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

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Each term and coefficient is described in detail below.

4. Functional forms and coefficient mappings

(a) Economic trust / grievance coupling

We use a smooth saturating function for the economic channel:

S_E(E;\theta_E,\kappa_E) \;=\; \dfrac{E^{\kappa_E}}{E^{\kappa_E} + \theta_E^{\kappa_E}},

\qquad \theta_E\in(0,1),\ \kappa_E>0.

is close to 0 for small and approaches 1 for large ; it is conveniently differentiable for Jacobian calculation.

(b) Elite cooptation function .
Elite cooptation reduces protest pressure by absorbing potential elite challengers and tying resources to the incumbent. We propose a bounded saturating functional form:

K(C,E_c,R) \;=\; K_{\max}\,\tanh\big(\rho_K\,[w_C\,C + w_{E}\,E_c + w_R\,R]\big),

\qquad K_{\max}>0,\ \rho_K>0,