NONLINEAR DYNAMICS OF POLITICAL INSTABILITY IN A REDUCED STRUCTURAL-DEMOGRAPHIC MODEL

Political instability often exhibits long-term oscillatory patterns, commonly referred to in Structural-Demographic Theory as secular cycles. Although existing computational approaches, such as the Multi-Path Forecasting framework, can reproduce historical instability trends through high-dimensional simulations, they often lack the analytical tractability needed to clarify the structural mechanisms underlying these oscillations. To address this limitation, we propose a reduced-dimensional continuous-time model consisting of three coupled ordinary differential equations that capture the nonlinear feedback among radicalization, collective violence, and accumulated structural pressure. Using a mean-field approximation for the population reservoir, we derive a minimal dynamical system that preserves the essential structure of the original framework. Stability analysis based on the Routh–Hurwitz criterion shows that equilibrium stability is governed by the rate of exogenous economic decline. We further establish the occurrence of a supercritical Hopf bifurcation, whereby sufficiently severe economic deterioration destabilizes the equilibrium and gives rise to a stable limit cycle representing persistent secular cycles of political violence. Numerical simulations support the analytical results and provide a rigorous mathematical foundation for the endogenous periodicity of social unrest.