IEEE Workshop on Nonlinear Circuit Networks
December 13-14, 2019

Synchronization Phenomena in Chaotic Complex
Networks with Coupling Strengthand Factor of Synchronization
Kyohei Fujii, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima 7708506, Japan
Email: {fujii, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract—We investigate the influence of directly coupling andindirectly coupling of chaotic circuits for synchronization state inthe network. The node of the network is expressed by a 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 the numberof couplings between the nodes. In this investigation, we find thatform of network is important.

I. I NTRODUCTION
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
are various investigations of synchronization using chaoticcircuits. However, few studies focus on the difference incoupling strength [3] - [5].
In this study, we focus on the path and coupling strength ofcoupled chaotic circuit network. Path is indirectly connection
formed by several circuits. Indirectly coupling refers to aconnection 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 nodein 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 a node. In this simulation, we investigate two computer
simulation using this circuit. First, we compare all the synchronization of the connections of the hub. Hub is a nodewith many links in the network. All couplings were classified
according to coupling strength. We make graphs for each valueof coupling strength and compare.

Second, we investigate synchronization on indirectly coupling. We focus on one coupling. The coupling strength waschanged of indirectly coupling. We investigate what kind ofsynchronization between the circuits.
II. C IRCUIT MODEL
Figure 1 shows the chaotic circuit which is three dimensional autonomous circuit proposed by Shinriki et al. This
circuit 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).

Fig. 1: Circuit model.
di1= V2n ,L


dt




dV1n
C1= gV1n − idn ,dt




dV

 C2 2n = idn − in .


dt

(1)

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

- 36 -
Gd(V1n − V2n − V ) (V1n − V2n > V ),0, (|V1n − V2n | < V ),idn = Gd(V1n − V2n + V ), (V1n − V2n < −V ).
(2)By changing the variables and parameters Eq. (3),
√

C2

V xn , V1n = V yn , V2n = V zn
in =

L



√
C2t = LC2 τ, α =,
C1

√
√
√



LL
1L

 β=Gd, γ =g, δ =.
C2C2R C2(3)Fig. 2: Original network model.

The normalized equations of this circuits are given asfollows:TABLE I: Coupling strength.
dx
= zn ,dτ dy= αγyn − αf (yn − zn ),dτdz dτ = f (yn − zn ) − xn .

Difference of Digree
Coupling strength
Difference of Digree
Coupling strength(5)This complex network is designed that each node has two
or more couplings with Scale-free property. Only three nodes
are set to the hubs (CC18, CC19, CC20). All coupling strength
values are fixed with a certain rule. This rule is based on the
difference of the node degrees between nodes. The value ofcoupling strength is determined by the number of degree of thenode. Namely, the coupling strength is set to the certain value
by calculating the difference of number of degree between thenodes and add 1.0 for every couplings. Eq. (6) is equation ofcoupling strength.
Coupling strength =1.0
1.0 + u50.1666
10.50040.2000
01.000The normalized circuit equation of these network models isgiven by the following equations.

III. S YSTEM MODEL{

60.1428
20.3333(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).

70.12530.2500

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

The parameter δ corresponds the coupling strength between
the circuits. We set the parameter as α = 0.5, β = 20.0 and γ= 0.5.

IV. S IMULATION 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

(6)

In this equation, u is difference of node degrees.
Initial value of all circuits have been changed. We have touse initial value sensitivity.

(7)(i, j = 1, 2, · · ·, 20)(8)

In Fig. 3, we sets of only coupling strength with degree
difference of 5. Simillaly, in Fig. 4, we sets of only couplingstrength with degree different of 6.

- 37 -In Fig. 5, some synchronization was greatly changed
when we multiplied low magnification to coupling strength.
These couplings are indirectry couplings for CC18-CC19, andstrongly couplings. Therefore, we estimate that synchronization are involved in the synchronization of CC18-CC19.
When some couplings affected to CC18-CC19, couplingstrength is 0 to 0.1. Next, we investigate synchronizationof CC18-CC19 when each coupling strength is changed.
We confirm coupling which changed CC18-CC19. When
synchronization rate reaches 50% to 70%, we judged thatsynchronization has changed.

Fig. 3: Synchronization rate of 5 degrees coupling.
In Fig. 3, lowest synchronization rate is 45% and highest
synchronization rate is 91%.

Fig. 6: Changed coupling.

Fig. 4: Synchronization rate of 6 degrees coupling.
In Fig. 4, lowest synchronization rate is 27% and highest
synchronization rate is 72%. In this result, even if several
couplings have same coupling strength, synchronization isdifferent.
We estimate this reason is indirectly connection. Next,
we focus on CC18-CC19. Some couplings were multipliedmagnification in network.
Figure 5 shows synchronization of CC18-CC19 when wechange the coupling strength to CC1-CC6, CC6-CC12 , CC5CC9 , CC11-CC20, CC10-CC19 and CC4-CC5.

Fig. 5: Synchronization when multiplied some magnification
to coupling.

In Fig 6, synchronization becomes constant on averagewhen indirectly couplings have a certain level of the couplingstrength. When some couplings open or extremely weakcoupling strength, synchronization is associated with someconnect will be weak. At the same time, synchronization
was not changed when we change coupling strength in otherindirectly coupling.
Next, we investigate this reason. When coupling strength
was changed, especially we open coupling, this phenomena
was caused. Therefore, link was reconnected which effect too
many coupling at other nodes. First, we choice too effective
links from all links. We choose three links which changes too
many nodes and number of links is not same. Then, link arereconnected on one side. In this study, CC9-CC20, CC7-CC18
and CC6-CC12 are chosen. These links are reconnected toCC5-CC20, CC7-CC2 and CC1-CC12. Each results is shownat Fig. 7 to Fig. 9. At this time, a number of links are changed.
However, coupling strength was not changed. We compare thisresult and original network result.

- 38 -In this result, the average difference of rank is 4.875. And
some coupling have too many differences. We understand that
effect of the coupling strength is important too.
V. C ONCLUSIONSIn this study, we investigated the influence of path byusing a scale-free coupled chaotic network by changing thecouplings strength. First, we compare synchronization on same
coupling strength in the network. Next, we compare somesynchronization when coupling strength was changed somevalues.
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 reason. However, because several situations
contradicted each other, we could not clarify the reason. In
addition, we find internet form is important.
In future work, we need to clarify the phenomenon ofthis time. Furthermore, we would like to verify only circuit
simulation. If we simulate society or nature network, weneed to investigate characteristic of the network. We need toconsider deciding corresponding parameter, defining criteria,
configuring network.

Fig. 7: Reconnect CC6-CC12 to CC1-CC12.

Fig. 8: Reconnect CC7-CC18 to CC7-CC2.

R EFERENCES
Fig. 9: Reconnect CC9-CC20 to CC5-CC20.
In this result, change is return ed. We understand that
network form effect to all synchronization. However, thephenomenon does not change when the network become
nearby form. Next, we investigate the effect of the couplingstrength. In this study, the coupling strength was changed.
the coupling strength was equalized and compared to original
network result. All coupling strength average value is 0.3953.
When we use this value and multiplied value 3.0, all couplings
synchronization rate become 99%. Therefore this value wasdecreased to 1.0. We set rank of synchronization rate andcompare from original network result. In this table, “sync”
mean synchronization rate.
TABLE II: Compare equalized and original network.

[1] M. Uchida, S. Shirayama, “Analysis of Network Structure and Model
Estimation for SNS” IPSJ journal, Vol. 47, No. 9, pp. 2840-2849, 2012[2] H. Kori, N. Masuda, “Synchronization of Coupled Oscillators on Complex
Networks” Journal of the Robotics Society of Japan, Vol.26, No.1, pp.
6-9, 2008.
[3] K. Ago, Y. Uwate, Y. Nishio, “Investigation of Synchronization in
Coupled Chaotic Circuit Network with Local Bridge”, Proc. of IEEE
Workshop on Nonlinear Networks, pp. 63-66, 2014.
[4] K. Oi, Y. Uwate, Y. Nishio, “Influence of Regional Change in Synchronization of Complex Networks in Coupled Parametrically ExcitedOscillators”, Proc. of IEEE Workshop on Nonlinear Circuit Networks,
pp. 75-78, 2016.
[5] S N. Kudoh, A. Kiyohara, T. Taguchi, “The Significance of “the smallworld” in a Neural Network.”, Cell Engineering RI, National Institute ofAdvanced Industrial Science and Technology (AIST) Vol. 14, No. 3, pp.
198-204, 2007.
[6] N. Inaba, T. Saito, S. Mori, “Chaotic Phenomena in a Circuit with a
Negative Resistance and an Ideal Switch of Diodes , Trans of IEICE,
vol.E70, no. 8, pp. 744-754, 1987.
[7] M. Shinriki, M. Yamamoto,S. Mori, “Multimode Oscillations in a Modified van der Pol Oscillator Containing a Positive Nonlinear Conductancee.”, Proc, IEEE, vol. 69, pp. 394-395, 1981.

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