The coupled differential equations in Section IV.B describe the dynamic interactions between voter rationalization (Ar), legislative behavior (Ap), democratic degradation (D), and system stability (S). To understand the long-term behavior of the system, we analyze equilibrium points and the conditions under which bifurcations---critical transitions in political stability---occur.
1. Equilibrium Points
Equilibrium occurs when the rates of change for all variables are zero:
\frac{dA_r}{dt} = 0, \quad \frac{dA_p}{dt} = 0, \quad \frac{dD}{dt} = 0, \quad \frac{dS}{dt} = 0
Substituting the dynamic equations from Section IV.B:
1. Voter Rationalization:
\alpha_1 U - \alpha_2 J + \alpha_3 C = 0 \quad \Rightarrow \quad A_r^* = f(U,J,C) = \frac{\alpha_2 J - \alpha_3 C}{\alpha_1}
2. Legislative Behavior:
\beta_1 A_r + \beta_2 ROI - \beta_3 J = 0 \quad \Rightarrow \quad A_p^* = \frac{\beta_3 J - \beta_2 ROI}{\beta_1}
3. Democratic Degradation:
\gamma_1 A_r + \gamma_2 A_p + \gamma_3 S_\text{meso} - \gamma_4 J = 0 \quad \Rightarrow \quad D^* = \frac{\gamma_1 A_r^* + \gamma_2 A_p^* + \gamma_3 S_\text{meso} - \gamma_4 J}{1}
