IEEE Workshop on Nonlinear Circuit Networks
December 11-12, 2015

Effect of the Hub in Complex Networksof Coupled Parametrically Excited Oscillatorswith Dispersion
Kosuke Oi, Kenta Ago, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {ooi,kentago,uwate,nishio}@ee.tokushima-u.ac.jp

I. I NTRODUCTION
Synchronization is one of the fundamental phenomena innature and it is observed over the various fields. Studies onsynchronization phenomena of coupled oscillators are extensively carried out in various fields, physics [1], biology [2],engineering and so on. The coupled van der Pol oscillator
is one of coupled oscillators, and synchronization generated
in the system can model certain synchronization of natural
rhythm phenomena. Parametric excitation circuit is one ofresonant circuits, and it is important to investigate various
nonlinear phenomena of the parametric excitation circuits for
future engineering applications. In simple oscillator including
parametric excitation, Ref. [3] reports that the almost periodic
oscillation occurs if nonlinear inductor has saturation characteristic. Additionally the occurrence of chaos is referenced inRefs. [4] and [5].
In our research group, we have investigated synchronizationof parametrically excited van der Pol oscillators [6]. By
carrying out computer calculations for two or three subcircuits
case, we have confirmed that various kinds of synchronization phenomena of chaos are observed. In the case of two
subcircuits, the anti-phase synchronization is observed. In
the case of three subcircuits, self-switching phenomenon ofsynchronization states is observed.
However, we have investigated the only simple networkmodels. It is important to investigate more complex network
for the broad-ranging future engineering application. In thisstudy, we challenge to investigate more complex network
using parametrically excited van der Pol oscillators with small
mismatch.
In our previous study, we have challenged to investigate
the synchronization and clustering in more complex network

using parametrically excited van der Pol oscillators with small
mismatch [7].
In this study, first, we compare synchronization phenomenaof parametrically excited van der Pol oscillators with small
mismatch in complex network with hub and without hub
in order to investigate effect of the hub in network of thesynchronization.
Next, we construct a larger scale network modified fromdolphin social network [8] and we investigate synchronizationand clustering phenomena of this network.
II. CIRCUIT MODEL
The circuit model of van der Pol oscillator under parametrically excitation is shown in Fig. 1.

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idnv

CLFig. 1. Circuit model.

1+
( )

Abstract—In this study, we investigate synchronization andclustering in several types complex network by using parametrically excited van der Pol oscillators including small mismatch. In
the case of random network of ten coupled oscillator with hubor without hub, we confirm the difference of synchronizationand clustering between the networks with hub and withouthub. Furthermore, we apply this circuit model to a larger scale
network modified from dolphin social network, and investigate
synchronization and clustering phenomena.

11-
0/

2 /

Fig. 2. Function relating to parametric excitation.

The circuit includes a time-varying inductor L whose characteristics are given as the following equation. The timevarying inductor is shown as Fig. 2.
L = L0 γ(τ ).

(1)γ(τ ) is expressed in a rectangular wave as shown in Fig. 2,
and its amplitude and angular frequency are termed α and ω,
respectively. The v − i characteristics of the nonlinear resistor
are approximated by the following equation.
id = −g1 vk + g3 vk .
By changing the variables and the parameters,√√g1
xnLCτ,v=t=0ng3√ω = ω0 L0 C√ √g1 Cynin =
g3 L0
√√
LL10 ε = g1, δ=CR C(2)(3)

A. Random network of ten coupled oscillator
In this section, we compare the number of synchronizationof each edges in random network with hub and without hub.
This network model is shown in Fig. 3. The 1st oscillator inFig. 3 (a) is the hub. The hub has the largest bond numberin the network and there is no dispersion in hub. There is nodispersion the 1st oscillator in Fig. 3 (b), however this is notthe hub because there is no largest bond number in the 1stoscillator. In the both of the random network, the number ofnode is 10 and the number of edge is 16. Each of the nodes
is connected by a resister. Figures 4 and 5 show attractors of
the nodes (when αn = 0.958 + mn , δ = 1.3 Sn = 10 inrandom network with hub) and lissajous of each internodes
for instance when full synchronous state.
Figure 6 shows the comparison the percent of the bondnumber. The horizontal axis denotes the coupling strength δand the vertical axis denotes the percent of the bond numbers[%]. This percent is 100 [%] when the network become fullsynchronous state. In this figure, we can observe the percentof the bond number become 100 [%] when δ = 0.7 in random
network without hub whereas the percent become 100[%] inrandom network with hub when δ = 1.0. From the above,
existence of the hub cut out to become full synchronization inthis case.

The normalized circuit equations are given by the following
equations.
∑dxn
(xn − xk )= ε(xn − x3n ) − yn − δ dτk∈Sn(4)dyn1

=xndτ
γ(τ )
where n is the count of the nodes. Sn is the set of the nodes
which are directly connected to the node n.

hub7
6

410

R5

In this research, we fix the the circuit parameters as ε = 1and ω = 1.00 for all circuits. Each circuit is given different
initial values for computer simulations.In this simulation,
all of the nodes without hub involve dispersion mn in α
(in Fig. 2) which is corresponding to the amplitude of thefunction relating parametric excitation within the compass of−0.01 ≤ mn ≤ 0.01.
In order to analyze synchronization states, we define thesynchronization by the following equation:(k ∈ Sn ).

813III. S IMULATION METHOD|xn − xk | < 0.1092(5)We fix the count of calculation as 100,000.

(a) With hub.

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1

3

7
10

6

45IV. EFFECT OF HUB IN COMPLEX NETWORKIn this section, we investigate effect of the hub in several
types complex networks contain dispersion using parametrically excited van der Pol oscillators by observing synchronization and clustering phenomena.

- 12 -(b) Without hub.
Fig. 3. Random network model.

Rx3x2x4x1x3x1x1x1x6x4x4x2x9x1x1x7x6x4x10x1x10x3x2x8x7x5x9x5x4x1x6x7

Fig. 4. Lissajous of each internodes.

x1x2x3

5x5x4

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223656y1y2y3y4

19y525

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44

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Fig. 5. Attractors when αn = 0.958 + mn , δ = 1.3 Sn = 10 in random
network with hub.

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1 21262714613223

Percent of the bond number [%]80Fig. 7. Complex network modified from ”dolphin social network”.

70605040

With hub

30

Without hub

20

100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0



Fig. 6. Comparison the percent of the bond number.

B. Large scale complex network
Next, we construct a larger scale complex network modified
from dolphin social network [8] and we investigate synchronization and clustering phenomena of this network.
Figure 7 shows the complex network modified from dolphinsocial network. The number of nodes is 62 and edges is 99.
This network is aligned average degree with the network ofFig. 3 (the average degree is about 3.20). The 15th node, the38th node and the 46th node are the hub. All of the hub has
no dispersion. Each of the nodes is connected by a resister.
The bond number of the hub is 11. There is coexistence of thenetwork with hub without hub in this network. In this section,
we investigate effect of more than one hub on synchronizationin the large scale complex network.
Figure 8 shows formation process of cluster by increasing
the coupling strength δ. The nodes colored express cluster.
When δ = 0.01, the formation of cluster sprawls aroundthe hub. From here onwards, the existence of more than
one hub makes it an open possibility to accomplishment ofsynchronization. When δ = 0.0175, this network is divided
into two clusters. When δ = 0.0180, this network becomesfull synchronous state.
From the above, we can confirm that the network with morethan one hub can induce synchronization and by increasing allof coupling strength, we can observe clustering phenomena.
V. C ONCLUSIONSIn this study, we have investigated effect of the hub inseveral types complex networks contain dispersion synchronization and clustering phenomena. In the case of random
network of ten coupled oscillator, we have confirmed that theexistence of the hub cut out to become full synchronization.

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For the future work, we would like to consider more elaborate investigation about clustering phenomena in the networkof Fig. 8.

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R EFERENCES

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(a) δ = 0.01

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43

31

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2332(b) δ = 0.0175

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4
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1 21
40

49
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7354

62

43

31

855
42

620

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1018
33

48

258

59
112950

35

26

27

1461

3223(c) δ = 0.0180
Fig. 8. Formation process of cluster.

Furthermore, we applied this circuit to a larger complex
network modified from dolphin social network. In this case
we have confirmed that the network with more than one hub
can induce synchronization and by increasing all of couplingstrength, we can observe clustering phenomena.

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