2007 International Symposium on Nonlinear Theory and itsApplications
NOLTA'07, Vancouver, Canada, September 16-19, 2007

Investigation of Asymmetrically Coupled Chaotic Circuitsand Chaotic MapsYukari Kowatari, † Yasuteru Hosokawa,† and Yoshifumi Nishio‡†Dept. of Information Science, Shikoku University
Furukawa, Ojin-cho, Tokushima, 771-1192 JAPAN‡Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506 JAPAN
Email: s1839028@keiei.shikoku-u.ac.jp, hosokawa@keiei.shikoku-u.ac.jp, nishio@ee.tokushima-u.ac.jp
Abstract—In our previous study, we observed interesting phenomena in asymmetrically coupled chaotic circuits.
Namely, the ratio of synchronization time of the one subsystem group increases in spite of increasing parametermismatches in the other subsystem. In this study, we investigate the observed phenomena from asymmetrically coupled chaotic circuits and asymmetrically coupled chaoticmaps.

1. Introduction
Coupled chaotic systems generate various kinds ofcomplex higher-dimensional phenomena such as spatiotemporal chaotic phenomena, clustering phenomena and soon. One of the most studied systems may be the coupled
map lattice proposed by Kaneko [1]. The advantage of thecoupled map lattice is its simplicity. However, many of
nonlinear phenomena generated in nature would be not sosimple. Therefore, it is also important to compare with thecomplex phenomena observed in natural physical systems
such as electrical circuits systems [2]-[6]. One of usefultools for investigating these phenomena is electrical circuits. There are many advantages of using electrical circuits. For instance, circuits are natural physical systems,
circuit elements are low price and high quality, circuit experiments are easy to match theory, and so on.
In our previous study [7][8], 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 parametermismatches 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.

Figure 1: Asymmetrically coupled system.
2. Asymmetrically Coupled System
Asymmetrically coupled system is shown in Fig. 1. Thissystem consists of two kinds of subsystems and couplingelements. Subsystems are coupled globally. An asymmetryof the system is realized by using two kinds of subsystems.
Some kinds of chaotic circuits or autonomous oscillators
were applied as subsystems in our previous studies. For instance, two diﬀerent coupling points of one chaotic circuit,
two diﬀerent parameters 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 parameter values. This system is based on globally coupled map (GCM) proposed by
Kaneko [9]. GCM is described by the following equation.
ε ∑f (xn ( j))N j=1
N

xn+1 (i) = (1 − ε) f (xn (i)) +(1)where n is a discrete time step and i is the index of theelements. f (x) is the function of the chaotic map.
In this study, the logistic map f (x) = 1 − ax2 is chosen.
The asymmetry of the system is realized by using two parameter values. Therefore, the function f (x) is shown asfollows.

- 465 -A-node

i dkvk1

B-nodeSubsystem B (m + 1 ≤ k ≤ m + n):ẋk = εβxk − εγ f (xk − yk )m+n∑
+εδ xi − (m + n)xk ,
i=1
ẏk = ζ {−zk + γ f (xk − yk )} ,
 żk = η(1 + qk )yk ,

vk2
ik-g

C1

C2

L

Figure 2: Chaotic Circuit.

Subsystem A ( 1 ≤ i ≤ p):(5)

where,fA (xn (i)) = 1 − (1 + Qa i)a1 xn (i)2
Subsystem B ( p + 1 ≤ k ≤ p + q):
fB (xn (i)) = 1 − {1 + Qb (i − p)}a2 xn (i)2

f (x) = x +(2)(3)

where i is the index number of the maps. n is the number ofiterations. a1 and a2 are the parameter of the logistic map.
p and q are the numbers of the elements in the subsystems
A and B, respectively. Qa and Qb are parameter mismatch
rates of the two subsystems.

(|x − 1| − |x + 1|).
2Figure 3 shows the voltage diﬀerences between the subsystems. Vertical axes show voltage diﬀerences and horizontal axes show time. Namely, in the case of synchronizing two subcircuits, the amplitude becomes zero. Firstgraph shows the voltage 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 at all. The third and fourthgraphs show the voltage diﬀerences between two subsystem B. Here we deﬁne the synchronization as following
equation and ﬁgure.
x1 − x2

3. Computer Simulations

x2 − x3
x3 − x4

3.1. Electrical Circuitsx4 − x5

First, we introduce the results observed from the systemusing chaotic circuits. The applied chaotic circuit is shownin Fig. 2. This chaotic circuit is a simple three-dimensional
autonomous circuit proposed by Shinriki et al. [10]. Anode is used as a coupling node. An asymmetry of thesystem is realized as a diﬀerence of parameters. Namely,parameters of subsystem A is diﬀerent from subsystem B.
Double scroll type attractors are observed from both of thesubsystems. Normalized circuit equations are described asfollows:
Subsystem A (1 ≤ k ≤ m):ẋk =ẏk = żk =

Figure 3: Voltage diﬀerences between two subsystems in
the case of chaotic circuit.

synchronization

Figure 4: Deﬁnition of the synchronization.

αβxk − αγ f (xk − yk )
m+n





∑
,
xi − (m + n)xk +αδ 
i=1−zk + γ f (xk − yk ),(1 + pk )yk ,|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 mismatch of subsystem B group, the synchronization time of- 466 -x1 − x2
x2 − x3
x3 − x4
x4 − x5(a) Qb = 0.002
x1 − x2
x2 − x3

Figure 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.

x3 − x4x4 − x5(b) Qb = 0.020
x1 − x2x2 − x3x3 − x4
x4 − x5(a) xn (1) vs. xn+1 (1)(b) xn (3) vs. xn+1 (3)(c) Qb = 0.040

Figure 6: Maps of (a) subsystem A and (b) subsystem B.
p = 2, q = 3, Qa = 0.050, a1 = 1.70, a2 = 1.98 andε = 0.40.
subsystem A group is increased. Namely, in spite of increasing small parameter mismatches of the system, thesynchronization time of subsystem A group is increased.
3.2. Chaotic Maps
We carry out computer simulations using Eq. (2) andEq. (3). Here we deﬁne the synchronization as the following conditional equation:(|xn−1 (i)−xn−1 (i+1)| < 0.05)∩(|xn (i)−xn (i+1)| < 0.05) (8)This equation shows that two consecutive closer values
than a given threshold value (0.05) deﬁne synchronization.
Figures 6 and 7 are examples of the computer simulation results. In Fig. 6, the horizontal axes show xn (i) andthe vertical axes show xn+1 (i). Figure 6(a) shows x1 andFig. 6(b) shows x3 . We can see the diﬀerence between thetwo maps. In Fig. 7, the horizontal axes show iterations
and the vertical axes show the values of xk − xk+1 . Fromthese results, we can see that subsystem A and subsystem
B are not synchronized at all and increasing Qb causes increasing a ratio of synchronization time of subsystem A.
Figure 8 is corresponding to Fig. 5. We can also see thesimilar result as Fig. 5. The chaotic circuit is continuous-Figure 7: Variable diﬀerences between two subsystems.
p = 2, q = 3, Qa = 0.050, a1 = 1.70, a2 = 1.98 andε = 0.40.
time system. The logistic map is discrete-time system. In
spite of this diﬀerence, we can observe the phenomena inboth systems. Therefore, we consider that the phenomena
are closely related to the structure of the system.
Next, we investigate the case of thirty maps. Figure 9shows the relationship of the ratio of the synchronization
time and small parameter mismatches. In this case, we canalso the same phenomena. Figure 10 shows the relationshipof the ratio of the synchronization time and the number ofthe maps in A group. This result shows that increasing rateof A group decrease the ratio of the synchronization timeof A group.
4. Conclusions
In this study, asymmetrically coupled chaotic maps areproposed and investigated. Asymmetry of the system is realized by using two parameter values. In the case of ﬁve
maps, it was conﬁrmed that ratios of synchronization timeof maps using one parameter set are increased by increasing a parameter mismatch rate of the other maps group.
We consider that this result is corresponding to the result- 467 -of our previous study. We also investigated the case ofthirty maps. In this case, we observed that increasing rateof A group decrease the ratio of the synchronization timeof A group. Investigating this phenomenon is our future
research.
References[1] K. Kaneko, “Spatiotemporal Intermittency in Coupled Map Lattices,” Prog. Theor. Phys. vol. 74, no. 5,pp. 1033-1044, 1985.

Figure 8: Relationship of the ratio of the synchronizationtime 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.

[2] L. M. Pecora and T. L. Carrol, “Synchronization in
Chaotic Systems,” Physical Review Letters, vol. 64, pp.
821–824, 1990.
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no. 5, pp. 7–18, 1978.
[4] D. A. Linkens, “Analytical solution of large numbers ofmutually-coupled nearly-sinusoidal oscillators,” IEEE
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[5] H. Sekiya, S. Moro, S. Mori, and I. Sasase, “Synchronization Phenomena on Four Simple Chaotic Oscillators Full-Coupled by Capacitors,” IEICE Trans. Fundamentals, vol. J82-A, No. 3, pp.375–385, 1999.
[6] M. Miyamura, Y. Nishio and A. Ushida, “Clustering in
Globally Coupled System of Chaotic Circuits,” IEEE
Proc. ISCAS’02, vol. 3, pp. 57–60, 2002.

Figure 9: Relationship between the ratio of the synchronization time and small parameter mismatches. p = 10,
q = 20, Qa = 0.002, a1 = 1.70, a2 = 1.95 and ε = 0.45.
The number of iteration is 1000000.

[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] K. Kaneko, “Clustering, Coding, Switching, Hierarchical Ordering, and Control in a Network of Chaotic Elements,” Physica D, vol. 41, pp. 137–172, 1990.
[10] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations in a Modiﬁed van der Pol Oscillator
Containing a Positive Nonlinear Conductance,” Proc.
IEEE, vol. 69, pp. 394–395, 1981.

Figure 10: Relationship of the ratio of the synchronization time and small parameter mismatches. The number ofmaps is 30. Qa = 0.020, Qb = 0.002, a1 = 1.70, a2 = 1.95
and ε = 0.45. The number of iteration is 1000000.

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