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

\delta_T &= \delta_0 + \delta_L\, L + \delta_N\, N,\\

\kappa_K  &= \kappa_0 + \kappa_{E}\, E_c + \kappa_C\, C - \kappa_R\,(1-R),\\

\lambda_E &= \lambda_0 + \lambda_G\, G + \lambda_M\, M,\\

\alpha_T &= \alpha_0\,(L + \sigma_N N + \sigma_M M),\\

\gamma_T &= \gamma_0\,(1-R),\\

\lambda_H &= \lambda_{H0}\, E_c.

\end{aligned}

These mappings ensure:

Higher generally increase stabilizing coefficients (trust recovery, economic damping, cooptation efficiency), thereby shifting the system toward resilience.

Lower (higher repression) increases short-term suppression but increases (backfire) which erodes faster given protests.

6. Boundedness and invariance