Because all state variables are normalized to [0,1][0,1][0,1], the model must ensure forward invariance of the unit cube. For this we choose coefficient values and saturating functions so that the vector field points inward on the boundaries T=0,1T=0,1T=0,1, E=0,1E=0,1E=0,1, P=0,1P=0,1P=0,1, H=0,1H=0,1H=0,1. In practice this is enforced by (i) logistic form for PPP with carrying capacity Pmax1P_{\max}\le1Pmax1, (ii) linear damping for EEE with sufficient E\lambda_EE, and (iii) multiplicative factors (1T)(1-T)(1T) in trust recovery.
7. Stochastic extension (optional for realism)
To model noise-induced transitions and to conduct ensemble simulations, we consider the stochastic system
dX=F(X;E,)dt+(X)dWt,X=(T,E,P,H),dX = F(X;\mu_E,\Theta)\,dt \;+\; \Sigma(X)\, dW_t, \qquad X=(T,E,P,H)^\top,dX=F(X;E,)dt+(X)dWt,X=(T,E,P,H),
where FFF is the deterministic vector field given above, \Theta denotes the vector of leadership-dependent parameters, WtW_tWt is a standard multi-dimensional Wiener process, and (X)\Sigma(X)(X) is a noise amplitude matrix (often taken diagonal with small entries). Stochastic simulations are used to study noise-induced tipping near critical parameter values Ecrit\mu_E \approx \mu_{\mathrm{crit}}Ecrit.
8. Fixed points and Jacobian (prelude to bifurcation analysis)
Fixed points X=(T,E,P,H)X^*=(T^*,E^*,P^*,H^*)X=(T,E,P,H) satisfy F(X;E,)=0F(X^*;\mu_E,\Theta)=0F(X;E,)=0. The Jacobian matrix J(X;E,)=F/XXJ(X^*;\mu_E,\Theta)=\partial F/\partial X\big|_{X^*}J(X;E,)=F/XX is a 444\times444 matrix whose spectrum determines linear stability. Explicit symbolic expressions for the Jacobian entries are straightforward to compute given the chosen functional forms (logistic, tanh\tanhtanh, Hill functions). For bifurcation analysis we examine how eigenvalues i(E,)\lambda_i(\mu_E,\Theta)i(E,) cross the imaginary axis as E\mu_EE (or other control parameters) vary:
Saddle-node (fold) bifurcation: a real eigenvalue crosses zero and a pair of fixed points collides/annihilates --- typically associated with abrupt tipping in PPP.
Hopf bifurcation: a complex conjugate pair crosses the imaginary axis small amplitude oscillations / protest waves may appear.
Using center manifold reduction and normal form calculations (standard textbook procedures), one can reduce the dynamics near a nonhyperbolic fixed point to canonical scalar forms such as x=r+x2\dot x = r + x^2x=r+x2 (fold) or z=(+i)z+z2z\dot z = (\alpha + i\omega) z + \beta |z|^2 zz=(+i)z+z2z (Hopf). In these reduced coordinates the bifurcation parameter rrr is an explicit function of E\mu_EE and the leadership-dependent coefficients \Theta; hence leadership parameters shift the location and nature of bifurcations.
9. Practical calibration and identifiability remarks
Calibration sources. Leadership parameter values C,L,M,N,G,Ec,RC,L,M,N,G,E_c,RC,L,M,N,G,Ec,R can be informed by expert coding, composite governance indices (e.g., World Bank governance indicators scaled), media sentiment measures, or historical qualitative coding. Coefficients such as G,E,0\lambda_G,\kappa_E,\delta_0G,E,0 are calibrated by fitting observed time series of protest incidence and economic indicators where available, using likelihood methods or Bayesian inference.
Identifiability. The mapping from leadership parameters to coefficients must avoid collinearity that compromises parameter identifiability; regularization and sensitivity analysis are recommended.
10. Summary of the mapping logic
Leadership parameters enter the ODE system by modifying coefficient functions that govern rates of trust recovery/deterioration, strength of cooptation, damping of economic stress, and suppression or facilitation of protest and black-horse emergence.
Analytically, this mapping makes the leadership vector =(C,L,M,N,G,Ec,R)\Theta=(C,L,M,N,G,E_c,R)=(C,L,M,N,G,Ec,R) a secondary control set: while E\mu_EE is the primary exogenous bifurcation parameter, \Theta determines crit()\mu_{\mathrm{crit}}(\Theta)crit(), the critical shock amplitude required to cause bifurcation.
The formalism therefore yields explicit, testable statements of the form:
 crit=(),\mu_{\mathrm{crit}} = \Phi(\Theta),crit=(),
 where \Phi is obtained by solving the fixed-point conditions and locating parameter values where the Jacobian acquires zero or purely imaginary eigenvalues.
This completes the formal presentation of how the seven leadership parameters are integrated into a mathematically precise ODE model for political unrest. The next logical steps (Section 3) are (i) fixed-point computation and Jacobian derivation in closed form for a reduced parametrization, (ii) normal-form reduction to obtain analytic expressions for in leading order, and (iii) numerical continuation and bifurcation diagrams to visualize how different leadership profiles (Soeharto, SBY, Jokowi, Prabowo) shift critical thresholds. If you wish, I will proceed immediately to (i) and (ii) with explicit symbolic derivations for a reduced three-variable core (e.g., ) to make the bifurcation analysis fully explicit.
