IEEE Workshop on Nonlinear Circuit Networks
December 15-16, 2017

Synchronization Phenomena in Weighted Complex
Networks Using Chaotic Circuits
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—In this study, we investigate synchronization phenomena in weighted complex networks using chaotic circuit. The
proposed chaotic circuits network has Scale-free distribution andthe chaotic circuits are coupled by using distance information.
Next, we focus on the role of hubs in the proposed networks. By
computer simulations, we observe that the synchronization rate
between hubs is high regardless of the coupling strength.

I. I NTRODUCTION
Many nonlinear phenomena in our society can be expressedusing complex networks. For example, computer network,neuron, firefly synchronization. In such complex networks,synchronization phenomena are very important things.
In this study, we investigate synchronization phenomena
in weighted complex networks using chaotic circuits. In thiscircuit system, the chaotic circuits are coupled with the distance information. Furthermore, we study influence of hub tosynchronization in complex network. We use a chaotic circuit
called “Mori-sinnriki Circuit” as a node. In this study, we use
weighted complex networks with two hubs.
In this simulation, we investigate two experiments using
this circuit. First, we focus on the distance between hubs. The
nodes other than hubs which are connected with fewer nodes.
We fix the position of hubs. Furthermore, we create anothernew hub by increasing links of another node. We investigate
the synchronization rate between the fixed hub and the newhub.
Second, we focus on a node which connect to the hub. We
build a network model with two hubs which are connecting
to different nodes. In this network, we can confirm effectof a node which connect to hubs. And we investigate thesynchronization rate between the hubs.
By computer simulations, we observe that the synchronization rate between hubs is high regardless of the couplingstrength.

Fig. 1: Circuit model.

di1 L dt = V2n ,
dV1n
C1= gV1n − idn ,dtdV2n
 C2 dt = idn − in .

(1)

The characteristic of nonlinear resistance which consists ofdual three diodes is following Eq. (2).

II. C IRCUIT MODEL
The chaotic circuit model which called Mori-Sinriki chaos
circuit is shown in Fig. 1. This circuit consists of one negative
resister, two capacitors, one inductors and dual-directional
three diodes. And this circuit equation is shown Eq. (1).

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Gd(V1n − V2n − V ) (V1n − V2n > V ),0, (|V1n − V2n | < V ),idn = Gd(V1n − V2n + V ), (V1n − V2n < −V ).

By changing the variables and parameters Eq. (3),(2)


√

C2

in =V xn , V1n = V yn , V2n = V zn
L




√
C2t = LC2 τ, α =,
C1


√
√
√



LL1L

 β=Gd, γ =g, δ =.
C2C2R C2(3)The normalized equation of this circuit is given as follows:dx= zn , dτ dy= αγyn − αf (yn − zn ),dτdz dτ = f (yn − zn ) − xn .

Fig. 2: Original network model.

(4)where f (yn − zn ) is described as follows :β(yn − zn − 1) (yn − zn > 1),





0, (|yn − zn | < 1),f (yn − zn ) = β(yn − zn + 1), (yn − zn < −1).

(a) Network with new hub:CC4.

(5)

III. S YSTEM MODEL
We investigate the synchronization rate using three networksbased on original network. The original network model isshown in Fig. 2. The network is composed of 10 chaotic
circuits and has Scale-free distribution. In this network, CC1and CC2 are fixed hubs.
Figure 3 shows three networks obtained from the original
network. First, Fig. 3 (a) shows the network that CC4 is added
4 links (connecting to CC3, CC5, CC6 and CC7). Namely,
CC4 becomes new hub in this network. Similarly, Fig. 3 (b)shows the network that CC8 is added 4 links (connectingto CC2, CC3, CC4 and CC5). Namely, CC8 becomes newhub in this network. In Fig. 3 (c), two hubs connect to thedifferent nodes. Also, the hub can connect only four nodes. If
the original network remains, there are other hubs left behind,which may affect two hubs. Accordingly, we decrease otherhub for focusing synchronization rate between hubs.
We compared the synchronization rate between hubs bychanging the coupling strengths.

(b) Network with new hub:CC8.

(c) Network with divide links.
Fig. 3: Proposed network.
The normalized circuit equation of these network models is
given by the following equations.

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

yn-yktFig. 5: Different wave form.
(6)

The parameter δ corresponds the coupling strength between
the circuits. And we set the parameter α = 0.500, β = 20.000
and γ = 0.500.

First, we consider a network model in Fig. 3 (a) and (b)and focus on CC1-CC2 to CC1-CC8. In Fig. 6, CC1-CC4
synchronization rate is higher than CC1-CC3 when spite ofcoupling strength is lower. Similarly, in Fig. 7, CC1-CC8
synchronization rate is higher than CC1-CC4. These two
synchronization rate is increased from the original networkmodel.

The coupling strength of each link is determined by thedistance. The coupling strength becomes weaker as the length
becomes longer. In Fig. 4, there are five kinds of the couplingstrength. We define shortest links as a base length (couplingstrength define 1.000). The coupling strength of other links
is defined by the reciprocal of the ratio of length to base.
An accurate value of the five kinds of coupling strength issummarized in Table 1.

y1-y3

y1-y4tFig. 6: Different wave form in Fig. 3 (a).

TABLE II: Synchronization rate of Fig. 6.
Links
Synchronization rate

CC1-CC332%

CC1-CC450%

y1-y4
Fig. 4: Coupling strength.

y1-y8tTABLE I: Coupling strength.
Fig. 7: Different wave form in Fig. 3 (b).
coupling strength

A1.000B0.5257

C0.3826

D0.3249

E0.3090

TABLE III: Synchronization rate of Fig. 7.
IV. S IMULATION RESULTS
Definition of synchronization in this study is determined avoltage waveform. We define synchronization as the following
Eq. (7).
Figure 5 is a different wave form which was observed.
Red line corresponds threshold which is given Eq. (7). It isdetermined that wave within red line which is the thresholdis synchronization. Also, we set the synchronization rate bycalculating the ratio at all measured points.
|yj − yi | < 0.03

(i, j = 1, 2, · · ·, 10)

(7)

Links
Synchronization rate

CC1-CC426%

CC1-CC840%

We assume that synchronization rate between hubs is determined by not only the coupling strength unconditionally. We
decrease the coupling strength between fixed and new hubs.
In Fig. 10, we set the coupling strength of CC1-CC4 to 0.
Similarly, in the Fig. 11, we set the coupling strength of CC1CC8 to 0. Although the coupling strength was set to 0, thesynchronization rate dose not change so much in Tables IVand V.

- 63 -y1-y3

In this result, the synchronization rate between the hubs ishigh regardless the coupling strength. We tried to investigate
reasons of high synchronization rate. As a result, in the case ofbetween hubs not connected to the same node, synchronization
rate is proportional the coupling strength. Furthermore, thisresult shows reason two synchronization rate between hubs
are differences in same coupling strength. We observe that it
occurs because the strongly connected nodes were different.
We seem the coupling strength between hubs involved to theconnected nodes.
In our study, we use only five coupling strength value
depending on the distance between chaotic circuits. Also, anetwork which used in this study is very small (N =10). We
need to use another type network and would like to investigate
the synchronization rate by using the same methods. Furthermore, so we should use bigger networks inspired from real
biological networks.

y1-y4tFig. 8: Different wave form in Fig. 3 (a).

TABLE IV: Synchronization rate of Fig. 8.
LinksSynchronization rate

CC1-CC329%

CC1-CC4
38%

y1-y4
y1-y8

R EFERENCESt[1] K. Ago, Y. Uwate and Y. Nishio “Synchronization of Coupled Chaotic
Circuits with Parameter Dispersion in Small-World Network”,Proceedingsof International Symposium on Nonlinear Theory and its Applications,pp.
431-434,Dec. 2015.
[2] H. Shuhei, T. Chikazawa, Y. Uwate and Y. Nishio, “Synchronization Phenomena of Complex Networks in Coupled Chaotic Circuits with Different
Degree Distribution”,Proceedings of RISP International Workshop onNonlinear Circuits, Communications and Signal Processing (NCSP’17),
pp. 333-336, Feb. 2017.

Fig. 9: Different wave form in Fig. 3 (b).

TABLE V: Synchronization rate of Fig. 9.
LinksSynchronization rate

CC1-CC424%

CC1-CC829%

Next, we split two hubs which links connect to divide nodes.
In Fig. 3(c), we set CC2 and CC7 as the hubs. And wechange the coupling strength between CC2-CC7. In Fig. 10,
we compare in case of two coupling strength (δ=0.2 and 0.0).
From this result, we can see that the synchronization rate
between hubs has relationship with the connecting nodes.

y2-y7
y2-y7tFig. 10: Different wave form in Fig. 3 (c).

TABLE VI: Synchronization rate of Fig. 10.
Coupling strength of CC2-CC7
Synchronization rate

026%2.06%

V. C ONCLUSIONSIn this study, we have investigated the influence of hub
by using various networks changing hubs position. Also, weinvestigate about reasons which synchronization rate between
hubs is higher in two methods. First, we change the couplingstrength between hubs. Next, we split two hubs by divide
connecting nodes.

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