2013 International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'13, Isrand of Hawaii, Hawaii, USA. March 4-7, 2013

Synchronization of an Array of Chaotic Circuits
Coupled by Different Coupling Strength
Koji Okamoto, Yuji Takamaru, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minami-Josanjima, Tokushima, 770-8506, Japan
phone:+81-88-656-7470, fax: +81-88-656-7471
Email:{okamoto, takamaru, uwate, nishio}@ee.tokushima-u.ac.jp

try, however the chaotic circuits are not synchronized whenthe chaotic circuits are arranged in asymmetry. The obtainedThis paper presents synchronization of coupled chaotic cir- synchronization states depend on the coupling strength andcuits as an array topology when coupling strength has differ- the circuit parameters.
ent value depending on distance between the chaotic circuits.
By using computer simulations, we observe interesting synchronization phenomena. Namely, several types of synchro- 2. Circuit model
nization are observed by depending on arrangement method
Figure 1 shows the model of the used chaotic circuit calledof chaotic circuits.
Nishio-Inaba circuit, investigated in [8]-[10].
Abstract

1. Introduction
Synchronization is one of the basic phenomena and it canbe observed everywhere in our living life [1]-[5]. Therefore,studies on synchronization phenomena have investigated in
the various fields. In particular, we are interested in synchronization phenomena observed in electronic circuits. We consider that there are many real physical phenomena exhibitingby coupled chaotic circuits, and it is important to investigate
synchronization phenomena observed from coupled chaoticcircuits for future engineering applications.
Recently, we have investigated synchronization phenomena observed from complex networks of the coupled chaoticcircuits [6], [7]. The chaotic circuits are placed on 2dimensional space and the coupled parameters depend on thedistance between the circuits. We have observed interestingclustering phenomena from this proposed system by computer simulations and circuit experimental results. In orderto understand the obtained results in detail, we need to consider more simple circuit system.
In this study, we investigate synchronization phenomena incoupled chaotic circuits as a ladder topology (1-dimensionalspace). In this circuit system, the chaotic circuits are arranged in one dimensional space irregularly. The value ofcoupling strength depends on the distance between the circuits. By using computer simulations, we confirm that thecoupled chaotic circuits are synchronized at in-phase or asynchronous states. The chaotic circuits are synchronized at inphase when the arrangement of chaotic circuits is symme-Figure 1: Nishio-Inaba chaotic circuit.

The circuit consists of a negative resistance, a nonlinear
resistance consisting of two diodes, a capacitor and two inductors. The following equation describes the approximatedI − V characteristic of the nonlinear resistance, where theparameter rd is the slope of the nonlinear resistance.
rdvd (i2 ) =2(VVi2 +− i2 −rdrd).

By using the variables and the parameters as follows;- 681 -√√C
L1 C
i1 =V x; i2 =V y; v = V z;
L1L2
√√C
L1
L1 C
r= α;
= β; rd= δ;
L1L2L2√dt = L1 Cτ ; ” · ” =dτ(1)

We obtain the normalized circuit equations.

3. Basic Synchronization phenomenaẋ = αx + z

First, we investigate the case of basic synchronization
phenomena. We arrange some chaotic circuits in onedimensional an array at regular intervals. In this simulation,
all chaotic circuits are connected to each other by resistors
where f (y) is described as follows:
and the coupling strength between two circuits is determined()by Eq. (6). Furthermore, the value of coupling strength deδ11f (y) =y+− y−.
(3)pends on the distance between the circuits.
2δδThe arrangement of chaotic circuits is shown in Fig. 3.
Figure 2 shows the chaotic attractor generated by theIn this figure, four chaotic circuits are coupled as the arraychaotic circuit, obtained from a computer simulation (Fig. 2topoloby. The distance between each chaotic circuits are dif(a)) and measured in a circuit experiment (Fig. 2 (b)). For theferent in each pattern. This distance for each pattern are decomputer simulation, we set the parameters as α = 0.460,
fined by d = 0.1, d = 0.2 and d = 0.3.
β = 3.0 and δ = 470. For the circuit experiment, the paFigure 4 shows the computer simulation results of therameters are fixed with L1 = 500[mH], L2 = 200[mH],phase differences between the adjacent chaotic circuits. FromC = 0.015[µF ], and rd = 1.46[M Ω].
these results, we confirm that the chaotic circuits are synchronized at in-phase when the distance parameter is set tod = 0.10. By increasing the value of d, the chaos synchronization is broken and asynchronous can be obtained when
the distance parameter is set to d = 0.30. For simulations,
we fix the distance parameter with d = 0.10 for synchronization.
ẏ = z − f (y)ż = −x − βy(a) Computer simulation.

(2)(b) Circuit experiment.
(a) d = 0.10.

(b) d = 0.20.

Figure 2: Chaotic attractor.
Next, we consider the following equations when eachchaotic circuits is coupled globally.
dxi= αxi + zidτdyi= zi + f (y)dτ(c) d = 0.30.

Figure 3: Arrangement of chaotic circuits.
(4)

N∑dzi
= −xi − βyi −γij (zi − zj )dτ
j=1(i, j = 1, 2, · · ·, N )where i in the equation represents the circuit itself, and j indicates the coupling with other circuits. The parameter γij
represents the coupling strength between the circuits. The
value of γij reflects the distance between the circuits in an
inverse way, described by the following equation:

γij =g.
(lengthij )2(a) d = 0.10.

(b) d = 0.20.

(c) d = 0.30.

Figure 4: Phase difference between the chaotic circuits.

(5)lengthij denotes the Euclidean distance between the i − th 4. Synchronization phenomena
circuit and the j − th circuit. The parameter g is a weighting
In this section, several types of the circuit arrangement areor scaling parameter that determines the coupling strengths.
considered as follows.
For our study, we set the parameter to g = 1.0 × 10−5 .

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4.1 Synchronization phenomena IHere, we investigate synchronization phenomena when thechaotic circuits are coupled at irregular intervals. The twotypes arrangements of chaotic circuits are considered symmetric or asymmetric model shown in Fig 5. In this figures, d = 0.01 is the most nearest distance between thechaotic circuits. The arrangement of Fig. 5 (a) shows thesymmetric arrangement across the coupling d = 0.10, while
Fig. 5 (b) shows the asymmetric arrangement across the coupling d = 0.10.

(a) d = 0.01.

(b) d = 0.1.

Figure 7: Phase difference of asymmetric arrangement.

the case of the symmetry case, all chaotic circuits are synchronized with in-phase as shown in Fig. 9 (c).
(a) Symmetric arrangement.

(b) Asymmetric arrangement.

Figure 5: Arrangement of chaotic circuits.
(a) 3-1

Figure 6 shows the simulation results of the phase differences between the chaotic circuits across the coupling d =0.10 and d = 0.01 as shown in Fig. 5 (a). Similarly, Fig. 7
shows the results of the phase differences between the chaoticcircuits across the coupling d = 0.10 and d = 0.01 as shownin Fig. 5 (b). From these results, all chaotic circuits are synchronized when the chaotic circuits are coupled by symmetry.
However, the case of that the chaotic circuits care coupled byasymmetry, asynchronous phenomena can be observed fromthe adjacent chaotic circuits between the coupling d = 0.10 .

(b) 3-2(c) 3-3

Figure 8: Arrangement of chaotic circuits.

(a) 3-1(b) 3-2(c) 3-3

Figure 9: Phase difference between the chaotic circuits (d = 0.10).
(a) d = 0.01.

(b) d = 0.1.

Figure 6: Phase difference of symmetric arrangement.

4.3 Synchronization phenomena III

4.2 Synchronization phenomena II
Next, the number of the chaotic circuits is changed asshown in Fig. 8. We consider three types of the arrangements
as symmetry and asymmetry cases; 3-1, 3-2 and 3-3.
Figure 9 shows the simulation results of the phase differences between the chaotic circuits via the coupling d = 0.10.
From these results, the chaotic circuits coupled by the coupling parameter d = 0.10 are not synchronized when thechaotic circuits are coupled by asymmetry (Fig. 9 (a)). In

Finally, we show some simulation results when the number of the chaotic circuits is increased. The results of fourtypes of the arrangements (4-1, 4-2 4-3, and 4-4) are shown in
Fig. 10. We observe chaos synchronization when the chaoticcircuits are arranged as 4-3 and 4-4. We first confirm thatchaos synchronization is occurred even if the chaotic circuits
are coupled by asymmetry.
Figure 11 shows the results of five types of the arrangements (5-1, 5-2 5-3, 5-4 and 5-5). Similarly, we observe chaos
synchronization when the chaotic circuits are arranged as 5-3,5-4 and 5-5. The adjacent chaotic circuits between the coupling d = 0.10 can be synchronized when the number of the- 683 -chaotic circuits is more than 3 for the one side group. Namely,
more than half number of the chaotic circuits against the total
number circuits in the one side group is needed to synchronize all circuits.
From these results, we confirm that the synchronization
phenomena changes due to the arrangement of the chaotic circuits when the number of chaotic circuits has large number.
Although all chaotic circuits are synchronized with in-phasewhen the arrangement of chaotic circuits is set to symmetry.

5. Conclusion
In this study, we have investigated the synchronization phenomena of coupled chaotic circuits as the array topology.
The arrangements for chaotic circuits are set to symmetricor asymmetric types via the coupling resistors. By carryingout computer simulations, we confirm that the chaotic circuits are synchronized at in-phase when the chaotic circuits
are coupled symmetry. Also, chaos synchronization could beobserved if the number of chaotic circuits has more than halfagainst the total number circuits in one side group.
In our future works, we would like to apply this proposedsystem for large-scale networks.
Acknowledgment(a) 4-1This work was partly supported by JSPS Grant-in-Aid forYoung Scientists 23700269.
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Figure 10: Phase difference between the chaotic circuits (d =0.10).

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Figure 11: Phase difference between the chaotic circuits (d =0.10).

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