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

Synchronization Phenomena of Chaotic Circuit
Networks with Distributed Hub
Shuhei Hashimoto, Takahiro Chikazawa, Yoko Uwate, and Yoshifumi Nishio
Dept. of Electrical and Electronic Eng.,
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 focus on network structure with ahub which is consisted of Scale-free network. We consider thesynchronization of coupled chaotic circuit network whose hubnodes are split two nodes and we investigate two network models.
In one of them, we replace a hub node with two distributed nodes
connected by a positive resistor. In another, we replace a hub nodewith two distributed nodes connected by a negative resistor. Fromthe simulation results, we can confirm that connection between
hubs plays an important role in network structure.

I. I NTRODUCTION
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.
Numerous researches on complex networks have been madeso far [1]-[7]. Especially, there are many studies related withhub nodes and these are very interesting. Because, in the realworld, large-scale complex networks often include huge nodes
with a large number of edges, and such “hub” nodes play key
roles in the networks.
In this study, we pay our attention on the behavior of
the networks whose hub nodes are split into several nodes.
Namely, when such a hub node is split into several nodes, howother nodes connected to the hub node change their behavior?
Furthermore, when such split hub nodes agree with or oppose
to each other, how the whole network behaves to cooperateor to compete? We construct the complex networks by using
coupled chaotic circuits and evaluate the behavior by usingsynchronization rate between a pair of the chaotic circuits. In
the first model, we replace a hub node with two distributed
hub nodes connected by a positive resistor. In the second
model, we replace a hub node with two distributed hub nodes
connected by a negative resistor. We evaluate these models
by synchronization rate and compare their behaviors. In distributed hubs connected by a positive resistor, we compare thesynchronization phenomena of the network whose hub nodereplaced with two distributed nodes and the network before
splitting a hub. Further, we investigate the effect of the network
structure around the hub node on the behavior of the wholenetwork.1

II. C IRCUIT MODEL
The chaotic circuit model which proposed by Shinriki etal. [9][10] is shown in Fig. 1. This circuit consists of a negative resistor, an inductor, two capacitors and dual-directional
diodes. In this study, we use a network in which each node isreplaced by a chaotic circuit. The circuit equation is shown inEq. (1).

Fig. 1.

Circuit model.

din

= v2nL
dt

1 ∑
dv1n
= gv1n − idn −(v1n − v1k )
C1dt
Rk∈S
n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).

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1 The extended version of this study is being reviewed for ISCAS’18 [8].

 Gd (n1n − v2n − V )0idn =Gd (n1n − v2n + V )(v1n − v2n > V )(|n1n − v2n | ≤ V )(v1n − v2n < −V ).

By changing the variables and parameters,
√
C2


V xn , v1n = V yn , v2n = V zni = nL√√C2LL
α=, β=Gd , γ =g,
C1C2C2

√

√

Ld
 δ= 1, t = LC2 τ, ” · ” = ,R C2dt

(2)III. N ETWORK MODEL
We made the underlying network which is 10 coupledchaotic cicuits and each circuit connected by a resistor. Thisnetwork model and the degree distribution of this network areshown in Fig. 3.

(3)

the normalized equations of chaos circuits are given as follows:ẋ = zn∑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 negativeresistor. The nonlinear function f (yn − zn ) corresponds tothe characteristics of the nonlinear resistor consisting of thediodes and described as follows: where f (y) is described asfollows : yn − zn − 1
f (yn − zn ) =0yn − zn + 1

Fig. 3.

In this study, we propose network model which a hub nodereplaced with two distributed nodes. The example of splitting
hub is shown in Fig. 4. Right network model in Fig. 5 is thenetwork that applied method shown in Fig. 4 to CC1 as a
hub of the Underlying network. Furthermore, we investigatedsynchronization of network in two different ways of connecting. In the first model, distributed hubs are connected by apositive resister and other nodes are connected by a positive
resistor each other. In the second model, distributed hubs areconnected by a negative resistor and other nodes are same at
the first model.

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

(5)

Fig. 4.

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.

Fig. 2.

AUnderlying network model.

Splitting a node.

Attractors with the different parameter.
Fig. 5.

- 66 -Network model with distributed hub.

IV. S IMULATION R ESULT
Definition of synchronization in this study is determined avoltage difference waveform. We define synchronization as thefollowing Eq. (6).

|zj − zi | < 0.15

(i, j = 1, 2, · · ·, 10)(6)

Fig. 7.

Fig. 6.

Average of difference in synchronization rate.

B. Synchronization phenomena of network with distributed
hub which connected by negative resistor

A difference waveform.

Figure 6 is a different waveform which was observed inthis simulation. The two lines in Fig. 6 correspond threshold
which is given Eq. (6). It is determined that wave within twolines 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 =1,000,000and step=0.02τ ) and investigate the synchronization rate in theentire network of 10 coupled chaotic circuits.
A. Comparing network with distributed hub and underlying
networkIn this simulation, we use two parameters of the strength ofnegative resistor γ. Attractors of these parameters are shownin Fig. 2. We compared synchronization rate of underlying network and network with distributed hub nodes whichconnected by positive resistor. Where the coupling strength
other than between distributed nodes is defined as δ and thecoupling strength between distributed nodes is defined as δ ′ .
In underlying network, we measured synchronization rate bychanging the coupling strength δ = 0.01 to δ = 0.1. In
network with distributed hub, we measured synchronization
rate by changing the coupling strength δ = 0.01 to δ = 0.01
and the coupling strength δ ′ = 0.02 to δ ′ = 0.2. Then we
calculated difference of synchronization rate of underlying
network and network with distributed hub nodes in eachcoupling strength δ. Figure 7 shows average of synchronization
rate of the coupling strength δ ′ in two parameters γ = 0.5
and γ = 0.65. In Fig. 7, we can observed that synchronization
rate is almost unchanged, even if a hub is replaced with two
distributed hub nodes.

In this section, we observed synchronization phenomena ofnetwork with distributed hub nodes which connected by negative resistor. We measured synchronization rate the network bychanging the coupling strength δ ′ = −0.02 to δ ′ = −0.2 in thecoupling strength δ = 0.1. Then, we grouped edges that are
similar in connection and we averaged synchronization rate foreach group. The first group is edges connected to CC1 which
is distributed hub. The second group is edges connected CC11
which is distributed hub. The third group is edges that have aclose relationship with node connected to CC1 of distributed
hub. The fourth group is edges that have a close relationshipwith node connected to CC11 of distributed hub. The fifthgroup is edges between node connected to CC1 and nodeconnected to CC11, that is, connection between nodes which
are coupled with nodes that are in anti-phase relationship.
Figure 8 shows averages of synchronization rate of eachgroup in the coupling strength δ ′ = −0.02 to δ ′ = −0.2.
In Fig. 8, we can observed that synchronization rates of thefirst group and the second group are similar and that of thethird group and the fourth group are similar. Synchronization
rates of the first group and the second group are lower as thenegative coupling is stronger. Synchronization rates of the thirdgroup and the fourth group are almost unchanged even if thenegative coupling changed. Synchronization rates of the fifthgroup are lower as the negative coupling is stronger, although
how to coupling of fifth group is similar to the third group andthe fourth group. This results are considered that fifth groupis connections between nodes which are coupled with nodes
that are in anti-phase relationship.

V. C ONCLUSIONSIn this study, we have investigated synchronization phenomena of network whose hub node is replaced with two
distributed hub nodes. Further, we connected distributed hubnodes in two ways. In the first model, distributed hub nodes- 67 -[5] K. Ago, Y. Uwate, Y. Nishio, “Investigation of Partical Synchronizationin Coupled Chaotic Circuit Network with Local Bridge”, Proc. of IEEE
Workshop on Nonlinear Cirsuit Networks (NCN’14), pp.63-66, Dec.
2014.
[6] T. Chikazawa, Y. Uwate, Y. Nishio, “Chaos Propagation in Coupled
Chaotic Circuits with Multi-Ring Combination”, Proc. of IEEE Asia
Pacific Conference on Circuits and Systems (APCCAS’16), pp. 65-68,
Oct. 2016.
[7] S. Hashimoto, T. Chikazawa, Y. Uwate, Y. Nishio, “Synchronization
Phenomena in Complex Networks of Coupled Chaotic Circuits withDifferent Degree Distribution”, Proc. of RISP International Workshop on
Nonlinear Circuits, Communications and Signal Processing (NCSP’17),
pp. 333-336, Feb. 2017.
[8] S. Hashimoto, Y. Uwate, Y. Nishio, “Synchronization Phenomena ofChaotic Circuit Networks with Distributed Hub Nodes Including Positiveand Negative Coupling”, IEEE International Symposium on Circuits andSystems(ISCAS), 2018. (being submitted)[9] M. Shinriki, M. Yamamoto and A. Mori, “Multimode Oscillations ina Modified van der Pol Oscillator Conraining a Positive NonlinearConductance”, Proc. IEEE, vol. 69, pp. 394-395, 1981.
[10] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena in a Curcuit witha Negative Resistance and an Ideal Switch of Diodes”, Trans. of IEICE,
vol.E70, no. 8, pp. 744-754, 1987.

Fig. 8.

Synchronization rate of each group.

connected by a positive resistor. In the second model, distributed hub nodes connected by a negative resistor.
In the result of the first model, when we compared synchronization rate of undelying network and network with
distributed hub, we can observed that synchronization rateis almost unchanged, even if a hub is replaced with two
distributed hub nodes. In the simulation of the second model,
we compare edges grouping similar connection. From thesimulation result, synchronization rates of group which isconnection between nodes which are coupled with nodes
that are in anti-phase relationship are lower as the negative
coupling is stronger.
As our future work, we investigate network with distributed
hub nodes coupled by negative resistor in detail. And we needto investigate synchronization phenomena of network withdistributed hub nodes in larger network.
R EFERENCES[1] D. J. Watts and S. H. Strogatz, “Collective Dynamics of Small-world”,
Nature, vol.393, pp.440-442, 1998.
[2] A. L. Barabasi and R. Albert, “Emergence of Scaling in Random
Networks”, Science, vol.286, pp.509-512, 1999.
[3] J. Aguirre, R. Sevilla-Escoboza, R. Gutirrez, D. Papo, and J.M. Buld,“Synchronization of Interconnected Networks: The Rolw of Connector
Nodes”, Phys. Rev. Lett. 112, 248701 Published 16 June 2014.
[4] K. Ago, Y. Uwate, Y. Nishio, “Influence of Bridge on Coupled Chaotic
Circuit Network”, Proc. of RISP International Workshop on Nonlinear
Circuits, Communications and Signal Processing (NCSP’14), pp. 417420, Mar. 2014.

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