Synchronization Phenomena in Coupled Chaotic Circuitswith Asymmetric Weights Depending on Length
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
of a node which connect to hubs. And we investigate thesynchronization rate between the hubs.
In this study, we investigate synchronization phenomena inBy computer simulations, we observe that the synchroweighted complex networks using chaotic circuit. The pro- nization rate between hubs is high regardless of the coupling
posed chaotic circuits network has Scale-free distribution and strength.
the chaotic circuit are coupled by using distance information. Next, we focus on the role of hubs in the proposed network. By computer simulations, we observe that the synchro- 2. Circuit model
nization rate between hubs is high regardless of the coupling
The chaotic circuit model which called Mori-Sinriki chaos
strength.
circuit is shown in Fig. 1. This circuit consists of one negative
resister, two capacitors, one inductors and dual-directional
1. Introduction
three diodes. And this circuit equation is shown in Eq. (1).
Synchronization phenomena can be observed everywherein our life. For example, we can confirm metronome, flashing firefly lights, beating rhythm of the heart and so on. Especially, synchronization phenomena of oscillatory network areinteresting. In addition, complex networks attract attention
from various fields. The feature of networks is characterized
by the degree distribution, the path length and the clustering
coefficient. Complex networks of chaotic circuits have beenstudied.[1] However, as research on coupling strength and thehub has not progressed yet.
In this study, we investigate synchronization phenomena inweighted complex networks using chaotic circuits. In this circuit system, the chaotic circuits are coupled with the distanceinformation. Furthermore, we study influence of hub to synchronization in complex network. We use a chaotic circuit
Figure 1: Circuit model.
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 usingdi1= V2n ,L
this circuit. First, we focus on the distance between hubs. Thedtnodes other than hubs which are connected with fewer nodes.
dV1n
We fix the position of hubs. Furthermore, we create another
C1= gV1n − idn ,(1)dt
new hub by increasing links of another node. We investigate


dV2n
the synchronization rate between the fixed hub and the new
 C2 dt = idn − in .
hub.
Second, we focus on a node which connect to the hub. WeThe characteristic of nonlinear resistance which consists ofbuild a network model with two hubs which are connecting
to different nodes. In this network, we can confirm effect dual three diodes is following Eq. (2).
Abstract- 553 -
Gd(V1n − V2n − V ) (V1n − V2n > V ),






0, (|V1n − V2n | < V ),idn =
 Gd(V1n − V2n + V ), (V1n − V2n < −V ).

(2)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 other hub for
focusing synchronization rate between hubs. In Fig. 3 (d),the network has twenty nodes. This networks was made toenhance hub nature from Fig 3 (c).
We compare the synchronization rate between nodes bychanging the coupling strengths.

By changing the variables and parameters Eq. (3),
√

C2

V xn , V1n = V yn , V2n = V zn
in =
L
√
C2,t = LC2 τ, α =C1√√√LL1L
 β=Gd, γ =g, δ =.
C2
C2
R C2(3)Figure 2: Original network model.

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 .

(4)(a) Network
new hub:CC4.

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).

with(b) Network
new hub:CC8.

with(5)(c) Network with divide links.

3. System model(d) Network with increase nodes.

We investigate the synchronization rate using three netFigure 3: Proposed network.
works based on original network. The original network modelis shown in Fig. 2. The network is composed of 10 chaotic
The normalized circuit equation of these network models
circuits and has Scale-free distribution. In this network, CC1
is given by the following equations.
and CC2 are fixed hubs.
Figure 3 (a) to (c) show the networks obtained from the dx
original network. Only Fig 3 (d) network is made by Fig 3 (c). = zn ,dτFirst, Fig. 3 (a) shows the network that CC4 is added 4 links ∑(connecting to CC3, CC5, CC6 and CC7). Namely, CC4 be-  dy(yn − yk ),= αγyn − αf (yn − zn ) − αδcomes new hub in this network. Similarly, Fig. 3 (b) shows(6)dτk∈S
nthe network that CC8 is added 4 links (connecting to CC2,  dz
CC3, CC4 and CC5). Namely, CC8 becomes new hub in  dτ = f (yn − zn ) − xn .
this network. In Fig. 3 (c), two hubs connect to the different- 554 -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.

y1-y4

y1-y8t

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 the coupling strength issummarized in Table 1.

Figure 5: Different wave form in Fig. 2.
Table 2: Synchronization rate of Fig. 5.
Links
Synchronization rate

CC1-CC426%

CC1-CC818%

y1-y3

y1-y4tFigure 6: Different wave form in Fig. 3 (a).
Table 3: Synchronization rate of Fig. 6.
LinksSynchronization rate

Figure 4: Coupling strength.

CC1-CC332%

CC1-CC450%

y1-y4
Table 1: Coupling strength.
strength

A
1.000

B
0.5257

C0.3826

D0.3249

y1-y8

E0.3090

tFigure 7: Different wave form in Fig. 3 (b).
Table 4: Synchronization rate of Fig. 7.

4. Simulation results
Definition of synchronization in this study is determined avoltage waveform. We define synchronization as the following Eq. (7).
In the following figures, red line corresponds threshold
which is given Eq. (7). It is determined that wave within ared line which is the threshold is synchronization. Also, weset the synchronization rate by calculating the ratio at all measured points.

|yj − yi | < 0.03

(i, j = 1, 2, · · ·, 10)

LinksSynchronization 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. 8, we set the coupling strength of CC1-CC4 to 0. Similarly, in Fig. 9, we set the coupling strength of CC1-CC8 to0. Although the coupling strength was set to 0, the synchronization rate decreases in Tables 5 and 6.

(7)

y1-y3
y1-y4

First, we consider a network model in Fig. 3 (a) and (b)and focus on CC1-CC2 to CC1-CC8. Fig. 5 shows CC1-CC4
and CC1-CC8 synchronization rate in the original network.
In Fig. 6, CC1-CC4 synchronization rate is higher than CC1CC3 when spite of coupling strength is lower. Similarly, inFig. 7, CC1-CC8 synchronization rate is higher than CC1CC4. These two synchronization rate is increased from theoriginal network model.

- 555 -tFigure 8: Different wave form in Fig. 3 (a).
Table 5: Synchronization rate of Fig. 8.
LinksSynchronization rate

CC1-CC329%

CC1-CC438%

y1-y4

5. Conclusions

y1-y8

In this study, we have investigated the influence of hub by
using various networks changing hubs position. Also, we investigate 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.
In this result, the synchronization rate between the hubs ishigh regardless of the coupling strength. We tried to investigate reasons of high synchronization rate. We investigate
in case of between hubs not connected to the same node. At
this time, synchronization rate is proportional the couplingstrength. We seem the coupling strength between hubs involved with the connected nodes. This is the same even if
a large-scale network is used. However, the synchronization
rate between hubs is high in large-scale network.
In our study, we use only five coupling strength value depending on the distance between chaotic circuits. Then, we
need to use another type network and would like to investigate
the synchronization rate by using the same methods. Furthermore, we should use large-scale networks inspired from real
biological networks.

tFigure 9: Different wave form in Fig. 3 (b).
Table 6: 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=0.0 and 2.0).
From this result, we can see that the synchronization rate between hubs has relationship with the connecting nodes.

y2-y7
y2-y7tFigure 10: Different wave form in Fig. 3 (c).

[1] S. Hashimoto, T. Chikazawa, Y. Uwate andY. Nishio,“Synchronization of Each Connection inCoupled Chaotic Circuits”,Taiwan and Japan Conferenceon Circuits and Systems (TJCAS’17), p. 63, Aug. 2017.

Table 7: Synchronization rate of Fig. 10.
Coupling strength of CC2-CC7Synchronization rate

06%

References

2.026%[2] K. Ago, Y. Uwate and Y. Nishio,“Synchronizationof Coupled Chaotic Circuits with Parameter Dispersion in Small-World Network”, Proceedings of InternaBased on Fig 10, we study the synchronization rate whentional Symposium on Nonlinear Theory and its Applicathe property as a hub is stronger in Fig 3 (c). In Fig 3 (d), thetions,pp. 431-434, Dec. 2015.
network has twenty nodes. We set CC3 and CC13 as the hubs.
Similarly to Fig 3 (c), we compare in case of two coupling [3] S. Hashimoto, T. Chikazawa, Y. Uwate and Y. Nishio,strength (δ=2.0 and 0.0) in Fig 11. From this result, even if“Synchronization Phenomena of Complex Networks inthe network became a large network and the number of nodes
Coupled Chaotic Circuits with Different Degree Distriincreased, the same tendency was seen.
bution”,Proceedings of RISP International Workshop on
Nonlinear Circuits, Communications and Signal Processing (NCSP’17), pp. 333-336, Feb. 2017.

y3-y13y3-y13tFigure 11: Different wave form in Fig. 3 (d).
Table 8: Synchronization rate of Fig. 11.
Coupling strength of CC3-CC13
Synchronization rate

017%

2.082%- 556 -