This integration yields several contributions. First, it enables the derivation of analytical conditions for bifurcation in which leadership parameters shift the critical threshold (_crit) of external shocks required to trigger unrest. For example, higher legitimacy and effective narrative control increase the system's resilience by raising _crit, while overreliance on repression may lower it, making unrest more likely even under moderate shocks.
Second, the formalism allows for comparative simulations across leadership profiles. By calibrating the seven parameters to empirical cases, such as Soeharto, Susilo Bambang Yudhoyono (SBY), Joko Widodo (Jokowi), and Prabowo Subianto, we can illustrate how different governance styles produce distinct bifurcation patterns. This demonstrates not only the explanatory power of the model but also its empirical relevance to real-world political trajectories.
Third, the inclusion of leadership parameters provides a bridge between political science and applied mathematics. It offers a framework in which qualitative assessments of leadership can be systematically translated into quantitative models, facilitating interdisciplinary dialogue and predictive capacity.
Finally, this contribution advances the study of black horse emergence. By explicitly linking leadership failures---such as weak legitimacy or poor elite management---to the growth potential of outsider figures, the model captures the nonlinear interaction between internal governance dynamics and systemic fragility.
In summary, the formalism developed here reframes leadership as a mathematical modulator of political stability. It demonstrates that the fate of regimes under economic or social stress cannot be understood without accounting for how leaders manage legitimacy, elites, narratives, and repression. This synthesis moves the field toward a more comprehensive and predictive science of political unrest.
II. Mathematical Formalism
A. Definition of State Variables: Trust (T), Economic Stress (E), Protest Intensity (P), and Black Horse Potential (H)
To construct a rigorous mathematical model of political unrest, we begin by formalizing the state variables that capture the dynamic evolution of societal stability. These variables are defined on normalized scales, bounded between 0 and 1, to ensure comparability across contexts and facilitate numerical simulation.
1. Trust (T)
Trust represents the degree of public confidence in governing institutions and political leadership. A value of T = 1 indicates complete trust and widespread legitimacy, while T = 0 reflects total erosion of confidence, often preceding systemic collapse. Trust functions as a stabilizing variable, exerting downward pressure on protest intensity and buffering the effects of economic stress. It evolves endogenously through feedback loops with both leadership parameters (e.g., legitimacy, narrative control) and external shocks.
2. Economic Stress (E)
Economic stress captures the level of socioeconomic pressure faced by society, such as inflation, unemployment, or income inequality. Unlike trust, which reflects perception and legitimacy, economic stress reflects material conditions. E = 0 denotes a state of economic ease or prosperity, while E = 1 corresponds to severe economic crisis. This variable is also directly influenced by exogenous shocks, represented by a control parameter (_E), making it a natural driver of bifurcation.
3. Protest Intensity (P)
Protest intensity measures the extent and persistence of collective action against the regime. It is conceptualized as a dynamic population-level response, encompassing demonstrations, strikes, or other forms of political mobilization. P = 0 signifies total quiescence, while P = 1 represents maximal unrest approaching revolutionary levels. Protest intensity grows with economic stress and declining trust, but is counteracted by leadership mechanisms such as consensus-building, repression, and elite cooptation.
4. Black Horse Potential (H)
Black horse potential refers to the latent probability of emergent outsider leaders or alternative elites gaining traction during instability. Unlike the other variables, H is not directly observable at all times but manifests strongly during moments of systemic weakness. H = 0 means no viable outsider figures exist or gain traction, while H = 1 corresponds to a scenario in which alternative leaders become dominant challengers to the regime. The growth of H is stimulated by high protest intensity and low trust, while being suppressed when leadership successfully maintains legitimacy and elite cohesion.
Together, these four variables form the core dynamic system. They are interconnected through nonlinear feedbacks: economic stress undermines trust, low trust amplifies protest intensity, rising protests open opportunities for black horse actors, and the success or failure of these processes depends on the embedded leadership parameters. By treating these variables as dynamical states, we lay the foundation for deriving the formal equations governing systemic stability.
B. Definition of Seven Leadership Parameters and Normalization
To integrate leadership quality into a mathematical model of political unrest, we define seven core parameters. Each parameter represents a distinct dimension of governance and is normalized to a continuous scale between 0 (absence of the quality) and 1 (maximum strength). This normalization facilitates both cross-leader comparison and mathematical embedding within the dynamical system.
