Filaments-nets Structure of the Phase Space of Coin Tossing Mechanism for Sensitivity and Complexity - Yue, Zengyuan (Discontinuity, Nonlinearity, and Complexity, 2026)
Detailed analysis of trajectories reveals a filaments-nets structure of the phase space of coin tossing, leading to a simple and unified explanation for the extremely sensitive dependence of the outcome, head or tail, on the initial state, for the extremely complex geometry of the cross sections of basins of attraction for heads and tails, and for the big difference between the transitional region and the “completely random region”. A “GDGC” (Great Differentiation & Great Combination) condition is proposed for the stability of statistical regularity, which can also be summarized by the following “Compensation Principle”: The more sensitive, i.e. the more unstable, the deterministic process is, the more stable, i.e. the more insensitive, the associated statistical regularity would be.