Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://uzmu.phys.msu.ru/issue/120102.pdf Äàòà èçìåíåíèÿ: Sun Sep 1 23:07:20 2013 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 22:33:20 2016 Êîäèðîâêà:

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2D- -
. . , , . , 119991, , , . 1, . 2. ( 9.05.2012; 3.07.2012) , - , . 2D- - , 1D- . , , , 2 , «-» , , . , , . 2D- - , .
PACS: 02.90.+p. : 530.1. : , , , , .

. .



, . , , , [1]. , . , - , , . [2, 3] , 1D- - -

: 1) «» ; 2) , «-» ; 3) ; 4) - ( ) [4]. , [5]. , ( , , ) (, WWW, , ) , [6], . , , , , . 2D- -



E-mail: kozlov@polly.phys.msu.ru

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[4, 7], . , , , .
1. 2D- -

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1D- , x
n+1

. 1: ( )

xn+1 (k ) = (1 - )f (xn (k ),a) + 2 (f (xn (k - 1),a) + f (xn (k + 1),a)) xn (k + m) = xn (k ) (m -- ). 2D- (6464) - , , . . ( , ) , : x
n+1

= f (xn ,a) = 1 - ax2 , n

(1)

xn (i, j ) (-1, 1), a (1, 2), x = (-1+ 1+ 4a)/2a, (. 1). 1D- , «-» , [4, 7]. (« » 2D-)

(i, j ) = (1 - )f (xn (i, j ),a)+ + [f (xn (i +1,j ),a)+ f (xn (i - 1,j ),a)+ 4 + f (xn (i, j + 1),a)+ f (xn (i, j - 1),a)].

(2)

, x 1 0, . . H (x(i, j ) - x ) , H (x) -- . , . . , , , . {x1 (i, j ),x2 (i, j ),x2 (i, j ),.. . ,xt (i, j ), .. .,xT (i, j )}, i, j 64x64, t ( 1 T ). x1 ,x2 , ... ,xT , , x1 , .. .,x . , [4, 7]: 1) x1 , ... ,x , p ( 1 /2), :
i=p+1

p =

(xi - x( -p

i mod p)

)

2

,

(3)

xp+1 ,.. . ,x x1 ,x2 ,. .., xp ,x1 ,x2 ,. .., xp , .... 2) -

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= min(1 ,2 ,.. . , ) = P P , , x1 , .. .,x . P , T . , . 1D- [4, 7], : 1) «» -- P , , ; 2) , «-» -- P , - (, «-» ) , «-» ; 3) - ( ) -- P , - , , . 2D- - , .. , «-» .
2. 2D-

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1D- ( «» , «-» , - ), , . 2D- = 0.1 a, ( H (x(i, j ) - x ), (i, j ) P (i, j )) . 1) C «» (. 2, a (1, 1.75)), , . 2) «» (. 3, a = 1.75), « » , «-» . 3) , « » (. 4, a (1.75, 1.9)), , «-» , . 4) (. 5, a = 1.9), - (, « »). 5) - (. 6, a (1.9, 2)), - , , , ( « » ). , , , (, . 2, 3 4 ), , . -,

, ,

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. 2: C «» (), ( ) P () a = 1.4 = 0.1 t = 15000

. 3: «» (), ( ) P () a = 1.75 = 0.1 t = 20000

. 4: « » (), ( ) P () a = 1.8 = 0.1 t = 5000

P ( 20), (

« »). , « » (. 2), . « » (. 3 4) . -,

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. 5: (), ( ) P () a = 1.9 = 0.1 t = 10000

. 6: - (), ( ) P () a = 1.905 = 0.1 t = 10000

P (. 2 3) , (. 3) (. 2). , , , . P , . . (. 2) P (. 3). , , P ( . 4). 2D- - 1D. = 0.1 a (1.75, 1.9) « » . ,

: 1) 10000 (. 4); 2) , (. 7). 2D- ( 1D-), - . , ( < 0.1) « » . , = 0.05 a = 1.65 (. 8). ~ P , . . - , . ( > 0.1) « » , . .

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. 7: « » (), ( ) P () a = 1.85 = 0.1 t = 60000

. 8: « » (), ( ) P () a = 1.65 = 0.05 t = 10000

2 . P , . . , , , (. 9).


(«-» : «» ) ( : «» ) . , , , ( -- , -- « »), .

, , ( < 0.1) « » a - , ( > 0.1) 2D- 1D- a . P . , , .. , . (

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. 9: 2 (), ( ) P () a = 1.95 = 0.2 t = 10000

= 20). .

, . . , , , 2011 .

[1] Pikovsky A., Rosenblum M., Kurths J. Synchronization: A Universal Concept in Nonlinear Science. (Cambridge University Press, 2003). [2] Kaneko K. Chaos. 3. P. 279. (1992). [3] Crutchfield J.P., Kaneko K. Phenomenology of spatiotemporal chaos. in: Directions in Chaos. (Singapore: World Scientific, 1987). [4] .., .., .. . . . 132. C. 105 (2002).

[5] Lai-Sang Young. Nonlinearity. 21. P. 245 (2008). [6] Reka A., Barabasi A. Reviews of Modern Physics. 74. P. 45 (2002). [7] .., .. . (, .-: , 2007).

Analysis of synchronization features in 2D-lattices of diffusively coupled quadratic maps by means of lo cal criterion of sto chasticity
A. A. Kozlov
Department of Polymer and Crystal Physics, Faculty of Physics, M.V.Lomonosov Moscow State University. Moscow 119991, Russia. E-mail: kozlov@pol ly.phys.msu.ru

In the field of synchronization processes coupled map lattices (CML) represent paradigmatic models of extended systems exhibiting spatiotemporal chaos, self-organization and synchronization due to the rich variety of global qualitative behavior. In this article dynamics on 2D-lattice of diffusively coupled quadratic maps with periodic boundary conditions is shown in all patterns of its behavior which are turned out to be quite similar to those ones of 1D-chain. However some of its specific features are also revealed and discussed, especially in a strong coupling regime when domains of differently oriented 2-element clusters involved zig-zag dynamics in domains are competing with the formation of chaotic elements on the borderlines of the domains. The comprehension of such cooperative behavior and other synchronization patterns mostly appears with emergent method, developed in the form of local criterion of stochasticity which allows investigating not only the local behavior of individual elements, but the evolution of the global dynamics of the entire phase space. Our results show the effectiveness of the criterion on 2D-lattice of diffusively

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coupled qaudratic maps and other examples of regular networks, so it is believed to provide a powerful framework to study various types of synchronizability for networks of coupled maps depending on their architecture. PACS: 02.90.+p. Keywords : coupled map lattices, logistic map, diffusion, synchronization, patterns. Received 9 May 2012.

1. -- , . . . ; .: (495) 939-51-56, e-mail: kozlov@polly.phys.msu.ru.

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