SG - model of SSN time series developed in [Kontor (2006, 2008a,2008b)] can be described as follows. The most convenient SSN time series to be considered firstly is the monthly averaged M-data (1749 - present), M(t).
It can be represented as: 
(1) , where "M(t)" is 0 or positive; "~" means some approximation; "U(t)" is the filtered M(t), corresponding to the frequency range [0 - fU:(1/10 months)]; and the last "σM" term is the estimation of the random scattering of M(t) around U(t), which is taken a constant and equal 11.83; the upper index (1) forewarns, that the equation (1) is just the first approximation (for details, see Appendix (A1)).
As for U(t), it can be represented by SG-model as some superposition of gaussians, i.e. G-peaks:
(2), where G U, i - peaks have the close parameters w and distances between peaks (Xc,i+1 - Xc,i), and a considerable variability of amplitudes, H i . U-data analysis shows, that G U,i - peaks are represented by two types of peaks, to wit by G U1- type and G UA- type. G U1- peaks have a small amplitude (H U1 < 20) and so, they correspond to single sunspot groups; G UA- peaks have amplitudes H UA > 20, which are changing in the broad range from ~ 20 to ~ 200.
So, U-data is an alternating sequence of groups, consisting of G U1- peaks or G UA- peaks. Groups of small G U1- peaks (they have the average length about 2.5 years) form the periods of very low solar activity ("microscopic grand minima", mGM) between the Schwabe cycles (S-cycles); in their turn, S-cycles (they have the average length about 8.5 years) are represented by groups of G UA- peaks with the changing amplitudes (see fig. 5 in [Kontor (2006)] ).
Analysis of K-data (K(t)), which is the very smoothed (filtered) M-data (it corresponds to the frequency range [0 - fK:(1/4.17 years)]), allows to find out both the envelope of G UA- peak maxima (H UA,i ) in each S-cycle (K-curve (t)) and the real shape of the every S-cycle (SC i (t)). K(t) can be represented as: 
(3),
where t 0,i is the date of the SC i (t) observed (or predicted) beginning; n (for M-data) = 23; and
(4).
It follows from (4), that a solar cycle (observed in K-data) is superposition of three gaussians (G P, G B, and G D).
Consequently, the S-cycle shape is determined by nine parameters: x P, x B, x D,i (which are counted off t 0,i ), w P, w B, w D (which are approximately equal and so designated by w), and H P,i, H B, H D ( amplitudes of P-, B-, and D- components of S-cycle). In accordance with SG-model, seven from nine free parameters can be considered as approximately constant, and only two parameters (H P,i and x D,i) are noticeably changed from one S-cycle to another. So far as x D,i anticorrelates with H P,i (for details see in Appendix (A2)), it is necessary to find out only the envelope of H P,i (GL-curve or GL(t)), and to get after that the complete description of S-cycle shape (4) and the forecasts for the next S-cycles (#24, #25, and so on).
The mathematical expression for that envelope can be received from analysis of some SSN time series segment, which covers three Gleissberg cycles (GL-cycles). It is the yearly averaged Y-data (1700 - present), and heavy smoothed (filtered) Y-data (E-data or E(t)) corresponding to the frequency range [0 - f E : (1/17.86 years)] (see for details [Kontor, 2006, 2008a]).
(5),
where x0 = 1700 (the beginning of WSN time series); n = 6; G GL,i (t) is some GL-peak; " Δ x i" is the temporal distance between the neighboring GL-peaks.
To receive an idea about variability of the amplitude S-cycle modulation, which is determined by GL(t), it is necessary to consider the SSN time series, which is longer than WSN time series, for example, LCN time series (9455 B.C. - 1995 A.D.). LCN-data is close to E-data because they have close Nyquist frequencies. Consideration of LCN-data from the point of SG-model's view shows, that three types of activity are realizing on the Sun: grand minima, when only single sunspot groups ( G U1- peaks ) are observed for the periods ~ hundred years (GMA-type); the low S-cycles, in which the stable B-component (4) dominates and the role of GL-modulation (5) of the P-component amplitude (H P,i) is inessential (BDA-type); and, at last, the high S-cycles, for which an amplitude is determined by GL-modulation (similar to (5), see for details [Kontor, 2006, 2008a, 2008b]). It is GLA-type of solar activity. Therefore any SSN time series (WSN time series, LCN time series and others) can be expressed as:
(6),
where SAE i (t) is a segment of SSN time series, which starts from GM i (t 0,i - the date of GM i beginning); after that, some low S-cycles (BDA), and a number of high S-cycles (GLA) are observed. SAE i (t) is finished when the next GM i+1 starts.
For example, WSN time series begins from 1610 A.D., but the new (and still current) SAE starts at 1645, when the Maunder Minimum begins (see fig. 1 in Kontor (2006)). It is followed by two low S-cycles (# -4, -3) and after that three GL-cycles are observed (the low S-cycles are also observed on "the GL-cycles junctions" (cycles # 5, 6 ; #14, 16, and, probably, # 24 , 25 ). So, the current SAE is not finished yet.
N.N.Kontor, e-mail: sgnnk@Live.com
published: 11/27/2008