The 15th IEEE International Workshop on Nonlinear Dynamics of Electronic Systems
NDES'07, Tokushima, Japan, July 23-26, 2007

Synchronization Phenomena on Asymmetrical
Globally Coupled Chaotic Systems
Using Circuits or Maps
Yasuteru Hosokawa

Yoshifumi Nishio

Dept. Information Science,
Shikoku University
Email: hosokawa@keiei.shikoku-u.ac.jp

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: nishio@ee.tokushima-u.ac.jp

Abstract— In our previous study, we observed the interesting
phenomena on asymmetrical chaotic coupled systems. It is thatthe ratio of synchronization time of the one subsystem groupincreases in spite of increasing parameter mismatches in the othersubsystem.
In this study, an asymmetrical globally coupled maps isproposed and investigated, in order to explain the these phenomena. Asymmetry of the system is realized by using twoparameters. In the case of ﬁve maps, it was conﬁrmed that ratiosof synchronization time of maps using one parameter set areincreased by increasing a parameter mismatch rate of the othermaps group. We consider that this result is corresponding toresults of previous study.

subsystem
Asubsystem
Asubsystem
Bsubsystem
BI. I NTRODUCTIONCoupled chaotic systems generate various kinds of complex higher-dimensional phenomena such as spatio-temporalchaotic phenomena, clustering phenomena and so on. Oneof the most studied systems may be the coupled map latticeproposed by Kaneko[1]. The advantage of the coupled map
lattice is its simplicity. However, many of nonlinear phenomena generated in nature would be not so simple. Therefore, it isimportant to investigate the complex phenomena observed innatural physical systems such as electric circuits systems[2][6].
One of useful tools for investigating these phenomena iselectric circuits. There are many advantages of using electriccircuits. For instance, circuits are natural physical systems,
circuit elements is low price and high quality, getting easily,
circuit experiments are easy to match theory, and so on.
In our previous study [7][8], some kinds of asymmetricalglobal coupled chaotic systems are investigated. Especially,
we paid attention to relationships between synchronization
phenomena and small parameter mismatches. In all systems,
an interesting phenomena are observed. The phenomena arethat a ratio of the synchronization time increases in spite ofincreasing parameter mismatches in the other system.
In this study, in order to explain the these phenomena, anasymmetrical globally coupled maps is proposed and investigated. Asymmetry of the system is realized by using twoparameters.

Fig. 1.

subsystem
B

Proposed system.

II. P ROPOSED S YSTEM
A proposed system is shown in Fig. 1. This system consistsof two kinds of subsystems and coupling elements. Subsystems
are coupled globally. An asymmetry of the system is realizedby using two kinds of subsystems. Some kinds of chaotic
circuits or autonomous oscillators were applied as subsystems
in previous studies. For instance, two different coupling pointsof one chaotic circuit, two different parameter of one chaotic
circuit, chaotic circuits and van der Pol oscillators and so on.
In this study, the logistic map is applied as subsystems. The
asymmetry is realized by using two parameters. The numbersof subsystem A and B are two and three. This system is basedon globally coupled map (GCM) proposed by Kaneko [1].
GCM is shown as following model.
ε ∑f (xn (j))
N j=1
Nxn+1 (i) = (1 − ε)f (xn (i)) +(1)where n is a discrete time step and i is the index of an element.
f (x) is applied one of chaotic maps.
In this study, the logistic map f (x) = 1−ax2 is chosen. The
asymmetry of the system is realized by using two parameter.

- 30 -A-node

vk1

B-nodei dk

Subsystem B (m + 1 ≤ k ≤ m + n):ẋ = εβxk − εγf (xk − yk )
 k

{m+n}
∑
+εδxi − (m + n)xk ,

vk2
ik-g

C1Fig. 2.

C2L
ẏk =żk =Chaotic Circuit.

i=1(5)

ζ {−zk + γf (xk − yk )} ,
η(1 + qk )yk ,where,

Therefore, function f (x) is shown as follows.

(|x − 1| − |x + 1|).
2
Figure 3 shows the voltage differences between each subsystems. Vertical axes show voltage differences and horizontal
axes show time. Namely, in the case of synchronizing twosubcircuits, the amplitude becomes zero. First graph shows thevoltage difference between the two subsystem A. Synchronizations and un-synchronized burst appear alternately in a random
way. The second graph shows the voltage difference between
subsystem A and subsystem B. These are not synchronized atall. The third and fourth graphs show the voltage differences
between two subsystem B.
f (x) = x +Subsystem A ( 1 ≤ i ≤ p):f (xn (i)) = 1 − (1 + Qa i)a1 xn (i)2(2)

Subsystem B ( p + 1 ≤ k ≤ p + q):
f (xn (i)) = 1 − {1 + Qb (i − p)}a2 xn (i)2(3)where i is a number of maps. n is a number of iterations. a1and a2 are corresponding to a parameter a of the logistic map.
p and q are numbers of subsystem A and B. Qa and Qb areparameter mismatch rates of each subsystems.

x1 − x2
x2 − x3
x3 − x4

III. C OMPUTER S IMULATIONS

x4 − x5

A. Electrical Circuits

Fig. 3.
circuit.

At ﬁrst, we introduce the case of the system applied chaoticcircuits. The applied chaotic circuit is shown in Fig. 2.
This chaotic circuit is a simple three-dimensional autonomous
circuit proposed by Shinriki et al.[9]. A-node is used as acoupling node. An asymmetry of the system is realized as adifference of parameters. Namely, parameters of subsystem A
is different from subsystem B. Double scroll type attractors areobserved on the each subsystems. Normalized circuit equations
are described as follows:
Subsystem A (1 ≤ k ≤ m):

i=1−zk + γf (xk − yk ),(1 + pk )yk ,Here we deﬁne the synchronization as following equationand ﬁgure.

synchronizationFig. 4.

ẋk = αβxk − αγf (xk − yk )
{m+n}∑
+αδxi − (m + n)xk ,ẏk =żk =Voltage differences between two subsystems in the case of chaotic

Deﬁnition of the synchronization.

|xk − xk+1 | < 0.01

(6)

Figure 5 shows ratios of the synchronization time and totaltime. Q is shown as following equation.
(4)

qk = Q(k − 1)

(7)

Q is corresponding to small parameter mismatches qk ofsubsystem B group. By increasing small parameter mismatchof subsystem B group, the synchronization time of subsystem- 31 -

(a) xn (1) vs. xn+1 (1)(b) xn (3) vs. xn+1 (3)Fig. 6. Maps of (a) subsystem A and (b) subsystem B. p = 2, q = 3,
Qa = 0.050, a1 = 1.70, a2 = 1.98 and ε = 0.40.

Fig. 5. Relationship of the ratio of the synchronization time and small
parameter mismatches in the case of System 2. m = 2, n = 3, pk =0.001(k − 1), α = 0.600, β = 0.500, γ = 20.0, δ = 0.070, ε = 0.6,
ζ = 1.5 and η = 0.5.

A group is increased. Namely, in spite of increasing small
parameter mismatches of the system, the synchronization timeof subsystem A group is increased.
B. Chaotic Maps
We carry out computer simulations using Eq. 2 and Eq. 3.
Here we deﬁne the synchronization as following conditional
equations:|xn−1 (i) − xn−1 (i + 1)| < 0.05

(8)

Fig. 8. Relationship of the ratio of the synchronization time and small
parameter mismatches. p = 2, q = 3, Qa = 0.050, a1 = 1.70, a2 = 1.98
and ε = 0.40. The number of iteration is 1000000.

and|xn (i) − xn (i + 1)| < 0.05

(9)These equations show that two consecutive closer values thana given threshold value(0.05) deﬁne synchronization. Figures 6and Figs. 7 are examples of the computer simulation results.
In Figs. 6, horizontal axes show xn (i) and vertical axes
show xn+1 (i). Figures 6(a) shows x1 and Fig. 6(b) shows
x3 . Figures 6(a) is smaller than Fig. 6(b). We can see thedifferece between the two maps. In Figs. 7, horizontal axes
show iterations and vertical axes show values of xk . Read areas
show asynchronous states and blue areas show synchronousstates. From these results, we can see that subsystem A andsubsystem B are not synchronized at all and increasing Qb
causes increasing a ratio of synchronization time of subsystem
A. Figures 8 are corresponding to Figs. 5. We can also see thesimilar result as Figs. 5.
The chaotic circuit is continuous-time system. The logistic
map is discrete-time system. In spite of this difference, wecan observe the phenomena in both systems. Therefore, We
consider that the phenomena are closely related to the structureof the system.
IV. C ONCLUSIONSIn this study, an asymmetrical globally coupled maps is proposed and investigated. Asymmetry of the system is realized

by using two parameters. In the case of ﬁve maps, it wasconﬁrmed that ratios of synchronization time of maps using
one parameter set are increased by increasing a parametermismatch rate of the other maps group. We consider that thisresult is corresponding to results of previous study.
Acknowledgments
This work was partly supported by Yazaki Memorial Foundation for Science and Technology.
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- 32 -x1 − x2x2 − x3x3 − x4x4 − x5(a)Qb = 0.002
x1 − x2x2 − x3x3 − x4x4 − x5(b)Qb = 0.025
x1 − x2x2 − x3x3 − x4x4 − x5
Qb = 0.050
Fig. 7.

Variable differences between two subsystems. p = 2, q = 3, Qa = 0.050, a1 = 1.70, a2 = 1.98 and ε = 0.40.

[7] 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.
[8] Y. Hosokawa and Y. Nishio, “Asymmetrical Chaotic Coupled System
Using Two Different Parameter Sets,” Proc. NOLTA’06, pp. 543–546,
Sep. 2006.
[9] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations ina Modiﬁed van der Pol Oscillator Containing a Positive NonlinearConductance,” Proc. IEEE, vol. 69, pp. 394–395, 1981.

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