平成19年度電気関係学会四国支部連合大会

380

17-50

Synchronization Phenomena in Globally Coupled Maps
Including Parameter Mismatches
T.Yamada† Y.Hosokawa† Y.Nishio‡(Shikoku University†

1. Introduction
One of famous coupled chaotic systems is the GloballyCoupled Map (GCM) proposed by Kaneko [1][2]. GCM
has been investigated by many researchers and many complex phenomena, for instance, spatio-temporal phenomena,clustering and so on, have been reported. However, most of
the works are found for the system without parameter mismatches in the maps.
In this study, we investigate synchronization phenomena
observed from GCM consisting of one-dimensional maps
described by a third-order polynomial function including
parameter mismatches.

Tokushima University‡)

red regions correspond to the 100% synchronization. The
darkness of the blue regions indicates that the rate of thesynchronizations becomes small. We noticed that the synchronization phenomena change around ε=0.2 regardlessthe number of the maps.
0.8-1
0.6-0.8x6-x7

0.4-0.6
0.2-0.4
0-0.2
00.1

0.2

0.3

0.4x1-x2ε2. System ModelFigure 1: Synchronization rate. Horizontal axis is ε. vertical axes are the differences between xn and xn+1. The redcolor indicates 100% synchronization.

The equation of GCM can be described as follows.
Nxn (t + 1) = (1 − ε ) f ( xn (t ) + (ε / N )∑ f ( xk (t ))(1)

4. Conclusions

k =1

In this study, a one-dimensional map described by a third-order polynomial function is used for the element ofGCM. This map has two control parameters and they affects the strength and the generating region of chaos. Namely, we can give the two different kinds of mismatches tothe map by changing the parameters. In order to make therole of the two parameter visible, we define the coordinateof the top of the function as P( p, q ). In this case, the third-order function is given as;f ( x) =[− (2 pq + 2 p) x
( p − 2 p + 1)
+ {(3 p − 1) q + 3 p − 1}x+ (2 pq + 2 p ) x − {(3 p − 1) q + p + p }]
1

4

3

2

2

2

2

2

4(2)

2

3. Computer Simulations
In this article, we show the computer simulated results
only for the case that the parameter mismatches aregiven as follows. The parameter value of q of the mapwith the index n is given by11+ (n − 1)(n = 1,2, L , N )(3)
22N
where N is the total number of the maps.
(4)
p = 1.2q − 0.92
The parameter value of p is given by Eq.(4). Figure 1
shows the rates of synchronization, when the couplingstrengthεand the number of the maps N are changed. The
qn =The following results have been obtained.
･Clustering phenomena are observed.
･Synchronization and anti-synchronization phenomena areobserved.
･Two different states are changed by the coupling parameter.
The border line of the two states is around ε=0.2.
In our future research, we investigate the detailed influences on the synchronization phenomena of the two parameter which are considered to affect the strength and thegenerating region of chaos independently.
References[1] K. Kaneko, “Clustering, Coding, Switching, Hierarchical Ordering, and Control in a Network of ChaoticElements,” Physica D, vol. 41, pp. 137–172, 1990.
[2] K. Kaneko, “Theory and Applications of Coupled Map
Lattices,” John Wiley & Sons, Chichester, 1993.
[3] T. Shibata and K. Kaneko, “Collective Chaos,” Phys.
Rev. Lett., vol. 81 pp. 4116–4119, 1998.
[4] C.W.Wu, “Global synchronization in coupled maplattices,” Proc. of ISCAS 1998, vol. 3, pp. 302–305,1998.
[5] M. C. Ho, Y. C. Hung and I. M. Jiang, “Phase synchronization in inhomogeneous globally coupled map
lattices,” Physics letter A, vol. 324, pp. 450–457, 2004.