via economy, narrative, crisis mgmt}} 

\;-\; \underbrace{\beta_T\, P}_{\text{trust erosion by protest}} 

\;-\; \underbrace{\gamma_T(1-R)\,P}_{\text{repression backfire term}} \\

\dot E &= \; \mu_E \;-\; \underbrace{\lambda_E(G,M)\,E}_{\text{damping by economic policy \& crisis mgmt}}

\;+\; \underbrace{\phi_P\, P}_{\text{protest worsens economy}} \\

\dot P &= \; \underbrace{\rho_P\,P\left(1-\dfrac{P}{P_{\max}}\right)}_{\text{intrinsic mobilization}} 

\;+\; \underbrace{\eta_E\, S_E(E;\theta_E,\kappa_E)}_{\text{economic grievance channel}} 

\;-\; \underbrace{\delta_T\, T}_{\text{trust dampening}} 

\;-\; \underbrace{\kappa_K\,K(C,E_c,R)\,P}_{\text{elite cooptation / suppression}} \\

\dot H &= \; \underbrace{\alpha_H\,P}_{\text{opportunity from protest}} \;+\; \underbrace{\beta_H\,E}_{\text{economic opening}} \;-\; \underbrace{\gamma_H\,T}_{\text{trust suppression}} \;-\; \underbrace{\lambda_H\,E_c}_{\text{elite containment}}.

\end{aligned}