State variables (all normalized to the unit interval):
T(t)[0,1]T(t)\in[0,1]T(t)[0,1] --- public trust / legitimacy,
E(t)[0,1]E(t)\in[0,1]E(t)[0,1] --- economic stress (0 = none, 1 = extreme stress),
P(t)[0,1]P(t)\in[0,1]P(t)[0,1] --- protest intensity (0 = quiescence, 1 = maximal unrest),
H(t)[0,1]H(t)\in[0,1]H(t)[0,1] --- black-horse emergence potential.
Leadership parameters (normalized to [0,1][0,1][0,1]; Section 2.B):
CCC --- consensus-building capacity,
LLL --- legitimacy (public trust baseline),
MMM --- crisis management effectiveness,
NNN --- narrative/communication control,
GGG --- economic governance effectiveness,
EcE_cEc --- elite management / coalition maintenance,
RRR --- repression vs. consensus balance (higher = more consensus, lower = more repression).
External control parameter:
ER0\mu_E\in\mathbb{R}_{\ge0}ER0 --- exogenous economic shock magnitude (driver of EEE); treated as a bifurcation parameter in bifurcation diagrams.
We introduce short-hand mappings from leadership parameters to model coefficients. The mapping is chosen to be monotonic and dimensionless; it is sufficiently general to be adapted or re-calibrated for empirical work.
2. Model structure --- core assumptions
1. Trust growth/decay. Trust increases via economic recovery and successful narrative/capacity to manage crises, and decreases with protest intensity and perceived repression/backfire. Leadership parameters L,N,M,GL,N,M,GL,N,M,G act to bolster trust; repression RRR modulates backfire effects.
2. Economic stress dynamics. Economic stress is driven by exogenous shocks E\mu_EE and endogenous feedback from protests (disruption of economic activity) and poor crisis management. Good economic governance GGG and effective crisis management MMM speed damping of EEE.
3. Protest dynamics. Protest follows logistic-style growth (capacity-limited mobilization) and is positively driven by economic stress and lowered trust; it is suppressed by elite cooptation KKK and by legitimacy LLL and narrative NNN. Repression (low RRR) can suppress protests short-term but produces legitimacy loss and thus can increase protest indirectly (backfire).
4. Black-horse dynamics. HHH grows when protests are sustained and trust is low; elite fragmentation (low EcE_cEc) and poor consensus capacity CCC raise the growth potential for outsider leaders. Strong elite management EcE_cEc and high LLL suppress HHH.
5. Saturation and nonlinearity. Many processes saturate (limited population to mobilize, limited trust ceiling). Sigmoidal or saturating functions (e.g., logistic, Hill functions, tanh\tanhtanh) are used to capture these effects.
6. Optional stochastic forcing. To capture real-world idiosyncratic shocks and noise-induced tipping, we allow for additive white noise terms in each equation; the deterministic skeleton below is our primary object for bifurcation analysis.
3. Explicit ODE system (deterministic skeleton)
We propose the following system (all time derivatives are ):
\boxed{%
\begin{aligned}
