is a key bifurcation parameter: large rr (fast/high repression relative to relief) tends to produce a regime with persistent PP and low TT; small rr tends to favour stabilization.
E\theta_E and \kappa in SES_E: determine how sensitive unrest is to economic stress (sharp thresholds for larger \kappa).
K\gamma_{\mathcal{K}} vs K\eta_{\mathcal{K}}: balance between repression's immediate suppression of protests and its erosion of trust.
P,E,T\alpha_P,\alpha_E,\beta_T: drive the growth of HH and hence the emergence probability of a black horse.
4) Equilibria and local stability (qualitative sketch)
Let x=(P,T,E,H,R)x^\ast=(P^\ast,T^\ast,E^\ast,H^\ast,R^\ast) be a fixed point solving the algebraic system where all time derivatives vanish. Local stability is governed by the Jacobian matrix J=xF(x)J=\nabla_x F(x^\ast). Bifurcations occur when eigenvalues of JJ cross the imaginary axis:
Saddle-node bifurcation: occurs when two equilibria (stable + unstable) collide and annihilate --- can explain sudden loss of a peaceful equilibrium. Condition: determinant criteria and transversality conditions (standard in bifurcation theory).
Hopf bifurcation: if a complex conjugate pair of eigenvalues crosses the imaginary axis, the system may develop sustained oscillations (cyclical unrest). This is relevant if feedback loops produce delayed over-compensation (e.g., strong periodic repression waves of protest).
Given the model's nonlinearities (sigmoids, tanh), both types are plausible as parameters (notably rr, \alpha, E\theta_E) are varied.
5) Stochastic dynamics and escape probabilities
Because i(t)\xi_i(t) are nonzero, the deterministic bifurcation picture must be augmented by stochastic theory:
Near a deterministic bifurcation, finite-amplitude noise can induce early transitions (noise-induced tipping). One can compute mean first-passage times for H(t)H(t) to cross HcH_c or P(t)P(t) to exceed a societal threshold PcP_c.
Large deviation theory / Freidlin--Wentzell formalism can be used to estimate exponential scaling of escape probabilities from basins of attraction as noise amplitude varies.
6) Operationalization and calibration
To make the model predictive:
1. Data mapping --- map observable indicators to state variables:
T(t)T(t): composite index from opinion polls, trust surveys, media sentiment (scaled 0--1).
E(t)E(t): CPI food inflation, unemployment rate, fuel price index, and commodity price shocks normalized.
P(t)P(t): event counts, protest size estimates, police reports, social-media mobilization metrics (virality, hashtag volume).
H(t)H(t): social media attention to alternative leaders, follower growth rates, fundraising metrics.
U(t),K(t)U(t),\mathcal{K}(t): budgeted relief amounts (per day) and recorded operational security intensity (number of deployments, curfews).
