平成19年度電気関係学会四国支部連合大会

381

17-51
Relationship Between Ratios of Synchronization Time
And Small Parameter Mismatches on Asymmetrically
Coupled Chaotic MapsYukari Kowatari, † Yasuteru Hosokawa,† and Yoshifumi Nishio‡(Shikoku University†, Tokushima University‡)

2. Asymmetrically Coupled System
This system consists of two kinds of subsystems and coupling elements. Subsystems are coupled globally. An
asymmetry of the system is realized by using two kinds ofsubsystems. In this study, the logistic map is applied assubsystems. The asymmetry is realized by using two parameter values.
SubsystemA (1 ≤ i ≤ p) :f A ( x n (i)) = 1 − (1 + Qa i) a1 x n (i) 2(1)SubsystemB ( p + 1 ≤ k ≤ p + q ) :onized at all and increasing Qb causes increasing a ratioof synchronization time of subsystem A. An experimentresult is corresponding to the experiment in the electriccircuit.The chaotic circuit is continuous time system. The
logistic map is discrete-time system. In spite of this difference, we can observe the phenomena in both systems.
Therefore, we consider that the phenomena are closely
related to the structure of the system.

3. Conclusions
In this study, asymmetrically coupled chaotic maps areproposed and investigated. Asymmetry of the system isrealized by using two parameter values. It was confirmedthat ratios of synchronization time of maps using one parameter set are increased by increasing a parameter mismatch rate of the other maps group. We consider that thisresult is corresponding to the result of our previous study.
Investigating this phenomenon is our future research.
10.9
0.8

Ratio of synchronization time

1. Introduction
In our previous study [1][2], some kinds of asymmetrically coupled chaotic systems are investigated. Especially,
we paid our attentions to the relationships between synchronization phenomena and small parameter mismatches.
In all the systems, the same interesting phenomenon isobserved. The phenomenon is that the ratio of the synchronization time increases in spite of increasing parameter mismatches in the other system.
In this study, in order to investigate these phenomena,
asymmetrical coupled chaotic maps are investigated. The
logistic map is used as a chaotic map. Asymmetry of thesystem is realized by using two parameter values. We pay
our attentions to the relationship between the ratio of synchronization time and parameter mismatches.

0.7
0.6
A-A
A-B
B-B

0.5
0.4
0.3
0.2
0.1
0f B ( x n (i)) = 1 − {1 + Qb (i − p )}a 2 x n (i) 2( 2)where i is the index number of the maps. n is thenumber of iterations. a1 and a 2 are the parameter ofthe logistic map. p and q are the numbers of the elements in the subsystems A and B, respectively. Qa andQb are parameter mismatch rates of the two subsystems.
We carry out computer simulations using Eq. (1) and Eq.
(2). Here we define the synchronization as the following
conditional equation:(| x n−1 (i ) − x n−1 (i + 1) | < 0.05) I (| x n (i ) − x n (i + 1) | < 0.05)(3)This equation shows that two consecutive closer values
than a given threshold value (0.05) define synchronization.
Figure 1 shows the relationship of the ratio of the synchronization time and small parameter mismatches. We
can see that subsystem A and subsystem B are not synchr-04.00
E-04

8.00
E-04

0.001

0.002

0.002

0.002

0.003

0.003

0.004

Qb(parameter mismatch)

Figure 1: Relationship between the ratio of the synchronization timeand small parameter mismatches. p = 10, q = 20, Qa = 0.002,
a1 = 1.70, a 2 = 1.95 and ε = 0.45. The number of iteration is1000000.

References[1] Y. Hosokawa, R. Tsujioka and Y. Nishio, “Relation between
Synchronous Rate and Small Variations on an Asymmetrical
Coupled Chaotic System,” Proc. NOLTA’05, pp. 166.169,
2005.
[2] Y. Hosokawa and Y. Nishio, “Asymmetrical Chaotic Coupled
System Using Two Different Parameter Sets,” Proc.
NOLTA’06, pp. 543.546, Sep. 2006.
[3] K. Kaneko, “Spatiotemporal Intermittency in Coupled
Map Lattices,” Prog. Theor. Phys. vol. 74, no. 5,pp. 1033-1044, 1985.
[4] L. M. Pecora and T. L. Carrol, “Synchronization inChaotic Systems,” Physical Review Letters, vol. 64, pp.
821–824, 1990.