2017 International Symposium on Nonlinear Theory and Its Applications,NOLTA2017, Cancun, Mexico, December 4-7, 2017

Synchronization Phenomena due to Changes in Network Structureof Coupled Chaotic Circuits in Complex Networks
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

Abstract
In this study, we consider the synchronization of coupledchaotic circuits which have different network structure. Especially, we focus on network structure with hub which isconsisted of Scale-free network. So, we compare synchronization phenomena of some network models created by threetypes of degree distribution and evaluate them by synchronization rate. From the simulation results, we can confirm
that connection between hubs is important network structure.
1. 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]-[3]. However, many researchers have not been researchedmore about synchronization phenomena in complex networks
of coupled chaotic circuits which compare degree distribution. In Previous study, we have investigated synchronization
phenomena of coupled chaotic circuis network which madeby three types of the degree distribution [4]. However, expected results were not obserbed.
In this study, we change the circuit which use in the research, and we compare synchronization phenomena of some
network models created by three types of degree distribution.
Furthermore, we focus on network structure like a hub. Complex networks of real world have characters like scale freeproperty, cluster property, small world property [2][3]. So,
we use 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 some

network models based on three types of degree distribution
by changing the coupling strength. Also we investigate synchronization phenomena of each connection of network withhubs.

2. Circuit model
The chaotic circuit model which proposed by Shinriki etal. [4][5] is shown in Fig. 1. This circuit consists of a negative resister, an inductor, two capacitors and dual-directional
diodes. In this study, we use network model which is 10 coupled chaotic circuits. The circuit equation is shown in Eq. (1).

Figure 1: Circuit model.
din
L= v2n

dt




dv1n1 X
C1= gv1n − idn −(v1n − v1k )
dt
R

k∈Sn



dv2n


 C2 dt = −in − idn ,(1)where n = 1, 2, 3, · · ·, 10 and Sn is the set of nodes which
are directly connected to the node n. The characteristic ofnonlinear resistor which consists of dual diodes is following
Eq. (2).
 Gd (n1n − v2n − V ) (v1n − v2n > V )0(|n1n − v2n | ≤ V )idn =Gd (n1n − v2n + V ) (v1n − v2n < −V ).

- 522 -(2)3. Network model
By changing the variables and parameters,r
C2
V xn , v1n = V yn , v2n = V zn
in =
L


rr

C2L
L
α=, β=Gd , γ =g,
CCC

122
r


p
Ld
 δ= 1, t = LC2 τ, ” · ” = ,R C2dt

In this study, we create networks which are 10 coupledchaotic circuits and each circuit connected by a resistor. Each
network is composed by each degree distribution. Networkmodel which used in this research is shown in Fig. 3.
(3)

the normalized equations of this circuit are given as
follows:
Figure 3: Network model.
ẋ = zn


X

ẏ = αγyn − αβf (yn − zn ) − αδ(yn − yk )
k∈Sn



ż = βf (yn − zn ) − xn .
(4)The parameter δ corresponds the coupling strength between the circuits. The parameter α is the strength of negative resistor. The nonlinear function f (yn − zn ) corresponds
to the characteristics of the nonlinear resistor consisting of
the diodes and described as follows: where f (y) is describedas follows :f (yn − zn ) =yn − zn − 10yn − zn + 1

Degree distribution A, B and C in Fig. 4 show the degree
distribution which used for composing network in this study.
Degree distribution A imitates the degree distribution of thepower law. Degree distribution B imitates the degree distribution of a random network. Degree distribution C is the degreedistribution of soaring. We fix that the number of circuits ineach network model is 10 and the number of edges in eachnetwork model is 16.

(yn − zn > 1)(|yn − zn | ≤ 1)(yn − zn < −1).

(5)This circuit generates asymmetric chaotic attractor in theparameter γ = 0.5 and symmetric chaotic attractor in theparameter γ = 0.65 as shown in Fig. 2.

Figure 4: Degree distribution.

Figure 2: Attractors with the different parameter.

The path length of each network model is nearly the samevalue. In addition to these networks model, we make othermodel with the different path length. These network modelsin Fig. 5 also created based on each degree distribution. The
path length of model (a) is longer than (a’). The path lengthof model (b) and (c) are shorter than model (b’) and (c’) eachother. The path length of each network is shown in Table 1.

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In this simulation, we use two parameters of the strength ofnegative resistor. Attractors of these parameters are shown inFig. 2. Synchronization rate which set the parameter γ = 0.5
is shown in Figs. 7 and the parameter γ = 0.65 is shown in
Fig. 8.

Figure 5: Network model with the different path length.
Table 1: Path length of each network model.
(a)
(a’)
(b)
(b’)
(c)
(c’)

Path length
1.822
1.689
1.844
2.156
1.889
2.0

Figure 7: Shynchronization rate in the parameter γ = 0.5.

4. Simulation Result
Definition of synchronization in this study is determined a
voltage difference waveform. We define synchronization as
the following Eq. (6).

|zj − zi | < 0.15

(i, j = 1, 2, · · ·, 10)(6)

Figure 8: Shynchronization rate in the parameter γ = 0.65.

Figure 6: A difference waveform.
Figure 6 is a different waveform which was observed inthis simulations. The two lines in Fig. 6 correspond threshold
which is given Eq. (7). It is determined that wave within twolines which is the threshold is synchronization. Therefore, wepropose 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 inthe entire network of 10 coupled chaotic circuits.

4.1 Comparing synchronization rate of network model

In the case of the parameter γ = 0.5, synchronization rateof the network model (a) is highest. Furthermore, when wecompare synchronization rate with different path length, network model with long path length obtains high synchronization rate than short one model based on the same degree distribution. In the case of the parameter γ = 0.65, it is almost
the same result above. However, synchronization rate of all
network models is lower and less difference except for network model (a).
4.2 Synchronization rate of each connection in networkmodel (a)In the section 5.1, we observed synchronization rate of network model (a) is highest even if changing the parameter.
So, we investigate synchronization rate of model (a) in detail. Figure 9 shows synchronization rate of each connection- 524 -

in network model (a). In Fig. 9, we can confirm that synchronization rate of one connection obtains high. This connection
corresponds that of between hub. So, synchronization rateof network which have connection between node with many
coupling is high.

cations and Signal Processing (NCSP’14), pp. 417-420,
Mar. 2014.
[2] K. Ago, Y. Uwate, Y. Nishio, “Investigation of Partical Synchronization in Coupled Chaotic Circuit Networkwith Local Bridge”, Proc. of IEEE Workshop on Nonlinear Cirsuit Networks (NCN’14), pp.63-66, Dec. 2014.
[3] T. Chikazawa, Y. Uwate, Y. Nishio, “Chaos Propagation in Coupled Chaotic Circuits with Multi-Ring
Combination”, Proc. of IEEE Asia Pacific Conferenceon Circuits and Systems (APCCAS’16), pp. 65-68, Oct.
2016.
[4] S. Hashimoto, T. Chikazawa, Y. Uwate, Y. Nishio,“Synchronization Phenomena in Complex Networksof Coupled Chaotic Circuits with Different DegreeDistribution”, Proc. of RISP International Workshop on
Nonlinear Circuits, Communications and Signal Processing (NCSP’17), pp. 333-336, Feb. 2017.

Figure 9: Shynchronization rate of each connection in network model (a).

[5] D. J. Watts and S. H. Strogatz, “Collective Dynamics ofSmall-world”, Nature, vol.393, pp.440-442, 1998.
[6] A. L. Barabasi and R. Albert, “Emergence of Scaling inRandom Networks”, Science, vol.286, pp.509-512, 1999.

5. 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 parameter. The network topology
structures are following degree distribution of three types.
When we compare network model in the parameter γ =0.5, we have confirmed that network model which have long
path length get high synchronization rate than short one.
When we compare network model in the parameter γ = 0.65,
the difference of synchronization rate become small except
network model (a). Then, we have investigated synchronization rate of each connection in network model (a). We haveconfirmed that connection between hub is important structure.
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.

[7] M. Shinriki, M. Yamamoto and A. Mori, “Multimode Oscillations in a Modified van der Pol Oscillator Conraining
a Positive Nonlinear Conductance”, Proc. IEEE, vol. 69,
pp. 394-395, 1981.
[8] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena in aCurcuit with a Negative Resistance and an Ideal Switchof Diodes”, Trans. of IEICE, vol.E70, no. 8, pp. 744-754,
1987.

Ackowledgment
This work was partily supported by JSPS Grant-in-Aid for
Scientific Research 16K06357.
References[1] K. Ago, Y. Uwate, Y. Nishio, “Influence of Bridge on
Coupled Chaotic Circuit Network”, Proc. of RISP International Workshop on Nonlinear Circuits, Communi-- 525 -