2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019

Synchronization Phenomena
in Coupled Chaotic Circuits with Coupling Strength
Depending on Difference of Number of Links on Nodes
Kyohei Fujii, Shuhei Hashimoto, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima 7708506, Japan
Email: {fujii, s-hashimoto, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract
We investigate the influence of directly coupling and indirectly coupling of chaotic circuits for synchronization state in
the network. The node of the network is expressed by chaoticcircuit. This complex network is designed that node has two
or more couplings except for one node with scale free property. The coupling strength is determined depending on thenumber of couplings between the nodes. In this investigation,
we also found synchronization phenomena other than indirect
coupling.

tion using this circuit. First, we compare all the synchronization of the connections of the hub. Hub is a node with many
links in the network. We classify all couplings according tocoupling strength. We make graphs for each value of coupling strength and compare. Second, we investigate synchronization on indirectly coupling. We focus on one coupling.
We change the coupling strength of indirectly coupling. We
investigate what kind of synchronization occur between thecircuits.

2. Circuit model
1. Introduction
In our living life, complex network can be seen in various
fields such as airport network, computer network and neurons in the brain. Furthermore, synchronization phenomena
observed in complex network have been investigated by many
researchers [1], [2]. Especially, synchronization phenomenaof coupled oscillatory systems are very interesting. There arevarious investigations of synchronization using chaotic circuits. However, few studies focus on the difference in coupling strength [3] - [5].
In this study, we focus on the path and coupling strengthof coupled chaotic circuit network. Path is indirectly connection formed by several circuits. Indirectly coupling refers toa connection through multiple circuits. We design a networkthat each node has different coupling strength depending thenumber of degree and the network has scale free property.
Degree refers to the difference in the number of couplings
between circuits that are coupled to each other. In order tomake the network which the total coupling strength of node in
the network has nearly uniform, we apply the following rule
to set the coupling strength. When the difference of numberof degree between two nodes is large, the coupling strength
between two nodes is set to small value.
We use a chaotic circuit called “Shinriki-Mori Circuit” as anode. In this simulation, we investigate two computer simula-Figure 1 shows the chaotic circuit which is three dimensional autonomous circuit proposed by Shinriki et al. Thiscircuit consists of one negative resister, two capacitors, one
inductor and dual-directional three diodes. This circuit proposed by Shinriki et al [6], [7]. This circuit equation is shownin Eq. (1).

558

Figure 1: Circuit model.

2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019din
= V2n ,Ldt
dV1nC1= gV1n − idn ,dtdV C2 2n = idn − in .
dt(1)are set to the hubs (CC18, CC19, CC20). All couplingstrength values are fixed with a certain rule. This rule is basedon the difference of the node degrees between nodes. The
value of coupling strength is determined by the number ofdegree of the node. Namely, the coupling strength is set tothe certain value by calculating the difference of number ofdegree between the nodes and add 1.0 for every couplings.
Equation (6) is equation of coupling strength.
{The characteristic of nonlinear resistance which consists ofdual three diodes is following Eq. (2).
Gd(V1n − V2n − V ) (V1n − V2n > V ),
0, (|V1n − V2n | < V ),idn =
 Gd(V1n − V2n + V ), (V1n − V2n < −V ).

(2)

Coupling strength =1.0
1.0 + u(6)

In this equation, u is difference of node degrees. Initialvalue of all circuits have been changed. The aim of using
different initial value is to aim initial value sensitivity.

By changing the variables and parameters Eq. (3),
√

C2

V xn , V1n = V yn , V2n = V znin =L
√
C2t = LC2 τ, α =,
C1


√
√√


LL1L
 β=Gd, γ =g, δ =.
C2C2R C2(3)The normalized equations of this circuits are given as follows:dx= zn ,dτ dy= αγyn − αf (yn − zn ),dτdz dτ = f (yn − zn ) − xn .
Figure 2: Original network model.

Table 1: Coupling strength.
Difference of Digree
Coupling strength
Difference of Digree
Coupling strength(4)70.12530.2500

60.1428
20.3333

50.1666
10.50040.2000
01.000where f (yn − zn ) is described as follows :The normalized circuit equation of these network models
is given by the following equations.

β(yn − zn − 1) (yn − zn > 1),





0, (|yn − zn | < 1),f (yn − zn ) = β(yn − zn + 1), (yn − zn < −1).

dx= zn ,
dτ∑ dy= αγyn − αf (yn − zn ) − αδ(yn − yk ),dτk∈Sndz dτ = f (yn − zn ) − xn .

(5)

(7)

3. System model
The parameter δ corresponds the coupling strength beThis complex network is designed that each node has two tween the circuits. We set the parameter as α = 0.5, β = 20.0
or more couplings with Scale-free property. Only three nodes and γ = 0.5.

559

2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019

4. Simulation results
We define synchronization as the following Eq. (8). Synchronization rate is ratio of number of synchronization to all
calculate point.
|yj − yi | < 0.03

(i, j = 1, 2, · · ·, 20)(8)

Figure 3 is graph focusing on all connection that areconnected to the hub. We multiply all couplings by samevalue. We investigate synchronization by changing the mutiply value.

Figure 5: Synchronization rate of 6 degrees coupling.
In Fig. 5, lowest synchronization rate is 27% and highest synchronization rate is 72%. In this result, even if several couplings have same coupling strength, synchronization
is different.
We estimate this reason is indirectly connection. Next, wefocus on CC18-CC19. We multiplied magnification to somecouplings in network. In Fig. 6, we selected direct, indirect,
irrelevant coupling to CC18-CC19.
Figure 3: Synchronization rate of same strength.
In this study, all coupling strength are multiplied by 3. We
measured the cases where the synchronization hardly occurs
when the magnification is low, the case where the synchroIn Fig. 3, when magnification of coupling strength is 3.0, nization hardly occurs when the magnification is low, andall synchronizations rate have different values. In Fig. 4, we the case where the synchronization does not change when thesets of only coupling strength with degree difference of 5. magnification is low.
Simillaly, in Fig. 5, we sets of only coupling strength withFigures 6 and 7 show synchronization of CC18-CC19
degree different of 6.
when we change the coupling strength to CC1-CC6, CC6CC12 , CC5-CC9 , CC11-CC20, CC10-CC19 and CC4-CC5.

Figure 4: Synchronization rate of 5 degrees coupling.

In Fig. 4, lowest synchronization rate is 45% and highest
synchronization rate is 91%.

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Figure 6: Selected coupling.

2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019

Figure 7: Synchronization when multiplied some magnification to coupling.

by using a scale-free coupled chaotic network by changing
the couplings strength. First, we compare synchronization onsame coupling strength in the network. Next, we compare
some synchronization when coupling strength was changedsome values.
In this result, we found that the strength of synchronization is different even with the same coupling strength. We
tried investigated reasons of different synchronization. We
investigate one synchronization when some coupling strength
was changed some values. At this time, synchronization wasweakened when some coupling strength was weakened. We
consider some reasons. However, because several situations
contradicted each other, we could not clarify the reason.
In future work, we need to clarify the phenomenon of thistime. Furthermore, we would like to verify only circuit simulation. If we simulate society or nature network, we need toinvestigate characteristic of the network. We need to consider
deciding corresponding parameter, defining criteria, configuring network.

In Figs. 6 and 7, some synchronization was greatly changed
when we multiplied low magnification to coupling strength.
These couplings are indirectry couplings for CC18-CC19,and strongly couplings. Therefore, we estimate that synchronization are involved in the synchronization of CC18-CC19.
When some coupling affected to CC18-CC19, couplingstrength is 0 to 0.1 . Next, we investigate synchronization ofReferences
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nect will be weak. At the same time, synchronization was1987.
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[7] M. Shinriki, M. Yamamoto,S. Mori, “Multimode OscilFigure 8: Changed coupling.

5. Conclusions
In this study, we have investigated the influence of path561

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