The result was a regime with an unusually wide basin of attraction: protest intensity () remained suppressed, and black horse potential () stayed low despite structural inequalities and intermittent crises. Numerical simulations show that for moderate increases in , trust erosion is offset by narrative reinforcement, and equilibrium stability is preserved.
However, the sensitivity analysis also reveals diminishing returns. Narrative's effect on stability is saturating: beyond a certain point, additional narrative control no longer significantly raises . If legitimacy declines---due to perceptions of democratic backsliding or overcentralization---then the system becomes vulnerable to an abrupt shift. In simulations, once trust () falls below a threshold, repression and narrative can no longer compensate, and the system tips into instability. At this point, black horse potential () rises rapidly, reflecting outsider figures gaining traction in the political landscape.
Thus, Jokowi's configuration exemplifies a stable-until-sudden-shift regime: stability is prolonged by strong narrative and elite consensus, but fragility accumulates silently when legitimacy wanes. In formal terms, the system hovers near the plateau of the narrative curve, where resilience no longer grows proportionally, leaving it susceptible to sharp bifurcations once critical thresholds are crossed.
D. Prabowo: low legitimacy, high repression low _crit, unstable under shocks
Prabowo Subianto's current configuration illustrates the vulnerabilities of combining fragile legitimacy with strong elite cooptation and coercive tendencies. Unlike Soeharto, who paired repression with long-term institutional embedding, Prabowo's legitimacy remains contested, deriving more from elite bargains than broad-based democratic mandate. In our formalism, this maps to a low legitimacy parameter, which depresses the stabilizing coefficient and lowers the critical stress threshold .
At the same time, Prabowo exhibits strong elite management and a proclivity toward repression. This places him on the descending slope of the repression curve: moderate repression may contain protests in the short term, but excessive reliance on it generates backlash, amplifying protest intensity () and feeding the black horse potential (). Simulations under Prabowo-like parameters show that even moderate increases in economic stress () can destabilize the system, causing trust () to erode rapidly. Once falls below a critical threshold, protests surge and outsider challengers gain immediate viability.
The consequence is a regime prone to early instability: lies much lower than in the cases of Soeharto, SBY, or Jokowi. Where those leaders could absorb considerable stress before tipping, Prabowo's system risks bifurcation under relatively small shocks---whether economic downturns, policy failures, or sudden legitimacy crises. In this sense, the black horse potential is already elevated ex ante, making the emergence of alternative figures not a rare event but an almost structural feature of the political landscape.
In summary, Prabowo's profile exemplifies a fragile equilibrium: elite control and repression appear to sustain stability, but without legitimacy as ballast, the equilibrium is shallow. Our formalism predicts that this configuration is highly sensitive to perturbations, with black horse dynamics activated at lower stress thresholds than in any of the preceding cases.
VI. Discussion
A. Theoretical implications: leadership as mathematical modulator
Our results demonstrate that leadership qualities are not merely normative or descriptive constructs; they can be formalized as mathematical modulators of systemic stability. By embedding seven leadership parameters into a nonlinear dynamical system, we show that attributes such as legitimacy, narrative capacity, coalition management, and repression function as control coefficients that shift bifurcation thresholds. In this sense, leadership acts as a form of "political elasticity," directly modulating how much exogenous stress a system can absorb before tipping into unrest.
