2017 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing (NCSP'17)
Guam, USA, February 28th to March 3rd, 2017

Synchronization Phenomena in Complex Networks of Coupled
Chaotic Circuits with Different Degree Distribution
Shuhei Hashimoto, Takahiro Chikazawa, Yoko Uwate, and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {s-hashimoto, Chikazawa, uwate, nishio}@ee.tokushima-u.ac.jp

3. Circuit model

1. Abstract
In this study, we investigate synchronization phenomena incomplex networks of coupled chaotic circuits with different
degree distributions. In addition, we investigate synchronization phenomena of each degree distribution by changing thepath length. Furthermore, we research difference of synchronization phenomena of each network.

The chaotic circuit model which called Nishio-Inaba circuit is shown in Fig. 1. This circuit consists of a negative resister, a capacitor, two inductors and dual-directional diodes.
And this circuit equation is shown Eq. (1).

2. Introduction
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 been studied[1]. However, many researchers have not been researchedmore about synchronization phenomena in complex networks
of coupled chaotic circuits which compare degree distribution.
In this study, we compare synchronization phenomena ofnetworks with three types of degree distribution. Complex
networks of real world have characters like scale free property, cluster property, small world property [2][3]. So, weuse degree distribution models based on the power law andthe normal distribution. Scale free network follows the powerlaw. Random network follows the normal distribution. Furthermore, we use the soaring distribution which is different
from the others. We compare synchronization rate of network based on three types of degree distribution by changing
the coupling strength. And we investigate synchronization
phenomena of network with different path length by changing the coupling strength.

Figure 1: Circuit model.
⎧di1⎪⎪⎪ L1 dt = αv + ri1
⎪⎪
⎪
⎪
⎪
⎨
di2= v − vdL2dt
⎪
⎪
⎪
⎪
⎪
dv⎪
⎪
⎪
⎩ C dt = −i1 − i2 ,(1)The characteristic of nonlinear resistance which consists ofdual diodes is following Eq. (2).
- -rd -V -- -V -vd =−i.
i+−222 rd - rd -(2)By changing the variables and parameters,- 333 -;⎧√CL1 C⎪⎪=Vx,
i=V yn , v = V zni⎪
1n2⎪⎪LL12⎪⎪⎪;√⎨CL1 CL1α=r, β=, γ = rd,⎪L1L2L2⎪⎪⎪;⎪⎪⎪⎪⎩ δ = 1 L1 , t = L1 C2 τ,R C(3)The normalized equation of this circuit is given as follows:⎧dx⎪⎪= αx + z⎪⎪dτ⎪⎪⎪⎪⎨ dy= z − f (y)dτ⎪⎪⎪⎪⎪dz⎪⎪⎪⎩ dτ = −x − βy,degree distribution of soaring. We ﬁx that the number of circuits in each network model is 10 and the number of edges ineach network model is 16.

(4)where f (y) is described as follows :f (yi ) =1
2- -
- -yi + 1 - − -yi − 1 - .
δδ-(5)4. System model
Figure 3: Degree distribution.
In this study, we create network which ten chaotic circuits
connect by resistances. Each network is composed by each
The path length of each network model is nearly the same
degree distribution. Network model which used in this revalue. In addition to these networks model, we make othersearch is shown in Fig. 2.
model with the different path length. Those network models
are shown Fig. 4.

Figure 2: Network model.
The normalized circuit equation of these network models
is given by the following equations.
⎧ dxi⎪= αxi + zi
⎪
⎪⎪ dτ
⎪
⎪
⎪
⎪ dyi = z − f (y)
⎨idτ
N

⎪
dzi
⎪
⎪
=−x−
βy−
δij (zi − zj )
⎪ii
⎪
⎪dτ
⎪
i,j=1
⎪
⎩
(i, j = 1, 2, · · ·, N ).

Figure 4: Network model with the different path length.

These network models in Fig. 4 also created based on each
degree distribution. The path length of model (a) is longer
than (a’). The path length of model (b) and (c) are shorter
than model (b’) and (c’) each other. The path length of each(6) network is shown in Table 1.
Table 1: Path length of each network model.

The parameter δ corresponds the coupling strength between the circuits. The parameter α is chaos degree and theparameter N is the number of circuits.
Figure 3 (a), (b) and (c) show the degree distribution whichused network in this study. The model (a) imitates the degree
distribution of the power law. The model (b) imitates the degree distribution of a random network. The model (c) is the- 334 -(a)
(a’)
(b)
(b’)
(c)
(c’)

Path length
1.822
1.689
1.844
2.156
1.889
2.0

5. Simulation Result
Deﬁnition of synchronization in this study is determined a
voltage difference waveform. We deﬁne synchronization as
the following Eq. (7).
|Zj − Zi | < 0.1(i, j = 1, 2, · · ·, 10)(7)

Figure 7: Shynchronization rate of network model (a) with different chaosdegree.

Figure 5: A difference waveform.
Figure 5 is a difference waveform which was observed inthis research. The two lines shown in Fig. 5 corresponds
threshold which is given Eq. (7). It is determined that wave
within two line which is the threshold is synchronization.
Therefore, we propose and investigate the synchronization
probability denoted the synchronization rate during a certain
time interval. in this research, we ﬁx a certain time interval as (τ =1,000,000 and step=0.02τ ) and investigate the synchronization rate in the entire network of 10 coupled chaoticcircuits.

5.1 Synchronization rate of network model with different
chaos degree
In this simulation, we use three parameters of chaos degree.
Attractor of these parameters shows Fig. 6. Synchronization
rate of network model (a), (b) and (c) with different parameteris shown in Figs. 7, 8 and 9. Synchronization rate of each network model at α=0.350 is highest. And synchronization rateof each network model at α=0.460 and α=0.490 are almost
the same. In the coupling δ=0.01, 0.02, network model (a)is most synchronize. However, when the coupling strength
increase, network model (b) is most synchronize.

Figure 6: Attractor of different chaos degree.

Figure 8: Shynchronization rate of network model (b) with different chaosdegree.

5.2 Synchronization rate of network with different path
lengthFigures 10, 11 and 12 show synchronization rate of network with different path length. The parameter α is ﬁxed at
0.460.
First, we compare synchronization rate of network model(a) and (a’), synchronization rate of network model (a) islower than network model (a’) in the weak coupling strength.
However, in the coupling strength δ = 0.08 to δ = 0.1, network model (a) more synchronized than network model (a’).
Second, we compare synchronization rate of networkmodel (b) and (b’), (c) and (c’). In the weak couplingstrength, synchronization rate of network model (b’) is lower
than network model (b). In the case of the coupling strengthδ = 0.08 to δ = 0.1, network model (b’) more synchronizedthan network model (b). When the weak coupling strength,
synchronization rate of network model (c’) is lower than network model (c). However synchronization rate of networkmodel (c’) is higher than network model (c) in the couplingstrength δ = 0.08 to δ = 0.1.

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Figure 9: Shynchronization rate of network model (c) with different chaos
Figure 11: Shynchronization rate with network model (b) and (b ).

degree.

Figure 10: Shynchronization rate with network model (a) and (a ).

Figure 12: Shynchronization rate with network model (c) and (c ).
Ackowledgment

6. Conclusion
In this study, we have investigated synchronization rate ofseveral network topology structures by changing the couplingstrength. Furthermore, we investigate synchronization phenomena with the different path length. The network topology
structures are following degree distribution of three types.
When we compare network of each degree distribution
with different chaos degree, we have conﬁrmed synchronization rate of each degree distribution at α = 0.350 is higher
than other chaos degree. And, we have conﬁrmed that synchronization rate of the models which imitate a random network is better than that of power law and soaring as the coupling strength increase.
When we compare synchronization rate in the different
path length, we have conﬁrmed synchronization rate of network which have long path length is lower than short onein weak the coupling strength. However, as the couplingstrength increase, network of long path length more synchronized than the short one.
For the future works, we would like to develop the networkmodel by adding a weight to the coupling. And we investigate
synchronization phenomena of larger network.

This work was partily supported by JSPS Grant-in-Aid for
Challenging Exploratory Research 26540127.
References[1] K. Ago, Y. Uwate, Y. Nishio, “Investigation of Synchronization in Coupled Chaotic Circuit Network with Local Bridge”, IEEE Workshop on Nonlinear Networks December 12-13, 2014.
[2] D.J. Watts and S.H. Strogatz, “Collective Dynamics ofSmall-world”, Nature, vol.393, pp.440-442, 1998.
[3] A.L. Barabasi and R. Albert, “Emergence of Scaling inRandom Networks”, Science, vol.286, pp.509-512, 1999.

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