\frac{\partial \dot{A_p}}{\partial A_r} & \frac{\partial \dot{A_p}}{\partial A_p} & \frac{\partial \dot{A_p}}{\partial D} & \frac{\partial \dot{A_p}}{\partial S} \\
\frac{\partial \dot{D}}{\partial A_r} & \frac{\partial \dot{D}}{\partial A_p} & \frac{\partial \dot{D}}{\partial D} & \frac{\partial \dot{D}}{\partial S} \\
\frac{\partial \dot{S}}{\partial A_r} & \frac{\partial \dot{S}}{\partial A_p} & \frac{\partial \dot{S}}{\partial D} & \frac{\partial \dot{S}}{\partial S}
\end{bmatrix}
Bifurcation points occur when the determinant of the Jacobian approaches zero or eigenvalues cross the imaginary axis ( i=0\lambda_i = 0i=0 or (i)=0\Re(\lambda_i) = 0(i)=0), ), signaling a critical threshold:
\det(J)|_\text{critical} = 0 \quad \text{or} \quad \Re(\lambda_\text{max}) = 0
Interpretation for TDD:
When monetary incentives U increase relative to perceived justice J beyond a critical threshold, the system may bifurcate toward authoritarian drift (S>0).).
When perceived justice J rises relative to U beyond a critical threshold, the system may bifurcate toward anarchy or unrest ((S<0S^* < 0S<0).
Thus, the model identifies sensitive parameters and non-linear thresholds that govern political stability, providing predictive insight into conditions under which transactional politics escalates into systemic democratic degradation.
D. Numerical Simulation: Scenarios of Varying Incentives and Justice Levels