Synchronization Phenomena of Chaotic Circuit
Networks with Distributed Hub Including Positive
and Negative Coupling
Shuhei Hashimoto, Yoko Uwate, and Yoshifumi Nishio
Dept. of Electrical and Electronic Eng.,
Tokushima University
2-1 Minami-Josanjima,
Tokushima 770-8506, Japan
Email:{s-hashimoto, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract—In this study, we focus on network structure with a
hub of Scale-free network. We consider the synchronization ofcoupled chaotic circuit network whose hub nodes are split twonodes and we investigate two network models. In one of them,
we replace a hub node with two distributed nodes connectedby a positive resistor. In another, we replace a hub node withtwo distributed nodes connected by a negative resistor. From thesimulation 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, ﬂashingﬁreﬂy 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 ofthe 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 secondmodel, 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 dis-978-1-5386-4881-0/18/$31.00 ©2018 IEEEtributed 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.
Next, as the third model, we replace a hub node withthree distributed hub nodes. We connect three distributed
hub nodes mutually and we make that connection ring hub.
In this study, we investigate a network of three distributed
nodes connected by positive/positive/negative resistors andpositive/negative/negative resistors. We investigate the effectof the connection type on the behavior of the whole network.
II. C IRCUIT MODEL
The chaotic circuit model which proposed by Shinriki etal. [8][9] 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⎪
⎪
L
= v2n
⎪
⎪
dt
⎪
⎪
⎪
⎪
⎨
dv1n1 C1
(v1n − v1k )= gv1n − idn −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 )0idn =⎩
Gd (n1n − v2n + V )(v1n − v2n > V )(|n1n − v2n | ≤ V )(v1n − v2n < −V ).

By changing the variables and parameters,
⎧C2⎪⎪⎪ in =V xn , v1n = V yn , v2n = V zn⎪
⎪L⎪⎪⎪⎨
LLC2α=, β=Gd , γ =g,⎪CCC⎪122⎪⎪⎪⎪⎪dL⎪⎩ δ= 1, t = LC2 τ, ” · ” = ,R C2dt(2)(3)
Fig. 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 γ is the bifurcation parameter ofchaotic circuit. The nonlinear function f (yn −zn ) correspondsto the characteristics of the nonlinear resistor consisting of thediodes and described as follows:⎧⎪⎨ yn − zn − 1
f (yn − zn ) =0⎪⎩
yn − zn + 1

III. N ETWORK MODEL
We made the basic network which is 10 coupled chaotic
circuits and each circuit connected by a resistor. This networkmodel and the degree distribution of this network are shownin 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 hubof the basic network. Furthermore, we investigated synchronization of network in two different ways of connecting. In
the ﬁrst model, distributed hubs are connected by a positive
resister and other nodes are connected by a positive resistor
each other. In the second model, distributed hubs are connectedby a negative resistor and other nodes are same at the ﬁrstmodel.

(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.

Basic network model.

Splitting a node.

Attractors with the different parameter.
Fig. 5.

978-1-5386-4881-0/18/$31.00 ©2018 IEEE

Neteork model with distributed hub.

IV. S IMULATION R ESULT
Deﬁnition of synchronization in this study is determined a
voltage difference waveform. We deﬁne synchronization as thefollowing Eq. (6).
|zj − zi | < 0.15

(i, j = 1, 2, · · ·, 10)(6)

Fig. 7.

Fig. 6.

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 =1,000,000
and step=0.02τ ) and investigate the synchronization rate in theentire network of 10 coupled chaotic circuits.
A. Comparing network with distributed hub and basic network
In this simulation, we use two bifurcation parameters ofchaotic circuit γ. Attractors of these parameters are shown in
Fig. 2. We compared synchronization rate of basic networkand network with distributed hub nodes which connectedby positive resistor. Where the coupling strength other thanbetween distributed hub nodes is deﬁned as δ and the couplingstrength between distributed nodes is deﬁned as δ  . In basicnetwork, we measured synchronization rate by changing thecoupling strength δ = 0.01 to δ = 0.1. In network withdistributed hub, we measured synchronization rate by changing
the coupling strength δ = 0.01 to δ = 0.01 and the couplingstrength δ  = 0.02 to δ  = 0.2. Then we calculated differenceof synchronization rate of basic network and network with
distributed hub nodes in each coupling strength δ. Figure 7
shows average of synchronization rate of the coupling strengthδ  in two parameters γ = 0.5 and γ = 0.65. In Fig. 7, we canobserved that synchronization rate is almost unchanged, even
if a hub is replaced with two distributed hub nodes.
B. Synchronization phenomena of network with distributed
hub which connected by negative resistor
In this section, we observe synchronization phenomena ofnetwork with distributed hub nodes which connected by negative resistor. We measure synchronization rate of the networkby changing the coupling strength δ  = −0.02 to δ  = −0.2
in the coupling strength δ = 0.1. Then, we grouped edges thatare similar in connection and we averaged synchronization

978-1-5386-4881-0/18/$31.00 ©2018 IEEE

Average of difference in synchronization rate.

rate for each group. The ﬁrst group is edges connected toCC1 which is distributed hub. The second group is edges
connected CC11 which is distributed hub. The third groupis edges between the nodes connected to CC1 of distributed
hub. The fourth group is edges between the nodes connected
to CC11 of distributed hub. The ﬁfth group is edges between
node connected to CC1 and node connected to CC11, that is,connection between nodes which are coupled with nodes thatare in asynchronous 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 asynchronous relationship.
C. Synchronization phenomena of extended network with ringhub
In this section, we investigated synchronization phenomenaof network which have three distributed hub nodes in extended
network (N = 29). Three distributed hub nodes are connetedeach other and they are connected by positive/positive/negative
resistors, positive/negative/negative resistors.
Figure 9 shows network model which used in this section.
We grouped edges that are similar in connection with theprevious section and we measure average synchronization rate
for each group. The ﬁrst group is edges connected to CC1
which is distributed hub. The second group is edges connected
CC28 which is distributed hub. The Third group is edges
connected to CC29 which is distributed hub. The fourth groupis the edges between the nodes connected to CC1 of thedistributed hub node. The ﬁfth group is the edges between

connected by negative resistor. From these results, groupsof edge which connected to distributed hub with negative
resistors obtained low synchronization rate like ten coupledchaotic circuits network.

Synchronization rate of each group in extended network(positive/positive/negative).

Fig. 10.

Fig. 8.

Synchronization rate of each group.

Synchronization rate of each group in extended network(positive/negative/negative).

Fig. 11.

V. C ONCLUSIONS

Fig. 9.

Extended network model (N = 29).

the nodes connected to CC of the distributed hub node. The
sixth group is edges between node connected to CC1 and nodeconnected to CC28. The seventh group is edges between nodeconnected to CC1 and node connected to CC29. The eighthgroup is edges between node connected to CC28 and nodeconnected to CC29.
Figure 10 and 11 show average of synchronization rate ofeach group. In Fig. 10, CC1 and CC28, CC1 and CC29 areconnected by positive resistor, CC28 and CC29 are connectedby negative resistor. In Fig. 11, CC28 and CC29 are connectedby positive resistor, CC1 and CC28, CC1 and CC29 are

978-1-5386-4881-0/18/$31.00 ©2018 IEEE

In this study, we have investigated synchronization phenomena of network whose hub node is replaced with twodistributed hub nodes. Furthermore, we connected distributed
hub nodes in two ways. In the ﬁrst model, distributed hubnodes 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 withdistributed 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 thatare in asynchronous relationship are lower as the negative
coupling is stronger. Furthermore, in the extended model, we
can conﬁrmed results similar to ten chaotic circuits network.
However, we need further investigation because these results
may change depending on the initial value.
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 with
distributed hub nodes in larger networks.

ACKNOWLEDGMENT
This work was partly supported by JSPS Grant-in-Aid for
Scientiﬁc Research 16K06357.
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 ConnectorNodes”, Phys. Rev. Lett. 112, 248701 Published 16 June 2014.
[4] K. Ago, Y. Uwate, Y. Nishio, “Inﬂuence of Bridge on Coupled ChaoticCircuit Network”, Proc. of RISP International Workshop on Nonlinear
Circuits, Communications and Signal Processing (NCSP’14), pp. 417420, Mar. 2014.
[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] 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.
[9] 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.

978-1-5386-4881-0/18/$31.00 ©2018 IEEE