2014 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'14, Honolulu, Hawaii, USA. February 28 - March 3, 2014

Synchronization of Coupled Chaotic Circuits Arranged in One-dimensional Coordinate
Depending on Network Structures
Jun Shiomoto1 , Yoko Uwate1 , Thomas Ott2 and Yoshifumi Nishio1
1Tokushima University
2-1 Minamijosanjima, Tokushima, 770-8506 Japan
Phone:+81-88-656-7470
E-mail: {shiomoto, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract
In this study, we investigate synchronization phenomena ofcoupled chaotic circuits. First, the chaotic circuits are combined by resisters in one-dimensional coordinate system. We
change the distance between the circuits to adapt the couplingstrength. We investigate the phase difference between thecentral circuits using computer simulation. From the computer simulation, the difference synchronization phenomena
are obtained between the full coupled system and the laddersystem.

2

Zurich University of Applied Sciences
Einsiedlerstrasse 31a, 8820 Waedenswil, Switzerland
E-mail: thomas.ott@zhaw.ch

this study, we use two systems. They are the full coupled system and the ladder system. In the full coupled system, chaotic
circuits are connected to all chaotic circuits. Figure 1 shows
the system model of the full coupled system. In the laddersystem, chaotic circuits are connected to only adjacent circuits. Figure 2 shows the system model of the ladder system.
We measure the phase difference between the central circuits
using computer simulations. We compare the results of thefull coupled system and the ladder system.

1. Introduction
Synchronization phenomenon is one of the typical pheFigure 1: Full coupled system.
nomena observed in nature. Recently, many studies have been
investigated synchronization of chaotic circuits [1]∼[5]. It is
focused how the differences of the network structure impact
on the whole circuits. Additionally, it is applicable to thefields of medical science and biology and so on.
In our research group, we have investigated the clustering
phenomena resulting from the synchronization phenomena
Figure 2: Ladder system.
observed in coupled chaotic circuits when the chaotic circuits
are arranged in two-dimensional coordinate[6], [7]. We observed that the chaotic circuits arranged in the near distanceare synchronized at in-phase state, and the coupled circuitswith the far distance could not be synchronized. From theresults we confirmed the relationship between clustering and2. Circuit Modelsynchronization phenomena.
The research about the system arranged in twoFigure 3 shows the circuit model. This is a chaotic circuit
dimensional coordinate is complicated for more detailed recalled
Nishio-Inaba circuit [8]∼[10].
searches. So in this study, we investigate the synchronizaThecircuit equations of this circuit are described as Eq. (1).
tion phenomena of coupled chaotic circuits arranged in onedimensional coordinate. We combine chaotic circuits by redi1= v + ri1
L1sister, and the circuits are arranged in one-dimensional coordt
dinate system. The number of the circuits is always an evendi2
= v − vd (i2 )(1)
L2number and we investigate symmetry systems. We investidt
gate synchronization phenomena by changing the distancedv
C= −i1 − i2
between the circuit and the number of coupling circuits. In
dt- 409 -α = 0.460
β = 3.0
δ = 470In this study, we use two systems. The full coupled systemand the ladder system. In the full coupled system, all circuits
are connected to all circuits by resisters. In the ladder system,
chaotic circuits are connected to only adjacent circuits.
When all circuits are connected to all circuits, the circuit
equations are shown in Eq. (5).

Figure 3: Circuit model.

dxi= αxi + zidτdyi= zi − f (yi )dτThe characteristic of nonlinear resistance is described asEq. (2).
rdVVvd (i2 ) =i2 +− i2 −(2)2rdrdNXdziγij (zi − zj )= −xi − βyi −dτj=1The circuit equations are normalized as Eq. (3) by changing
the variables as below.
√C
L1 CV x; i2 =V y; v = V z;
i1 =L1L2r√
L1C
L1 C= α;= β; rd= δ;r
L1L2L2pdt = L1 Cτ ; “ · ” =;dτ(i, j = 1, 2, · · ·, N )Where 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:g.
(6)γij =(dij )2rẋ = αx + zẏ = z − f (y)ż = −x − βy(5)(3)

dij denotes the Euclidean distance between the i-th circuitand the j-th circuit. The parameter g is coupling coefficientthat determines the coupling strengths. In this study, we set
the parameter as g = 1.0 × 10−2 .
3. Simulation Method

The value of f (y) is described as Eq. (4).
δ
11f (y) =− y−y+2δδ
We use the chaotic attractor as shown in Fig. 4.

z(4)In this study, we use two systems arranged in onedimensional coordinate. There are the full coupled systemand the ladder system. We use ten circuits in all simulations.
We divide into the two symmetric groups, and there are five
circuits in one side of the group. The network structure of theladder system is shown in Fig. 5.

Figure 5: Network structure.

xFigure 4: Chaotic attractor.
The parameters of this attractor are described as below.

We change the distance between the circuits by changing
the coupling strength. The value of d is the distance of the circuits in the group. After that, we change the distance between
the central circuits. We define the distance between the central circuits as dcenter . We change the value of center from
0.1 to 13.0. We investigate the phase difference between thecentral circuits. We compare synchronization phenomena of
the full coupled system and ladder system.

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4. Simulation result
Figures 6 ∼ 8 show the simulation results when d is fixedwith 0.1, 0.3 and 0.5.
From the simulation results, the ladder system becomesasynchronous quickly in Figs. 6 ∼ 8. The central circuits
become asynchronous around dcenter = 2.0. However, in thefull coupled system, the way of a change in the phase difference is different by the value of d. So we focus on the fullcoupled system as follows.
In the case of d = 0.1, the phase difference increases insmall steps, and the central circuits become asynchronousaround dcenter = 11 ∼ 12. In the case of d = 0.3, the phasedifference increases with the case of d = 0.1. Although thecentral circuits in the case of d = 0.3 become asynchronousaround dcenter = 9 ∼ 10. In the case of d = 0.5, the phase difference increases quickly. The central circuits become asynchronous around dcenter = 6 ∼ 7.
From these results, the relationship between the network
structure and synchronization phenomena on each coupledsystem is made clear. We show some examples of the graphicof phase difference in Fig. 9.

Figure 8: Phase difference (d=0.5).

(a) phase difference = 0(b) phase difference = 10(c) phase difference = 30(d) phase difference = 60

Figure 6: Phase difference (d=0.1).

(e) phase difference = 90

Figure 7: Phase difference (d=0.3).

Figure 9: Graphic of phase difference.

- 411 -5. Conclusions[8] R. Stoop, P. Benner and Y. Uwate, “Real-World Existence and Origins of the Spiral Organization of ShrimpShaped Domains,” Phys. Rev. Lett., 105, 074102, Aug.
2010.

In this study, we have investigated the synchronization phenomena in coupled chaotic circuits networks. We also investigated the phase difference between the central circuits.
From computer simulation, we obtained the various results [9] C. Bonatto and J. A. C. Gallas, “Periodicity Hub andNested Spirals in the Phase Diagram of a Simple Resisby changing the distance between the central circuits and thetive Circuit,” Phys. Rev. Lett., 101, 054101, Aug. 2008.
value of d.
In the ladder system, we obtained the result that the cen[10] R. Stoop, P. Benner and Y Uwate, “Real-World Existral circuits become asynchronous around dcenter = 2.0 in thetence and Origins of the Spiral Organization of Shrimpall results. After that, the central circuits are asynchronous
Shaped Domains,” Phys. Rev. Lett., 105, 074102, Aug.
even by increasing the value of dcenter . From this result, in2010.
the ladder system, we can see that the central circuits are notimpacted from other circuits.
On the other hand, in the full coupled system, the way of achange in the phase difference is different by the value of d.
As enlarging the value of d, the value of dcenter is graduallyclose until the state of asynchronous. From this result, thedifference synchronization phenomena are obtained between
the full coupled system and the ladder system.
In our future works, we would like to investigate asymmetric system and more large scale system.
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