The Stability and Instability of Total Power: A Nonlinear Dynamic Analysis of Power Accumulation and Collapse

This paper models the cyclical rise and fall of total power - an aggregate of political, economic, military, and ideological strength-through the interplay of power accumulation, consumption, and endogenous non-linear feedback effects such as corruption and inefficiency. By framing total power as a dynamic system, we derive an optimal logistic decision rule for power accumulation from an infinite-horizon optimization problem where the state maximizes long-term utility from power and consumption. A key finding is that the state's time preference (discount factor β) intrinsically determines the logistic map's growth parameter (A). Analyzing this logistic rule using bifurcation diagrams, Lyapunov exponents, and the Feigenbaum constant, we demonstrate how decreasing patience (lower β, thus higher A) drives transitions from stable equilibria through period doubling cascades (limit cycles) into chaotic regimes, leading to collapse. Finally, we simulate historical power collapses, including those of the Roman Empire, the Soviet Union, and the Ming Dynasty, showing that all state collapses follow the same universal mathematical path - from order to chaos - driven by shifts in effective growth parameters.