Synchronization Phenomena of Complex Networks
by Using Parametrically Excited Oscillators
Kosuke Oi

Yoko Uwate

Yoshifumi Nishio

Dept. Electrical and Electronic Eng.
Tokushima University
Email: ooi@ee.tokushima-u.ac.jp

Dept. Electrical and Electronic Eng.
Tokushima University
Email: uwate@ee.tokushima-u.ac.jp

Dept. Electrical and Electronic Eng.
Tokushima University
Email: 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 on
synchronization phenomena of coupled oscillators are extensively carried out in various fields, physics [1], biology [2],engineering and so on. We consider that it is important toinvestigate the synchronization phenomena of coupled oscillators for the future engineering application. The coupled van der
Pol oscillator is one of coupled oscillators, and synchronization
generated in the system can model certain synchronizationof natural rhythm phenomena. The van der Pol oscillator
is studied well because it is expressed in simple circuit.
Parametric excitation circuit is one of resonant circuits, and
it is important to investigate various nonlinear phenomena ofthe 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. Additionallythe occurrence of chaos is referenced in Refs. [4] and [5].
In our research group, we have investigated synchronizationof parametrically excited van der Pol oscillators [7]. 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 applications. In
our previous study, we have challenged to investigate more
complex network using parametrically excited van der Pol
oscillators with small mismatch. First, two oscillators are
combined by resisters in one-dimensional coordinate system.
We have investigated synchronization between two oscillator
by changing the value of coupling strength. We also haveinvestigated synchronization phenomena of complex networkby applying this circuit model to ten coupled oscillators as
random network model with hub.
In this study, we compare synchronization phenomena ofparametrically excited van der Pol oscillators with small
mismatch in complex network with hub and without hub inorder to investigate effect of hub for the synchronization andclustering.
II. C IRCUIT M ODEL
Figure 1 shows the parametrically excited van der Poloscillator.
iidnv

CFig. 1.

L

Parametrically excited van der Pol oscillator.

1+
( )

Abstract— Recently, many studies of synchronization of chaoticcircuits have been investigated. Synchronization phenomenon is
one of the typical phenomena observed in nature. And, it isapplicable to the fields of medical science and biology and soon. In this study, we investigate synchronization phenomena ofcomplex networks by using parametrically excited oscillators.

11-
0Fig. 2.

/

2 /



Function relating to parametrically excitation.

The circuit includes a time-varying inductor L whose characteristic is given as the following equation. The operation ofthe time-varying 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 .

(2)

The normalized circuit equations are given by the followingequations.
∑dxn= ε(xn − x3n ) − yn + δ(xk − xn ) dτk∈Sn(3)dyn1
=xn(n = 1, 2, 3, · · · , 10))dτγ(τ )Sn is the set of nodes which are directly connected to the noden.
III. R ANDOM N ETWORK W ITH H UB
We apply the circuit model to ten coupled oscillators as
random network model with hub. The ten coupled oscillator
model is shown in Fig. 3. In this circuit system, there is a hub(1st oscillator) which is connected to many oscillators. Here,the small mismatch is added to α which is corresponding to theamplitude of the function relating to parametrically excitation.
The mismatch is generated by random and the range of themismatch is set to [-0.01:0.01]. Table I shows the patternof five types of the small mismatch used in this computer
simulations.
92hub

8

1

3

7
10

6

4

R

5Fig. 3.

A random network model with a hub.

IV. R ANDOM N ETWORK W ITHOUT H UB
Next, we focus on comparison with synchronous state ofthe random network without hub. Figure 5 shows a random
network model without hub. We investigate the synchronous
state of random network model without hub in order tocompare with synchronous state of random network modelwith hub. Table II shows the pattern of five types of thesmall mismatch without hub used in this computer simulations.
Figure 6 shows synchronous state of random network model
without hub. In this graph, the lower area of each line denotes
synchronous area and the upper area of each line denotes
unsynchronous area in common with Fig. 4. In Fig. 7, wedraw a comparison between Fig 4 and Fig 6. The synchronous
state of random network model without hub is smaller than
the synchronous state of random network model with hub. In
these results, we have confirmed that the synchronous state
depends on the network structure.

TABLE I
w=10
-0.009-0.006-0.001
0.009-0.00200-0.001
0.005

w=200.008-0.006-0.0050.007-0.01-0.001-0.004
-0.01
0.009

w=3
00.008
0.009
-0.01
-0.01
-0.004-0.0030.005
0.009-0.00992

SMALL MISMATCH

small mismatch
α1α2α3α4α5α6α7α8α9α10Synchronous state with δ.

Fig. 4.

w=40-0.01
0.009-0.0030.001-0.0020.007-0.006-0.01-0.009w=50-0.009
0.01-0.007-0.001
0.004
0.009
0.007
0.005
-0.001

81

37
6

4

10

5Fig. 5.

RA random network model with a hub.

V. C LUSTERING P HENOMENA
Figure 4 shows the simulation results. The horizontal axis
denotes the coupling strength δ, and the vertical axis denotes
α of the second oscillator. In this graph, the lower area of eachline denotes synchronous area and the upper area of each line
denotes unsynchronous area. By increasing the value of thecoupling strength, the synchronous area becomes large. We
can see that the synchronous area is determined by the small
mismatch pattern. Namely, several types of synchronization
states can be observed from ten coupled oscillators with thesmall mismatch.

We investigate clustering phenomena of ten coupled oscillators as random network with hub and without hub. Figure
8 shows the clustering phenomena (how do the oscillatorschange the fully synchronization state) in random networkwith hub and without hub when the coupling strength ischanged. The horizontal axis is the coupling strength, and thevertical axis denotes the node number between the oscillators.
By comparison Fig. 8 (a). and Fig. 8 (b), we confirm thedifference of clustering phenomena. In Fig. 8 (a), the network
completely synchronize in δ = 0.60. However, in Fig. 8 (b),TABLE II
SMALL MISMATCH ( WITHOUT HUB )small mismatch
α1α2α3α4α5α6α7α8α9α10

w=10-0.009-0.006-0.001
0.009-0.002
00-0.0010.005

w=2-0.0010.008-0.006-0.0050.007-0.01-0.001-0.004
-0.01
0.009

w=3
0.009
0.008
0.009
-0.01
-0.01
-0.004-0.0030.005
0.009-0.009

w=4
0.003
-0.01
0.009
-0.0030.001-0.0020.007-0.006-0.01-0.009w=5-0.006-0.0090.01-0.007-0.001
0.004
0.009
0.007
0.005
-0.001
Fig. 7.

Comparison of synchronous state.

510

Circuit number

2
3
6
4
98
7
0.60

0.38

0.30

0.02
0.07
0.09
0.24

1



(a) Clustering phenomena with hub
Fig. 6.

Synchronous state with δ (without hub).

5VI. C ONCLUSIONSIn this study, we have investigate synchronization phenomena of parametrically excited van der Pol oscillators withsmall mismatch. In the case of two subcircuits, we haveconfirmed that the two subcircuits are synchronized at inphase state when the adding mismatch is small. By increasing
the small mismatch, we observe unsynchronous phenomena.
Furthermore, we have applied this circuit model to ten coupledoscillators as random network model with hub and random
network model without hub. In this result, we have confirmedthat the synchronous state depend on the net work structure.
For the future work, we would like to consider the influenceof the small mismatch for synchronization state of other typenetwork or more large scale networks.
ACKNOWLEDGMENT
This work was partly supported by JSPS Grant-in-Aid for
Challenging Exploratory Research 26540127.

3246
970.70

0.60

0.35

0.10

10.08

80.06

network completely synchronize in δ = 0.70. Basically, thepresence or absence of hub effect of process of completesynchronization. In addition, we characterize Fig. 9 and Fig. 10in order to investigate more further process of complete
synchronization. In Fig, 9 (c), the value of bond number forsynchronization is 12. However in Fig. 10 (c), the value ofbond number for synchronization is only 5. In these results,
clustering phenomena depend on the presence or absence ofhub.

Circuit number10(b) Clustering phenomena without hub
Fig. 8.

Clustering phenomena with δ (w=1).

R EFERENCES[1] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synchronization andgraph topology,” Int. J. Bifurcation and Chaos, vol.15, no.11, pp.34233433, Nov. 2005.
[2] J. Cosp, J. Madrenas, E. Alarcon, E. Vidal and G. Villar, “Synchronizationof nonlinear electronic oscillators for neural computation,” IEEE Trans.
Neural Networks, vol.15, no.5, pp.1315-1327, Sep. 2004.
[3] C. Hayashi, “Nonlinear Oscillations in Physical Systems,” Chap. 11,
McGraw-Hill, New York (1964).
[4] C. Hayashi, M. Abe, K. Oshima and H. Kawakami, “The method ofmapping as applied to the solution for certain types of nonlinear differential equations,” Ninth International Conference on Nonlinear Oscillations,Kiev (Aug.-Sept.1981).
[5] M. Inoue, “A Method of Analysis for the Bifurcation of the Almost Periodic Oscillation and the Generation of Chaos in a Parametric Excitation
Circuit,” Trans. of IEICE, vol. J68-A, no. 7, pp. 621-626, 1985.

(a) δ = 0.01(b) δ = 0.10

92
hub

813

710

64

:synchronous state5(c) δ = 0.25
Fig. 9.

(d) δ = 0.60
Formation process of cluster (with hub).

:synchronous state(a) δ = 0.01

:synchronous state(b) δ = 0.10

:synchronous state(c) δ = 0.25
Fig. 10.

:synchronous state(d) δ = 0.60
Formation process of cluster (without hub).

[6] H. Kumeno, Y. Nishio, “Synchronization Phenomena in Coupled Parametrically Excited van der Pol Oscillators,” Proc. NOLTA’08, pp. 128-131,
Sep. 2008.
[7] K. Oi, Y. Uwate, Y. Nishio, “Synchronization and Clustering in Coupled Parametrically Excited Oscillators with Small Mismatch,” Proc.
ISCAS’15, May. 2015 (accepted).