We will use  from the three ways of expressing dates  mainly "the mathematical" and "the historical" way.
historical astronomical mathematical  1.4.2005 AD 2005(.25) 2005.25 1.4. 1 AD 1(.25) 1.25 1.4. 1 BC 0(.25) 0.25 1.4. 2 BC 1(.25) 0.75 1.4.2005 BC 2004(.25) 2003.75
Basic orbital periods of bodies are given with regard to stars. These are so called sidereal periods.
In the next computations we will use mean periods according to VSOP87 (Bretagnon, Variations Seculaires des Orbites Planetaires).
Outer planets 
Inner planets 
J 11.8620 years ( 4332.59 days) S 29.457159 years (10759.23 days) U 84.020473 years (30688.48 days) N 164.770132 years (60182.29 days) 
M 0.2408467 years ( 87.96926 days) V 0.6151973 years (224.70080 days) E 1.0000174 years (365.25636 days) R 1.8808480 years (686.97973 days) 
Conjunction is a close apparition (join, alignment) of two or more bodies. For simplicity we will consider only alignment of bodies in plane perpendicular to approximate plane of planetary motion (i.e. conjunction in longitude). Exact alignments in one line are sporadic and they are studied in connection with other phenomena (eclipses of Sun and Moon, transits of Mercury and Venus through solar disc…)
If bodies are observed from the Sun, we speak about
heliocentric conjunction (conjunction with the Sun).
E.g. if bodies ordered SunVenusEarthMars are in a straight line,
we say VenusEarth, EarthMars, VenusMars and VenusEarthMars are
in conjunction (seen from the Sun).
Practical astronomy understands by conjunction usually geocentric conjunction. If SunVenusEarthMars are aligned, it is said, Venus is in conjunction and Mars in opposition (with the Sun, seen from the Earth).
Mean period, with which (helio)centric conjunctions of two
bodies repeat, is called synodic period.
Synodic (relative) period of two periods P,Q is period:
We designate synodic period with round brackets ().
For any periods A,B and constant k it holds:
(A,B) = (B,A) (k∙A,k∙B) = k∙(A,B) ((A,M),(B,M)) = (A,B).
In practical astronomy it is implied, that one of the period is
orbital period of the Earth. E.g. synodic period of Jupiter is
determined to be c. 399 days. It is period of Jupiter with regard
to Earth: (E,J) = (365.256,4332.59) = 398.9 days.
Se synodickou period se postupně rozvírá (a pak zase přivírá)
úhel PSQ; S is centrum, bod (center of gravity systems) okolo kterého
tělesa P,Q obíhají.
With synodic period the angle PSQ gradually opens (and then
closes); S is the point (centre), around which motion of bodies P,Q
happens.
Outer planets 
Inner planets 
(J,N)= 12.7822 years ( 4668.69 days) (J,U)= 13.8120 years ( 5044.81 days) (J,S)= 19.8589 years ( 7253.45 days) (S,N)= 35.8699 years (13101.47 days) (S,U)= 45.3602 years (16567.82 days) (U,N)=171.4443 years (62620.01 days) 
(M,R)= 0.276217 years (100.8882 days) (M,E)= 0.317255 years (115.8775 days) (M,V)= 0.395801 years (144.5662 days) (V,R)= 0.914227 years (333.9215 days) (V,E)= 1.598690 years (583.9214 days) (E,R)= 2.135349 years (779.9361 days) 
Synodic day is rotational period Tr measured with regard to orbital period T, i.e. synodic period (Tr,T).
In case of planet of the Sun we speak about “solar day”, e.g. if Earth as rotational period Tr =1 stellar (sidereal) day, is its solar day equal to (1.0, 365.256) = 1.0027 stellar days. We divide solar day to 24 hours.
Let us have a number of small periods Pi: (P0, P1,..., Pn),
and greater period Q. If periods Pi have common multiple P, which
is approximately equal to Q, it could appear, all the system has
period P.
But evidently it is not true. Deviations of periods (PQ) will
gradually accumulate; during synodic period (P,Q).
Such accumulation of deviations can appear as transformation of cycle P, as change of its polarity.
E.g. periods 3,4 and 13 years makes common multiple c. 1213 years. It can be well approximated by period P=12 years. But after a longer time we will register beats with period c. (12,13)=156 years.
In solar system periods (U,N)= 171.44 years and 9∙(J,S)=178.730 differ by more than 7 years. Though they are usually covered by so called 180years period. Deviations of periods oscilates with period c. (9∙(J,S),(U,N))=(178.7, 171.4)=4200 years.
I.Charvatova has found period c. 4400 years (resp. its aliquots 2200 years and 1100 years) in motion of the Solar system gravity centre. According to I.Charvatova is the basic interval made of c. 55 conjunctions (J,S), i.e. 1100 years. Observed deviations of motional characteristics are in turns positive and negative (intervals in years): (2200,1100) +; (1100,0) ; (0,+1100) +; (+1100,+2200) .
Cycle of great conjunctionsMentioned definition of conjunctions is not physically correct, it is only geometrical construction. We neglect finite speed of light as well as timespace relations and so on.
E.g. to compare of conjunctions VE (period 1.6 years) and conjunctions JN (period 12.8 years), we have to consider space distance about 29 AU. (Light run this distance c. 4.3∙10^{12} m during c. 14300 seconds, i.e. c. 4 hours…).
Fouryears period was found in a number of biological and economic phenomena. It was used by ancient Greeks (the Olympics period) as well as by Mayan astronomers (cycle of 4 naming days). According to Wood, there is cycle of Moon tides of length 4.001 years.
By „internal cycle” we mean cycle of inner planets with period I=6.4 years.
Resonance 3:4:7 of synodical periods Venus, Earth and Mars:
Conjunctions VenusEarthMars repeats approximately every 6.4 years. During 4 conjunctions VE, 3 conjunctions ER appear.
This motion repeats regularly like clockwork.
VE ER   1886.140 1886.181 1887.728 1888.281 1889.335 1890.406 1890.929
VE ER   1892.528 1892.593 1894.132 1894.806 1895.720 1896.949 1897.330
VE ER   1898.923 1899.055 1900.520 1901.147 1902.127 1903.241 1903.715
VE ER   1905.322 1905.352 .... ....
If conjunctions could be heard, the sound would remind a drummer that play 3 triplets during 4 beats (beats are conjunctions VenusEarth).
(E,R)/(V,E)= 779.9361 days/583.9214 days= 2.1353487 years/1.5986896 years= 1.335687.
VE ER   2039.6081 2040.0078 2041.2152 2042.1051 2042.8032 2044.1968 2044.4104   2046.0066 2046.2968 2047.6000 2048.4269 2049.2099 2050.6200 2050.7979   2052.4023 2052.8267 ...
After c. 150 years regularity of conjunctions VER quite disappear (see outline on the right):
Entire cycle repeats with period c. 300 years.
Ratio (E,R)/(V,E) is not 4/3 (192/144), but rather 191/143. After c. 300 years one conjunction VE and one ER "drops out": 191∙ 1.5986896 years = 305.350 years, 143∙ 2.1353487 years = 305.355 years
So: (E,R)/(V,E) = (4∙481)/(3∙481); 48∙((V,E),(E,R)) = 48∙ 6.361133 years = 305.334 years
Mayan evaluate period: 140*(E,R)= 140*780 days = 109200 days
(299 years=23*13 years), or 187*(V,E).
In this case: (E,R)/(V,E) = (4*471)/(3*471).
Likewise: (4∙(E,R))/((V,E),(E,R)) = (4∙121)/(3∙121) or R/Ey=(2∙541)/54=107/54, see Ecliptic year.
Instants, when configuration EarthVenusMars makes a line, repeat also (like conjunctions) approximately with period 6.4 years. Besides, another structure appears here:
Mars stays four times (three periods R) at the same place.
Later (after 6.4 years from the beginning) the same pattern starts again
(phase shift F, c. 250300 days).
Let: 3*R + F = 6.4 y. Then F = 6.4  3*1.880848 = 0.757 years = 276.7 days.
Date (Interval) Time  1885 Mar 29 (1.88364) 1885.24820 1887 Feb 9 (1.86721) 1887.11545 1889 Jan 3 (1.90007) 1889.01556 1889 Oct 1 (0.74196) 1889.75754 1891 Aug 23 (1.89185) 1891.64943 1893 Jul 8 (1.87543) 1893.52490 1895 Jun 5 (1.90828) 1895.43322 1896 Feb 10 (0.68446) 1896.11770 1897 Dec 29 (1.88364) 1898.00138 1899 Nov 17 (1.88364) 1899.88506 1901 Oct 9 (1.89185) 1901.77696 1902 Jul 31 (0.80767) 1902.58464 1904 Jun 18 (1.88364) 1904.46832 ...
Every c. 300 years one period R "vanishes". Complete cycle 6.4 y vanishes after c. 1020 y (300 y*6.4/1.88)?
Period 13*18*20 days = 4680 days (12.81 years).
13 Mayan tuns = Double period of conjunctions VER:
Because frac(6.4/1.88)= 2/5, observer of Mars with period 6.4 years realizes, that Mars makes pentagon like as conjunctions VE, frac(1.6/1.0) = 3/5. Oppositions VE, Mars at perihelion:
Rozdíl Datum Lv Le Lr LvLe  ( 31.98) 1715.86: 47.6 228.2 326.0 180.6 ( 31.98) 1747.84: 38.7 219.9 326.6 181.1 ( 31.98) 1779.81: 29.9 211.5 327.2 181.6 ( 31.98) 1811.79: 21.2 203.2 327.8 182.0 ( 31.97) 1843.76: 12.1 194.2 328.2 182.1 ( 31.98) 1875.74: 3.5 185.8 328.8 182.3 ( 31.98) 1907.72: 354.9 177.4 329.4 177.5 ( 31.98) 1939.69: 346.3 168.9 330.0 177.4 ( 31.98) 1971.67: 337.8 160.5 330.6 177.3
Conjunctions VE repeat their position relative to Earth every 8 years and conjunctions VER occur with a period 6.4 years. Thus conjunction VER occurs approximately at the same location relative to the Earth after 32 years: 32 years = 3∙10.67 years = 4∙8.00 years = 5∙6.40 years.
Seen from Mars: Venus moves forwards with period (V,R/3)=32.819643 y, meanwhile Earth moves backwards with period (E,R/2)= 15.780949 y.
Resultant motion makes period: [32.819643, 15.780949] =
10.656765 years. Five periods (E,R): 5*(E,R) = 10.676744 years.
Equality of these periods gives unstable resonance:
5/V  6/E  4/R = 0 (1139 years)
Note: modulation (10.656765, 302.4347) = 11.046 years.
Mayan period. 13 Mayan katuns make 256.3 years.
Triple conjunctions JS (observed from the Earth) returns after 257258 years; e.g. 333, 411, 452, 710, 967, 1008, 1306, 1425 AD.
Integer ratios of pairs MercuryEarth (E/M=4/1) and VenusMars (R/V=3/1) lead to beats
:Lx+45  1909,62  1922,79  1935,95  1949,12  1962,28  1975,45  1988,62  2001,78  2014,95  
0  0  29,2  353,3  0,3  16,6  333,2  347,2  26,8  341,0  352,3 
+1,88  0  2,5  353,7  23,6  359,9  336,7  4,0  359,2  341,0  17,6 
+3,76  0  330,4  9,7  26,4  342,5  0,9  8,1  330,3  355,1  23,1 
+5,64  0  339,0  23,4  351,4  355,7  23,3  341,5  337,9  7,6  350,2 
+7,52  0  3,4  14,4  331,3  15,4  20,7  334,0  0,9  0,0  330,9 
+9,40  0  25,3  340,1  341,3  27,2  345,4  354,9  20,2  327,1  341,7 
+11,28  0  17,9  338,5  2,1  7,1  335,4  18,4  10,4  327,3  2,3 
5*Lv15*Lr  1909,62  1922,79  1935,95  1949,12  1962,28  1975,45  1988,62  2001,78  2014,95  
0  0.00  345,9  341,3  347,9  351,3  347,9  357,7  353,0  358,0  1,3 
+1,88  102,86  89,2  85,1  92,1  93,2  92,1  100,5  95,2  102,9  103,5 
+3,76  205,71  193,0  187,8  196,7  194,9  196,0  203,0  198,3  207,5  205,2 
+5,64  308,57  295,3  289,7  300,5  296,5  300,7  305,7  301,6  311,1  307,0 
+7,52  51,42  37,3  34,5  43,9  39,1  45,5  47,5  45,2  54,2  49,4 
+9,40  154,28  138,7  138,8  147,1  140,3  149,8  149,0  149,3  157,2  151,8 
+11,28  257,14  240,8  242,1  249,5  244,6  253,9  250,9  253,8  259,3  254,6 
Planet Mars seems to be completely meaningless by
its size, but it appears in time of solar maxima and proton
events in conjunction with Jupiter more often, than would be expected
from the accidental layout.
It applies, for example, to proton flares:
1.9.1859, 28.9.1870, 10.9.1908, 9.8.1917,
7.3.1942, 25.7.1946, 4.5.1960,
28.1.1967, 10.4.1969.
Proton eruption 7.3.1942
Regarding the small period of conjunctions RJ and the fact, that 5*(R,J) = 5*2.235 years = 11.18 years ~ W, it still may be only random phenomenon.
Two bodies P and Q repeat their positions (e.g. in conjunction at the same place), if q periods P is equal to p periods Q, so if: q*P = p*Q, i.e. P/Q=p/q, where p,q are whole numbers.
Let
Period I is called period of inequality (or inequality period, inequality):
Usually I is on order of greater then P and Q (I>>P, I>>Q).
The place, where planets repeat their positions move with period I.
Conjunction of planets J and S appears on an average every
19.859 years. During this time Jupiter get approximately:
(J,S)/J *360° = 1.67416*360 = 360+242.698°. And Saturn
approximately (J,S)/S *360° = 0.67416*360 = 242.698°.
Because 240°=(2/3)*360°, conjunction places make equilateral
triangle, so called "big trigon". During 19.859 years this trigon
moves by
((J,S)/S  2/3)*360° = 242.698240 = 2.698°.
After 120°/2.698 * 19.859 y, i.e. c. 900 years (great inequality) the second apex appear at the starting point, and after c. 1800 years the third apex. The whole triangle returns to its original position after c. 2700 years.
If there was no rotation, conjunction line would be oriented
in the same direction every 3*(J,S). So, also after 42*(J,S),
45*(J,S) and 48*(J,S). During this period (c. 900 years) trigon
takes approximately 120° forward. Therefore conjunction line is
oriented in the same direction every c. 43, 46, and 49
conjunctions.
43∙(J,S) = 853.9 years, 46∙(J,S) = 913.5 years, 49∙(J,S) = 973.1 years
Lambert, Johann Heinrich, 17281777 German physicist, mathematician and astronomer. Dealt with perspective, spherical trigonometry, cartography, photometry, reflection and dispersion of light, algebra. 
J.H.Lambert has noted, that mean speed of Saturn increased compared to speed from Galileo’s measurements. This deviation was later make clear by Laplace with help of effect of small denominators.
Great inequalityValue of inequality JupiterSaturn (so called "great inequality", longperiod inequality, Laplace's period, ...) is not known with a good precision. It is assumed, the period is "about 900 years" (840960 years?).
From Bretagnon data (J=11.861983 years, S=29.457158) we have: I = (J/2,S/5) = 883.3 years.
S/1 
S/2 
S/3 
S/4 
S/5 
S/6 
S/7 
S/8 

J/1 
19.859 
60.947 
57.013 
19.422 
11.705 
8.376 
6.522 
5.340 
J/2 
7.426 
9.929 
14.978 
30.474 
883.27 
28.507 
14.487 
9.711 
J/3 
4.567 
5.405 
6.620 
8.538 
12.024 
20.316 
65.464 
53.556 
J/4 
3.298 
3.713 
4.249 
4.965 
5.971 
7.489 
10.042 
15.237 
J/5 
2.580 
2.828 
3.128 
3.500 
3.972 
4.591 
5.438 
6.670 
J/6 
2.119 
2.284 
2.475 
2.703 
2.976 
3.310 
3.729 
4.269 
According to Ptolemaio's values (J=11.862923, S=29.465040) is: I = (J/2,S/5) = 909.0 years.
Earth swings with a period P, c. 2550026000 years. Therefore motion of planet appears to be distorted. Corresponding "distorted" periods are called tropical periods. Let P=25750 years. Then J' = (11.861983, 25750) = 11.85652 years and S' = (29.457158, 25750) = 29.42350 years.
During 19.859 years trigon move by ((J',S')/S'  2/3)*360° = 242.976°240° = 2.976°.
After 120°/2.976 * 19.859 y, i.e. c. 800 years (i.e. great inequality seen from the Earth) the second apex appear at the starting point, and after c. 1600 years the third apex. The whole triangle returns to its original position after c. 2400 years.
S/1 
S/2 
S/3 
S/4 
S/5 
S/6 
S/7 
S/8 

J/1 
19.8589 
61.0913 
56.7618 
19.3784 
11.6836 
8.3628 
6.5120 
5.3319 
J/2 
7.4241 
9.9294 
14.9870 
30.5457 
800.940 
28.3809 
14.4464 
9.6892 
J/3 
4.5654 
5.4039 
6.6196 
8.5412 
12.0347 
20.3638 
66.1358 
53.0054 
J/4 
3.2962 
3.7120 
4.2479 
4.9647 
5.9725 
7.4935 
10.0541 
15.2728 
J/5 
2.5792 
2.8270 
3.1274 
3.4994 
3.9718 
4.5916 
5.4406 
6.6748 
J/6 
2.1184 
2.2827 
2.4747 
2.7019 
2.9751 
3.3098 
3.7293 
4.2706 
Seen from the Earth tropical periods seem to be the only true and correct periods. But we should be careful while computing derived periods. A slight difference of tropical and sidereal period causes in our example a quite different results (2400 y vs. 2700 y, see above).
(Some values in the table, e.g. 3.4994, 14.9870, come out near to integer fraction of terrestrial year. Value (J/3,S/4)=8.5412 years is equal to ecliptic year Y).
Babylonian period is distinct period of motion of inner planets. It makes approximately 427 years.
Ray Tomes included Babylonian period into his schemes (see Theoretical Cycles Periods ):
427.0 213.5 106.8 53.38 26.69
142.3 71.17 35.59 17.79 8.897 4.448 2.224
Timo Niroma created the following schema (see StuiverBraziunas analysis ):
1. 415425 years 2. 305314 years 3. 260280 years, affects moisture 4. 177227 years, almost all, very pronounced, an intensity cycle, affects temperature (median 202 years) 5. 154157 years, a length cycle, 13 Jovian years 6. 143148 years, a length cycle, 13 average cycles 7. 104105 years, half of the 200year cycle 8. 85 90 years 9. 78 79 years, the Gleissberg cycle 10. 63 67 years, 18341901, (1954?) 11. 57 59 years 12. 51 52 years, 17831834, 19011954 13. 43 45 years
It holds approximately: B/2 = 25*Y = 100*(E,R) = 225*(E,Ln) = 18*J, so
Simulation of motion of RJS makes patterns with period 853.9 years.
Great inequality (J/2,S/5) makes c. 2*B=72*J=29*S=400*(E,R).
Let us assume I= 2∙B = 72∙J = 29∙S' = 43∙(J,S').
Then using J=11.8620 we get: I=854.06 years, B=427.03 years, S' =
29.450441 years. Synodic period (J,S') = 19.861925 years.
If value of great inequality JS was just value I= 2*B=
854.06 years, then derived periods differ from the Bretagnon’s
periods by ratio:
S/S'=29.457158/29.450441 = 1.000228;
(J,S')/(J,S)=19.861925/19.8588709= 1.000154.
Synodic period is mean period of repetition of conjunctions. In reality conjunctions occur (in consequence of elliptical orbits and nonuniform motion of bodies) in irregular intervals.
E.g. conjunctions JS, in years 19402000: 1940.85, (20.41), 1961.26, (20.01), 1981.28, (19.15) and 2000.43 (in parenthesis are intervals):.
Intervals of conjunctions JupiterSaturn repeats in triads, i.e. with mean period c. 60 years (Chinese astrological cycle), 3*(J,S) = 3*19.859 y = 59.577 y.
In every 900 years (great inequality) phase shift appears and new sequence of triads begins.
E.g. maximum separation c. 20.5 years appears before conjunctions in years 750.03, 1723.11 and 2636.65.
I. y. 750.03
0: ( 20.49) 750.03 ( 19.59) 769.62 ( 19.47) 789.08 1: ( 20.48) 809.56 ( 19.69) 829.25 ( 19.38) 848.63 2: ( 20.48) 869.11 ( 19.77) 888.88 ( 19.32) 908.20 3: ( 20.45) 928.65 ( 19.86) 948.51 ( 19.25) 967.76 4: ( 20.42) 988.18 ( 19.96) 1008.14 ( 19.19) 1027.34 5: ( 20.38) 1047.72 ( 20.04) 1067.76 ( 19.15) 1086.91 6: ( 20.34) 1107.26 ( 20.11) 1127.37 ( 19.14) 1146.50 7: ( 20.29) 1166.79 ( 20.18) 1186.97 ( 19.12) 1206.09 8: ( 20.22) 1226.31 ( 20.26) 1246.57 ( 19.12) 1265.70 9: ( 20.15) 1285.85 ( 20.31) 1306.16 ( 19.15) 1325.32 10: ( 20.07) 1345.39 ( 20.36) 1365.74 ( 19.19) 1384.93 11: ( 19.97) 1404.91 ( 20.41) 1425.32 ( 19.25) 1444.57 12: ( 19.88) 1464.44 ( 20.45) 1484.90 ( 19.30) 1504.20 13: ( 19.78) 1523.98 ( 20.48) 1544.46 ( 19.37) 1563.83 14: ( 19.70) 1583.53 ( 20.48) 1604.01 ( 19.47) 1623.47 15: ( 19.59) 1643.06 ( 20.49) 1663.56 ( 19.55) 1683.11
II. y. 1723.11
0: ( 19.51) 1702.61 ( 20.49) 1723.11 ( 19.64) 1742.75 1: ( 19.41) 1762.16 ( 20.49) 1782.66 ( 19.73) 1802.38 2: ( 19.34) 1821.73 ( 20.47) 1842.19 ( 19.84) 1862.03 3: ( 19.26) 1881.29 ( 20.44) 1901.73 ( 19.92) 1921.65 4: ( 19.21) 1940.85 ( 20.41) 1961.26 ( 20.01) 1981.28 5: ( 19.15) 2000.43 ( 20.37) 2020.80 ( 20.08) 2040.88 6: ( 19.14) 2060.02 ( 20.31) 2080.34 ( 20.16) 2100.50 7: ( 19.11) 2119.61 ( 20.25) 2139.86 ( 20.23) 2160.09 8: ( 19.12) 2179.22 ( 20.18) 2199.39 ( 20.30) 2219.70 9: ( 19.12) 2238.82 ( 20.11) 2258.93 ( 20.34) 2279.27 10: ( 19.16) 2298.44 ( 20.01) 2318.45 ( 20.40) 2338.85 11: ( 19.22) 2358.07 ( 19.92) 2377.99 ( 20.44) 2398.43 12: ( 19.27) 2417.70 ( 19.82) 2437.52 ( 20.47) 2457.99 13: ( 19.34) 2477.33 ( 19.73) 2497.06 ( 20.49) 2517.55 14: ( 19.43) 2536.98 ( 19.63) 2556.61 ( 20.49) 2577.10
III. y. 2636.65
0: ( 19.51) 2596.61 ( 19.53) 2616.15 ( 20.51) 2636.65 1: ( 19.60) 2656.26 ( 19.45) 2675.71 ( 20.49) 2696.20 ....
Spörer Gustav, 18221895, amateur astronomer, discovered independently from R.Ch.Carrington the time dependence of the occurrence of spots on the distance from the equator  'Butterfly diagram' and expressed assumption that the Solar activity before r.1716 was very weak. 
Phase of new cycle is always shifted by on an average
(J,S)=19.859 y with regard to the previous cycle.
Phase shifts cause, that the mean cycle of conjunctions (with
regard to intervals of separation) seems to be (statistically,...)
little bit longer, approximately 19.859*(n+1)/n, where n is c.
4248.
Triads then lasts c. 61 years, i.e. approximately (J,S/2)=60.95
years (1/7 of Babylonian period 427 years).
Greatest deviations from mean intervals of conjunctions of planet Jupiter and Saturn are caused:
Mean cycles of these changes are:
Let us try to put these two cycles together.
With regard to phase shifts, we will write one function for
each interval (c. 8001000 years).
The exact instants of shifts are not clear  perhaps they even
do not exist  one cycle fades while the other one become
stronger:
Interval Function  (2000,1000) A_{p}∙sin(2π(t1111)/P)+A_{q}∙sin(2π(t1157)/Q) (1000, 0) A_{p}∙sin(2π(t1131)/P)+A_{q}∙sin(2π(t1157)/Q) ( 0, 1000) A_{p}∙sin(2π(t1151)/P)+A_{q}∙sin(2π(t1157)/Q) ( 1000, 2000) A_{p}∙sin(2π(t1171)/P)+A_{q}∙sin(2π(t1157)/Q) ( 2000, ... ) A_{p}∙sin(2π(t1191)/P)+A_{q}∙sin(2π(t1157)/Q) ....
Here P=3*(J,S)=59.58 y, Q=(U,N)/2=85.72 y, Ap,Aq are
constants and t is time [years]. Values 1111, 1131, 1151, 1171
AD follows c. 20 years phase shifts.
The value 1157 AD was selected 20 years after conjunction (U,N).
Egyptian minimum (1300,1200), Homér's minimum (800,700)
Greek minimum (450,350)
Middle age minimum (650,705)
Middle age maximum (1120,1280), Wolf's minimum (1280,1340)
Maunder, Edward Walter
Maunder, Edward Walter, 18511928 English astronomer, dealt with spectroscopy, photography of sunspot, observations of comets, Mars and the like. He confirmed the Spörer's assumption that Solar activity before r.1716 was very weak. 
Sporer's minimum (1400,1500), Maunder's minimum (1645,1715), minimum (1870,1930), end of millenium maximum (19302030)
Eddy, John A.
Eddy, John A., 19312009, American astronomer, collected data from a variety of sources and reconstructed retrospectively values of solar activity. He specified period of so called Maunder's minimum (16451715) and defined so called Spörer's minimum (14601550). 
Comments:
Maunder's minimum started about 1650 AD. An analogous "zeropoint" was e.g. at 800 AD, i.e. 850 years (2*B, i.e. 12 Bruckner's cycles) before Maunder's.
Other data:
Minima  Maxima 
m21: 1040 m22: 1160 m23: 12701330 m24: 14101500 m25: 1670 m26: 1820 m27: 1880 
M21: 1130 (11001150) M22: 1190 (11701220) M23: 1370 M24: 1610 M25: 17201780 M26: 1860 M27: 1960? 
Both mentioned cycles coincide with Babylonian period c. 427 years: 61.0 ∙7 = 427.0 years, 85.5 ∙5 = 427.5 years.
Derived periods
[(J,S/2), (U,N)/2]= [60.95, 85.72]= 35.6 years (Bruckner's period),
((J,S/2), (U,N)/2)= (60.95, 85.72)= 210.9 years (half of Babylonian period).
Are these cycles synchronized? Let us assume, it holds:
P=a1*P1+F1 = a2*P2+F2, where a1=14, P1=59.577 years, F1=19.859
years and a2=5, P2= 171.44 y, F2=?
Then P = 14*59.577 + 1*19.859 = 43*19.859 = 853.94 y
(2*B) and phase shift F2 = P 5*171.44 = 3.27 y.
If F2= 0, then (U,N) = 2*B/5 = 2*36*J/5 = 427.031/5=170.813
y.
And hence Neptunian period: N = ((U,N),U)= (170.813, 84.020)=
165.358 y (deviation 0.4 % from Bretagnon mean period 164.770 y).
For years 20002700 AD.
There would be next "Maunder's minima" around years 2150, 2350 and 2500?!.
Conjunctions ER appears close to geometrical axis JS with period B/4. E.g. (with accuracy 5°) in years 1636.3,1743.2,1850.0 or 1869.1 and 1975.9.
It holds:
i.e. [J,S]/2, 4∙(E,R))= (16.9132418/2, 4∙2.1353487)= (8.45662, 8.54140)= 852.04 years= 2∙426.02 years.
Lt us assume two relations:
From equations (1/J+1/U)(1/S+1/N) = 1/(3J) and (1/J+1/N)(1/S+1/U) = 1/(4W) it follows W= 3/4(J/2,S/3)= 11.23375 years = [J,212.1 years].
And hence:
(But in comparison with the mean period of Solar cycle Ŵ=11.011.1 years is value 11.23 years too high).
Babylonian counted also with the cycle B/6 = 6*J = 71.17 years. (Volcanic eruptions 1812,1883 and 1831,1902 remain this period).
65 oscillatory years give c. 71 years, i.e. 6 orbital period Jupitera.
1.091854 years: 9/2*(E,R/2)/65 = 9/2*(0.999979,1.880711/2)/65 = 70.97052/65 = 9/2*15.7712270/65
Extrapolation of low solar activity from 18001830 to 19802010 went wrong, 180 years cycle failed. But in this case also 320 years cycle go wrong, because of Maunder’s minimum in years 16601690.
Schove, Derek JustinSchove, Derek Justin , (19131986) English meteorologist and astronomer engaged in dating historical events in relation to natural phenomena (eclipses, auroras, floods, storms, etc.). Author of many papers, articles and books. Reconstructed retrospectively values of solar activity. 
In Schove’s dates periods of high (SSS,SS,S) and low (WW,W) solar activity seems to change with period more than 180 years. E.g. progress with step 205 years (about half of Babylonian period) we get e.g. years: 350, 555, 760, 965, 1170, 1375, 1580, 1785, 1990. Nearby (±40 years) of these dates relatively high maxima appear. (Tidal action of planet JVE cannot make such differences…!?)
Simulation of motion of inner planets (with Bretagnon’s data) establishes especially part 2/3 of Babylonian period, i.e. period c. 284 years
I have noted the possibility of phase shifts of conjunction cycles while reading texts (paragraph about period 10201030 years) of climatologist Timo Niroma:
http://personal.inet.fi/tiede/tilmari/sunspots.html .
More notable extremes of solar activity seems to appear every c. 1025 years, i.e. 6*(U,N), 12B/5:
"The auroral data of G. L. Siscoe of the years 4501700 (Rev. Geophysics and Space Physics 18, 1980) give another chance to try to calculate a value for the 1000year cycle. The lowest superminimum (smoothed) between 450 and 1450 appeared from 620 to 680. It precedes the lowest superminimum of this millennium, the Maunder Minimum in 16401700 by 1020 years. The next superminimum after this preMaunder is in 780800, which apparently corresponds to the superminimum in 18001820 both by duration and relative height with a 1020 year delay. The third superminimum in the Siscoe data is in 850880 corresponding to the superminimum in 18801920 about 1030 years later. The Siscoe supermaxima in 740770, 820850, and 900930 correspond to supermaxima beginning 1030, 1010, and 1050 years later, so that a supercycle of 10201030 years in average length is rather apparent." ( Timo Niroma, "Sunspots: Sunspot cycles and supercycles and their tentative causes")
Period G.L.Siscoe 1025 years: 5125/5;
minima v Eddyho diagramu Solar activity (9625,8600,7575,6550,5525,4500,3475, 2450,1425, 400, +625, 1650, 2675, 3700)
Data in the following table progress in the rough with Babylonian period 427 years. On the left are instants of conjunctions ELnR and instants of Jupiter in perihelion.
ELnR Jp UN  2021.55 2022.63 1594.48 1595.57 * (1605.74) 1167.41 1168.50 740.34 741.44 * ( 745.20) 313.26 314.37 113.81 112.69 * ( 111.19) 540.88 539.76 967.94 966.83 * ( 965.47) 1395.01 1393.90 1822.08 1820.97 * ( 1821.01) 2249.15 2248.04 2676.22 2675.11 * ...
Possible correlations with conjunctions UranusNeptune are marked on the right.
If mean period (U,N) was 170.813 years, then it would be equal to 20* Y= 80* (E,R)= 240 z. It would have integer ratio to Babylonian period (2:5), to 1025 years period of polar lights G.L.Siscoe (1:6) and also to Mayan period M=5125 years (1:30).
The following data (with period M/5 = 5125/5 = 1025 y)
seams to appear near to minima on Eddy diagram of solar
activity:
9625,8600,7575,6550,5525,4500,3475, 2450,1425, 400,
625, 1650, (2675), (3700).
These data (with period M/3 = 5125/3 = 1708 y) remind
great eruptions of volcanoes:
5049 ?, 3341 Avelino, 1633 Thera, 75 Vesuvius, 1783 Laki, 3491 ?.
Cycle of approximately 900years (840960 years?) created by the socalled great inequality of the planets Jupiter and Saturn. G. Beutler in the book Methods of Celestial Mechanics gives period 890 years, prof. A.E.Roy writes in the book Orbital motion about 900year oscillation. Last extremes of Laplace's cycle was approximately in years 1560 (minimum) and 2000 (maximum).
Mayan calendar round (52 years) and computation of tuns (118) makes period 9 Aztec centuries i.e. 9*104 = 936 years.
With the same period perihelia of Jupiter and Saturn synchronize. Value (J_{a}/2,S_{a}/5) computed from anomalistic period J_{a},S_{a} makes c. 938939 years.
Laplace's periodPeriod was observed in the following phenomena:
The values nP=360/P are generally called the Mean
Angular Velocities (MAV), see
Angular
velocity.
If period P is given in days, we speake about
Mean Day Motion (MDM).
If in given time planet run just with mean velocity, its
motion is sometimes called "true motion" (=mean motion).
If a planet is in state of true (mean) motion, its actual
period is the mean period.
Actual velocity of motion deviates from velocity of
true (mean) motion.
Precision of antique astronomical
instruments was not better than c. 10’ (0.167 dg)
 in comparison with that precision of
values in age of Edmund Halley achieved 10”.
However the precision of some results in antique was very high,
e.g. Hipparchos determined MDM of the Earth with the
precision 0.0435” (0.000012 dg). Values of MDM
given in the Almagest (by Claudius Ptolemaios) lead
 after correction of precession  to the following MDM
for Jupiter and Saturn: nJ=299,104581“ nS=120,422528“.
These values corresponds to periods
J=3600/299,104581 * 360/365,25 = 11.862923 years and
S=3600/120,422528 * 360/365,25 = 29.465040
years.
Bretagnon theory (VSOP) makes these MDM values: <nJ>=299.1283“ and
<nS>=120.4547“, i.e.
periods
:
J=11.861983 years, S=29.457159 years.
For year 2000 NASA provides these actual MDM
values: nJ=299.1124“ and nS=120.4943“.
The period (I) of the Jupiter –Saturn inequality period gets from ratio S:J near to resonance 5:2: I(J,S) = 1/(5/S2/J) . Because the denominator is small, the resulting value strongly depends on precision of mean orbital periods of Jupiter and Saturn. From Bretagnon data I(11.861983, 29.457158) = 883.3 years, from Ptolemaio’s data I(11.862923, 29.465040) = 909.0 years.
Bretagnon values
Mean motion of Jupiter is in Bretagnon VSOP82 theory
defined by this row: LM = { 0.5995465, 52.969096500,15e7,
0.0}.
Here the value 52.9690965 determine change of mean longitude for
century in radians:
Then J = 360 * 100 / (52.9690965 * 180/PI) =
36000/3034.905674=11.8619832 years.
Value 15e7 is a second order term (longterm change).
Perturbations are computed separately.
In theory VSOP87 is for the same (including perturbations) 915
terms, that have to be computed and summarized for one position of
Jupiter. Here are the first 3 terms, where starting row with
longitude 0.599546 corresponds to the value LM above.
In the second row is the great inequality term, the third row
belongs to resonance S/J = 2/1:
J S A B C 1/ 0 0 0.59954649739 0.00000000000 0.00000000000 2/ 2 5 0.00573506125 1.44396306420 7.11354700080 3/ 1 2 0.00062308554 3.41857056095 103.09277421860
Then other rows follow.
The 17th row represents the mean motion of Jupiter, where the value 529.69096509460
(in radians for millennium) is.
J S A B C 17/ 1 0 0.00001824700 5.72883078185 529.69096509460
From this we get mean period of Jupiter:
J= 360 * 1000 / (529.69096509460 * 180/PI) =
36000/3034.905674=11.8619831585 years.
Then yet about 900 other terms follows. The values A, B
and C in row are substituted to the formula v= A* Cos(B+ C*t)
and summarized, so values with small A does not influence result
too much.
So, periods can be computed from the values C in rows according to
relation:
T = 2000PI/ C = 6283.1853/ C.
For the great inequality we so get: 6283.1853/7.113547 = 883.27
years.
For the resonance S/J = 2/1 is 6283.1853/103.09277 = 60.9469
years
(note, it is the Chinese 60years period modulated by complete
cycle of the trigon of conjunctions JS:
1/60.9469 = 1/59.579 – 1/2649.1).
English astronomer
Jeremiah Horrocks (1619 – 1641) found (in the year 1637)
random departures in mean motion of Jupiter and Saturn.
Edmunt Halley (16561742) noted
that MAV of the Jupiter (Saturn) increased (decreased) compared to
antique values. The theory of planetary
motion based on Newton laws then resulted into formula of
continual change:
nJ = 299.1283611” – 0.000 000 089 81” T, nS = 120.4546453” + 0.000
002 836” T ,
where T is time measured in Julian centuries (36525 days) from 0
January 1850 (the antique values for T=20).
Further Edmund Halley clarified value for Saturn to
nS=120.4054”, confirmed that MDM of Saturn (in comparison with
antique) decreased and concluded that this change is due to mutual
gravitational perturbations of Jupiter and Saturn.
Johann Lambert (17281777) realized, that MDM of Saturn increased
compared to Halley value by 0,02” (Saturn accelerated) 
and from this started to be clear, that the observed phenomenon is
not continual change, but harmonic wave.
According to observations of the 18. century, there was
nJ=299,128361“, nS=120,454645“ and so 2nJ5nS=4,016503”
(=0,0000194725 rad).
PierreSimon Laplace derived the longterm harmonic function
(with value 2nJ5nS in denominator)
and computed, that theoretical results (that not include this
function) can differ from real planetary positions by 20’ (0.33 dg)
for Jupiter and by 50’ (nearly 1 dg) for Saturn.
Let us assume, that both Jupiter and Saturn was in the state of
their true (mean) motion at the year <t> = 1780 and that
they influence each other with Laplace period I = 4*220 = 880
years. Bretagnon''s values of mean motion
are <nJ>=299.1283“, <nS>=120.4547“.
We can assign these values of MDM to all the years:
3500,2620,1740,860,20,900,1780,2660,…
and also to theirs centers:
3060,2180,1300,420,460,1340,2220,…
The year 2000 is 220 years after 1780 and for
this date we have NASA values
(nJ=299.1124“, nS=120.4943“).
We can expect the same values also in
years: 3280,2400,1520,640,240,1120,2000,2880,…
Let as
define functions:
Q = sin[ (t  <t>) *
2PI / I ],
nJ = <nJ> 
<aJ> * Q,
nS = <nS> +
<aS> * Q
and evaluate amplitudes from the above adjusted
values (for 2000 and 1780):
<aJ> = 299.1283“
299.1124“ =0.0159“ and <aS> = 120.4943“
120.4547“ =0.0396“.
Now we substitute all the known values:
Q = sin[ (t  1780) * 2PI / 880 ],
nJ = 299.1283“  0.0159“ * Q,
nS = 120.4547“ +
0.0396“ * Q
For years 1340, 1560, 1780 and 2000 we get these MDM values and actual (osculating) periods.
Year  Q  nJ  nS  Years  J  S 
1340  0  299.1283“  120.4547“  3060,2180,1300,420,460,1340,2220  11.8620 y  29.4572 y 
1560  1  299.1442“  120.4151“  2840,1960,1080,200,680,1560,2440  11.8614 y  29.4669 y 
1780  0  299.1283“  120.4547“  2620,1740,860,20,900,1780,2660  11.8620 y  29.4572 y 
2000  1  299.1124“  120.4943“  2400,1520,640,240,1120,2000,2880  11.8626 y  29.4475 y 
If amplitudes <aJ>=0.0159“ and <aS>=0.0396“ are changes of MDM, then after a year they reach about: <AJ>=0.0159“ * 365.25 =5.81“ and <AS>=0.0396“ * 365.25 = 14.46“
Maximum departure from a mean value appeara c. after 220
years:
<AJ,max>= 5.81“*220 =1278“ and <AS,max>=
14.46“*220=3181“
These values corresponds to 1278“/60=21.3‘ for Jupiter and
3181“/60=53.0‘ for Saturn.
M.Somerville evaluate it to be 19.78‘ and 48.04‘,
Paul Schlyter gets 19.92‘ and 48.72‘.
Some scientists consider about connection of Laplace's cycle
with observed millennial cycle of solar activity.
When Jupiter get  for its own accerelation  some
extra energy from the Sun, then some lost of the solar
energy would be observed. Years in which Q take the values
1:
9000,8120,7240,6360,5480,4600,3720,2840,1960,1080,200,680,1560, 2440,…
The year 1560 corresponds to so called Suess' minimum.
There was oposition of Uranus against other outer planets in years 691 and 1561 (difference 870 years):
Synodical rotation period of the Sun seen from the Earth
was estimated to be c.27.275 days.
In some years before (~Maunder minimum) it was lower:
161113  26,163 days 
162526  24,913 days 
164244  24,300 days 