Two Regimes in the Regularity of Sunspot Number

Astrophysical Journal. 2013. Vol. 779. P. 108–116.

Shapoval Alexander, Le Mouel J., Shnirman M. G., Courtillot V.

Sunspot number WN displays quasi-periodical variations that undergo regime changes. These irregularities could indicate a chaotic system and be measured by Lyapunov exponents. We define a functional l (an “irregularity index”) that is close to the (maximal) Lyapunov exponent for dynamical systems and well defined for series with a random component: this allows one to work with sunspot numbers. We compute l for the daily WN from 1850 to 2012 within 4-year sliding windows: l exhibit sharp maxima at solar minima and secondary maxima at solar maxima. This pattern is reflected in the ratio R of the amplitudes of the main vs secondary peaks. Two regimes have alternated in the past 150 years, R1 from 1850 to 1915 (large l and R values) and R2 from 1935 to 2005 (shrinking difference between main and secondary maxima, R values between 1 and 2). We build an autoregressive model consisting of Poisson noise plus an 11-yr cycle, and compute its irregularity index. The transition from R1 to R2 can be reproduced by strengthening the autocorrelation a of the model series. The features of the two regimes are stable for model and WN with respect to embedding dimension and delay. Near the time of the last solar minimum (~2008), the irregularity index exhibits a peak similar to the peaks observed before 1915. This might signal a regime change back from R2 to R1 and the onset of a significant decrease of solar activity.

Keywords: chaos, solar cycles, irregularity index, Lyapunov exponent