Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://chaos.phys.msu.ru/loskutov/PDF/Math_mod_paper.pdf Äàòà èçìåíåíèÿ: Thu Feb 5 11:59:59 2009 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:58:27 2012 Êîäèðîâêà: Windows-1251

2004 , 16, 12, c.109-122

:

c

.. , ..

...,
, , . , , "" . () , .. , (). . , . , . .
TO THE SELF-ORGANISATION PROBLEM: A MODEL OF FORMATION OF THE COMPLICATED FUNCTIONAL SYSTEMS

A. Loskutov, K. Vasiliev Physics Faculty, Lomonosov Moscow State University A model describing the forming of some patterns with prescribed properties which consist of the given types of elements is proposed. This model is based on the approach when the final complicated composition corresponds to a dynamical system formed from "elementary" subsystems. This composition (cascade) can be both a homogeneous one, i.e. it can consist of identical components, and it can have certain nonhomegeneities (defects). As a set of components one-dimensional maps with chaotic behaviour are considered. Therewith, it is required that the whole cascade has the prescribed type of the behaviour. It is shown that only a small part of the obtained nonhomogeneous cascades can possess the given evolutionary regime. On the basis of these results we draw a parallel with the self-organisation problem and formation of complex molecules.

1. . , (, ), , ( ) . , , "" . . , ( , ) . , , . .


110

.. , ..

. . , , , . ( ). , , . (., , [1-9] ) , , . , () . , . () , .. , . ( ) . , , . , . 2. , 1 . , , , . 2.1. . Ta I , Ta : I I , Ta : x - f (x, a), (1)

a A. [11-13]. Ta , . , I . , - I , , . Ta ( ) . , Ta S f = f /f -3(f /f )2 /2 < 0 , xc , , (. [12, 13] ). G : A A, G : a - g (a), a A A. (2)

1 (. [10]). . .


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111

(1) ? T : y - h(y ) , (3)

y M × A, y = (x, a), h(y ) = (f (x, a), g (a)). g , . (3) . . , (2) : ai+1 = g (ai ), i = 1, 2, . . . , - 1, a1 = g (a ), ai = aj i = j, ai A, 1 i . , - ^ ^ a = (a1 , . . . , a ), a IR . ^ ^ A = {a A A ž ž ž A : a = (a1 , . . . , a ), ai = aj , 1 i, j , i = j , ^
times

^ a1 , . . . , a A}, A IR , , A. , g (a) (2) . , g {a} (a1 , a2 , . . . , a ). (. [7, 14]), t (3) , t = k , k . , fi = f (x, ai ), i = 1, 2, . . . , . , F1 = f f -1 ž ž ž f2 f1 , F2 = f1 f f -1 ž ž ž f2 , . . . , F = f -1 f -2 ž ž ž f1 f , 1 ^ Ta : x - F1 (x, a) , ^ 2 T : x - F2 (x, a) , ^ a ^ ?^ Ta = (4) ............. , Ta : x - F (x, a) . ^ ^ , , (3) . , . 1. , (3) : 1) M , x1 , x2 a A, f (x1 , a ) = x2 ; 2) xc , f (x, a)/ x Dx f (xc , a) = 0 a A.
x=xc

x2 , x3 , . . . , x x1 a1 , a2 , . . . , a , ?^ (x1 , x2 , . . . , x ) Ta a = ^ (a1 , . . . , a ). . x1 , x2 , . . . , x . 1) f (x1 , a1 ) = x2 , f (x2 , a2 ) = x3 , . . . , f (x , a ) = x1 a1 , a2 , . . . , a a = (a1 , a2 , . . . , a ). ^ , (x1 , x2 , . . . , x ) = p ?^ Ta a = (a1 , a2 , . . . , a ). p ^ , x1 xc ,


(p) =
i=1

Dx f (xi , ai ) Dx f (xc , a) = 0 a.

| (p)| < 1. 1 .


112

.. , ..

, , , . , (. [10]). ?^ 2. f (x, a) C 2 [M × A] Ta a = (a1 , a2 , . . . , a ) , p = (x1 , x2 , . . . , x ). , ^ |ai | a = 1


, S
i x

tS

-1 a LSx

i=1

i = 1, 2, . . . , , Sa = max |Da f (x, a)|, L = max |D2 f (x, a)|, Sx = max |Dx f (x, a)|, x
x,a x,a x,a

p = (xc + x1 , x2 + x2 , . . . , x + x ) a = (a1 + a1 , a2 + a2 , . . . , a + a ), ^ |xi | x = 1 LSx
-1

.

. , ai , ai = ai + ai . x1 = x1 - xc . x1 T1 (. (5)), .. x1 = F1 (x1 , a1 , a2 , . . . , a ).


xc + x1 = F1 (xc , a1 , a2 . . . , a ) + Dx F1 (xc , a)x1 + ^
i=1

Dai F1 (xc , a)ai . , ^

xc = F1 (xc , a) Dx F1 (xc , a) = (p) = 0 , x1 = ^ ^ Dx f (xl , al )Da f (xi , ai )ai . ,
i=1 l=i+1

|x1 |

a i=1 l=i+1

Dx f (xl , al ) Da f (xi , ai ) (5)
i x

a S

a i=1

S.

, : (p ) - (p) = (p ) =
i=1

Dx f (xi , ai ) =

D2 f (xi , ai ) x





Dx f (xl , al )xi +
l=1 l =i

D2 f (xi , ai ) ax



Dx f (xl , al )ai .
l=1 l =i

i=1

i=1

, Dx f (x1 , a1 ) = Dx f (xc , a1 ) = 0. (p ) = D2 f (xc , a1 )x1 + D2 f (xc , a1 )a1 x ax D2 f (xc , a1 ) = Da Dx f (xc , a) ax |x1 | D2 f (xc , a1 ) x
l =2 a=a1

Dx f (xl , al ). l =2

= Da (0) = 0. , | (p )| =

Dx f (xl , al ) . l =2 Dx f (xl , al ) |x1 |LSx -1

|x1 | D2 f (xc , a1 ) x

< 1. , |x1 |

x = 1/(LSx -1 ). , x1 x , . x1 a (5). a

a S

a i=1

i Sx =

1 LSx

-1


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113

a =
tSa LSx

1
-1

. S
i x

i=1

2 . . , . Ac A , a Ac (1) . , (2), ( ) . , (1) a Ac G, Ac , G : Ac Ac , ( ). (. [7, 14]) , ^ ^ ^ . Ad Ac , a Ad , ^ . , , ^ a A, ^ , . qa : [0, 1] [0, 1], qa : x - ax(1 - x), (6)

a A = (0, 4]. , (6) . , , (6) , (., , [15, 16] ). (6) . , , a, , (. [13]). [17]: F , C 3 x x(1 - x), a0 a , a0 F (xc ) = 1. Ac = {a (0, a0 ] Fa : x aF (x) } . , (6) , . , (6) (. [5, 9]) . , ^ ^ a Ad Ac , ^ . , , .


114

.. , ..

2.2. . , , , . , xn+1 xn (. (1)). . m , (1)
i Ta : x - i (x, a), i = 1, 2, . . . , m .

(7)

1 2 Ta , Ta , 1 2 2 Ta Ta , Ta 1 1 2 Ta . , Ta Ta 2 1 F2 = 1 2 , Ta Ta F1 = 2 1 . 2 K2 = F2 F2 ž ž ž K2 = F1 F1 ž ž ž . 1 2 3 Ta , Ta , Ta . F1 = 3 2 1 , F2 = 1 3 2 , F3 = 2 1 3 , K3 = Fj Fj ž ž ž , j = 1, j = 2 j = 3. m m F1 , F2 , . . . , Fm , m. , , (1), , m = 1, K1 = ž ž ž . , , , (7). . [18] . , : ? [18] , . () [19]. , (7). , (2). - : ai+1 = g (ai ), i = 1, . . . , - 1, a1 = g (a ), ai = aj , i = j ai A. , Ta1 : x - (x, a1 ) f1 , Ta2 : x - (x, a2 ) f2 , ?a = T^ (8) ................. , Ta : x - (x, a ) f .

, fi , i = 1, . . . , , F1 = f f
-1

ž ž ž f2 f1 , (9)

................... , F = f
-1

f

-2

ž ž ž f1 f .


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F1 , F2 , . . . , F , , fi , i = 1, 2, . . . , . , K = Fj Fj ž ž ž , j (1, 2, . . . , ). , . , , ?^ ai , i = 1, . . . , (.(8)). , Ta Ti , i = 1, 2, . . . , (. 2.1):
1 Ta : x - F1 (x, a) , ^ ^ 2 Ta : x - F2 (x, a) , ^ ^

(10) ............. ,
Ta : x - F (x, a) , ^ ^

x1 = f1 (x0 ), . . ., x -1 = f -1 (x -2 ). , , . , , , k Ta (10), 1 k , t, ^ m , Ta , m = k + 1 (mod ), ^ k m t. , Ta , Ta ^ ^ . P , P = k , k . [18] . , ( ), (.. ) ? , a Ac , a Ac A, m (7). i , i = 1, 2, . . . , m, Km ? , a ^ Ad Ac K . 2.3. . . Fi , , "" . Fi , : l = Fi Fi . . . Fi Fj . ,
l -1

Fi Fi Fj Fi Fi Fj ž ž ž = 3 3 ž ž ž

(11)

Fi Fi Fi Fj Fi Fi Fi Fj ž ž ž = 4 4 ž ž ž .

(12)

, ? (.. (1) (2)) , fi fj "" (11), (12) .


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[20], , (11) (12) . () Fi Fj , i = j , , () l . , (11) l = 3, (12) l = 4. : ? , l , ( ) ? , . . , , . (10) . , , Fi , , , . l, , Fi1 Fi2 ž ž ž Fil-1 Fj . 3. , . , . , l l . {l }, ( , ""), i1 ,...,ik = i1 ž ž ž ik . K (i1 ,...,ik ) i1 ž ž ž ik i1 ž ž ž ik ž ž ž .
k

k

, , . , , i = {i1 , ..., ik }, i1 , ..., ik ( , ) i1 , . . . , ik , , . 3.1. . = 2 (, , ). ( ): F1 = f2 f1 F2 = f1 f2 . , 2 : f1 f2 f1 f2 ž ž ž f1 f2 ž ž ž f2 f1 f2 f1 ž ž ž f2 f1 ž ž ž . (14) (13)

Fi F2 F2 F2 ž ž ž F1 F1 F1 ž ž ž .


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(13) (14), i = 1, j = 2. l , . . f1 f2 , .2.1, , (. [20]): , f1 f2 , P = 2k , k = 1, 2, . . . . , , , , . F , fi fj , (6). , , ik , .. i1 ,i2 , i1 ,i2 ,i3 , i1 ,i2 ,i3 ,i4 . , . , i1 ,...,ik 4950 (a1 , a2 ), a1 , a2 [3, 8; 4], f . , i = {i1 , ..., ik } 2, 3, 4 ik 2 10. , i1 ,i2 = i1 i2 , i1 ,i2 ,i3 = i1 i2 i2 , i1 ,i2 ,i3 ,i4 = i1 i2 i3 i4 , l = Fi Fi . . . Fi Fj .
l -1

. , i , , k (max i) k , k i ( 2, 3 4). 4752000. i1 ,...,ik , . 344. , a1 = 3.907 a2 = 3.922 ( , (6) ) : 3,2 2,5,4 2,5,6,4 3,3,6,6

3 2 2 5 4 2 5 6 4 3 3 6 6

-0, 127 -0, 130 -0, 065 -0, 025

, , , . i1 ,i2 , , 87 (.. 0.00048 4950(10 - 1)(10 - 2)/2 = 178200 ). i1 ,i2 ,i3 = 94 (0, 00011 4950(10 - 1)(10 - 2)(10 - 3)/3 = 831600). i1 ,i2 ,i3 ,i4 163 (.. 0, 00004 4950(10 - 1)(10 - 2)(10 - 3)(10 - 4)/4 = 3742200). . . , ,


118

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(6). , , a1 , a2 , , , , . . - a1 = 3, 67857336 . . ., a2 = 3, 97459125 . . ., a1 , a2 Ac , ( , ) (.1). , , l > 105 .
0,500 0,450 0,400

ïîêàçàòåëü Ëÿïóíîâà

0,350 0,300 0,250 0,200 0,150 0,100 0,050 0,000 1 51 101 151 201 251 301 351 401 451 501

äëèíàêàñêàäà
. 1. - a1 = 3, 67857336, a2 = 3, 97459125.

(a1 = 3, 978, a2 = 4, a1 , a2 Ac ) , l = 3n + 3, n = 2, 3, . . . , (.2). n = 0 (l = 3) n = 1 (l = 6), , . a a = 4, 0. , . .3. , a1 = 3, 752685, a2 = 3, 955848, a1 , a2 Ac (.4). , l . , , l (, , -). , . , , 2, .


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119

0,70

ïîêàçàòåëü Ëÿïóíîâà

0,60 0,50 0,40 0,30 0,20 0,10 0,00 2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50 53 56 59 62

äëèíàêàñêàäà
. 2. a1 = 3, 978, a2 = 4.
0,60

0,50

0,40

ïîêàçàòåëü Ëÿïóíîâà

0,30

0,20

0,10

0,00 2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50 53 56 59 62

äëèíà êàñêàäà
. 3. , .2, a1 = 3, 5699456 a2 = 4.

4. , . -


120
0,6 0,5 0,4 0,3

.. , ..

ïîêàçàòåëü Ëÿïóíîâà

0,2 0,1 0
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46

-0,1 -0,2 -0,3 -0,4

äëèíà êàñêàäà
. 4. , .2, a1 = 3, 752685 a2 = 3, 955848.

(, ). , . , , , . , . . , , , . , , . - . , , . , "", ? . , ( ) . f1 , f2 ab, abb, abbb, abbbb, . . . , (15)

a F1 , a F2 . A, B , C , . . . , .. . , . , ababbbbabbabbb (ADB C ). , (15). .

48


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, . , - , . 5. . , , . . , . ( ) ( , ), . . , , .. , . , . , . -, , , , . , . , . . -, - . , .
1. .., ... . , 1987, .293, .1346-1348. 2. E.Ott, C.Grebogi, J.A.Yorke. Controlling chaos. Phys. Rev. Lett., 1990, v.64, p.1196-1199. 3. L.Fronzoni, M.Geocondo, M.Pettini. Experimental evidence of suppression of chaos by resonant parametric perturbations. Phys. Rev. A, 1991, v.43, p.6483-6487. 4. F.J.Romeiras, E.Ott, C.Grebogi, W.P.Dayawansa. Controlling chaotic dynamical systems. Physica D, 1992, v.58, p.165-192. 5. .., ... . . , 1993 .48, .169-170. 6. T.Shinbrot, C.Grebogi, E. Ott, J.A.Yorke. Using small perturbations to control of chaos. Nature, 1993, v.363, p.411-417. 7. A.Loskutov, A.I.Shishmarev. Control of dynamical systems behavior by parametric perturbations: an analytic approach. Chaos, 1994, v.4, p.351-355. 8. ... . II. . . . -, p. .-., 2001, No 3, c.3-21. ?


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9. ... . . . . .., ... - .: , 2001, .163-216. 10. A.Loskutov and S.D.Rybalko. Some properties of one- and two-dimensional perturbed maps. Proc. of the 5th Int. Conf. on Difference Equations and Applications, ICDEA'2002, Temuco, Chile, January 3-7, 2002. Taylor and Francis Publ., London, 2002, p.207-230. 11. A.Lasota, M.C.Mackey. Chaos, Fractals and Noise. Stochastic Aspects of Dynamics. Springer, Berlin: 1994. 12. A.Katok, B.Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Cambridge Univ. Press, Cambridge, 1995. 13. W.de Melo, S.van Strien. One-Dimensional Dynamics. Springer, Berlin: 1993. 14. A.Loskutov, V.M.Tereshko, K.A.Vasiliev. Stabilization of chaotic dynamics of one-dimensional maps. Int. J. Bif. and Chaos, 1996, v.6, p.725-735. 15. .., .., ... . - : , 1986. 16. .. . . - .: , 1988. 17. M.V.Jakobson. Absolutely continuous invariant measures for one-parameter families of one- dimensional maps. Commun. Math. Phys., 1981, v.81, No 1, p.39-88. ? 18. M.V.Hirsch. Convergent activation dynamics in continuous time networks. Neural Networks, 1989, v.2, p.331-349. 19. H.L.Smith. Convergent and oscillatory activation dynamics for cascades of neural nets with nearest neighbor competitive or cooperative interactions. Neural Networks, 1991, v.4, p.41-46. 20. A.Loskutov, V.M.Tereshko, K.A.Vasiliev. Predicted dynamics for cyclic cascades of chaotic deterministic automata. Int. J. Neural Syst., 1995, v.6, p.216-224. 03.12.2003.