a. Compute Ai(E)A_i(E)Ai(E) symbolic expressions from definitions in Section 3 (requires SE(E)S_E(E)SE(E), K(E)K(E)K(E), and mapping of leadership parameters to coefficients).
b. Solve discriminant equation (10): A1(E)24A2(E)A0(E)=0A_1(E)^2 - 4 A_2(E) A_0(E) = 0A1(E)24A2(E)A0(E)=0 for EE^*E in (0,1)(0,1)(0,1). Each real root EE^*E is a candidate fold height.
c. For each candidate EE^*E, compute P=A1(E)/(2A2(E))P^* = -A_1(E^*)/(2 A_2(E^*))P=A1(E)/(2A2(E)) (must be 0\ge00 and Pmax\le P_{\max}Pmax). Then compute TT^*T from (5) and finally compute E\mu_EE via (6). That E\mu_EE is crit()\mu_{\mathrm{crit}}(\Theta)crit() associated with that fold.
d. Check Jacobian at (T,E,P)(T^*,E^*,P^*)(T,E,P): compute JJJ in (11). Verify zero eigenvalue (within tolerance) and check that nullspace has dimension 1 (simple eigenvalue). Compute the nondegeneracy coefficients aaa and bbb as in Section 2.D to ensure classical saddle-node.
e. If symbolic closed forms are desired: compute resultant eliminating PP^*P between (8) and its derivative (9b) to obtain a polynomial in EE^*E alone (this is exactly the discriminant), then solve symbolically if possible (SymPy). Otherwise solve numerically with reliable root solvers and continuation.
8. Additional remarks (interpretation & practicalities)
The reduction to a quadratic (8) is central: it demonstrates why fold bifurcations are generic here --- the protest equation includes a logistic (quadratic) nonlinearity and linear couplings to TTT and EEE, so multiplicity of roots (coexisting low- and high-PPP equilibria) is expected. The leadership parameters enter A1,A0A_1,A_0A1,A0 through KKK, SES_ESE, T\delta_TT, etc., and thereby modulate the discriminant (E)\Delta(E)(E). In particular, increasing legitimacy LLL generally increases BLB_LBL, decreasing the TBP/(TBLSE) \delta_T B_P /(\alpha_T B_L S_E)TBP/(TBLSE) term and thereby shifting the discriminant to favor single-root (more stable) regimes.
When KKK does not depend on EEE (pure leadership control), A1A_1A1 simplifies and the discriminant becomes easier; if KKK depends on EEE (e.g., elite cooptation weakens with economic stress), that coupling can create richer algebraic structure (multiple fold points).
For the full 4D augmented model (including GGG), a similar elimination procedure can be attempted but will produce cubic or quartic algebraic relations requiring higher-order resultants. Practically, use continuation tools to track folds and Hopf curves in parameter space.
B Jacobian derivation and eigenvalue conditions
1. Notation and setup
We consider the reduced dynamical system (restating for convenience)
T=F1(T,E,P;,E),E=F2(T,E,P;,E),P=F3(T,E,P;,E),\begin{aligned} \dot T &= F_1(T,E,P;\Theta,\mu_E),\\ \dot E &= F_2(T,E,P;\Theta,\mu_E),\\ \dot P &= F_3(T,E,P;\Theta,\mu_E), \end{aligned}TEP=F1(T,E,P;,E),=F2(T,E,P;,E),=F3(T,E,P;,E),
where \Theta denotes the vector of seven leadership parameters (Consensus CCC, Legitimacy LLL, Crisis management MMM, Narrative NNN, Economic stability ESESES, Elite coordination EMEMEM, Repression vs consensus RRR) and E\mu_EE is the exogenous economic stress (control parameter). Let X=(T,E,P)X=(T,E,P)^\topX=(T,E,P) and write F(X;,E)F(X;\Theta,\mu_E)F(X;,E) for the vector field.
An equilibrium satisfies F(X;,E)=0F(X^*;\Theta,\mu_E)=0F(X;,E)=0. Local linear stability is governed by the Jacobian matrix evaluated at the equilibrium:
J(X;,E)=DXF(X;,E)(the 33 matrix of partial derivatives).J(X^*;\Theta,\mu_E) \;=\; D_X F(X^*;\Theta,\mu_E) \qquad\text{(the 33 matrix of partial derivatives).}J(X;,E)=DXF(X;,E)(the 33 matrix of partial derivatives).
Below I give explicit expressions for JJJ in terms of the model functions used in Section 2/4.A, then derive eigenvalue conditions and the bifurcation criteria.
2. Jacobian --- explicit partial derivatives
