Observation-based Iterative Map for Solar Cycles. II. The Gnevyshev–Ohl Rule and its Generation Mechanism

Abstract The Gnevyshev–Ohl (G-O) rule, or even-odd effect, is an important observational phenomenon in solar cycles, originally suggesting that even-numbered cycles are typically followed by stronger odd-numbered ones. However, subsequent studies have reported varied forms and often conflicting manifestations of this rule, along with diverse interpretations of its physical origin. Using an observation-based iterative map, we investigate these different forms of the G-O rule and propose a more general underlying rule: statistically, a given solar cycle is more likely to be followed by a stronger one, regardless of even-odd numbering. This transition asymmetry arises from the map’s inherent asymmetry relative to the diagonal. Over timescales comparable to historical observations, both the G-O rule and its reversal can arise randomly, without a consistent preference. The short-term behavior of the rule is sensitive to the initial cycle, the selected time interval, and the parameters of the recursion function. These findings reconcile previously conflicting reports and point to a general generation mechanism: G-O-like behavior arises naturally from nonlinear stochastic dynamics. While different recursion parameters may lead to varying short-term patterns and statistical strengths, the emergence of G-O-like features appears to be a generic property of such systems.