2015 International Symposium on Nonlinear Theory and its Applications
NOLTA2015, Kowloon, Hong Kong, China, December 1-4, 2015

Synchronization Phenomena in Complex Networksby Using Parametrically Excited Oscillatorswith Involvement of Small Mismatch
Kosuke Oi† , Yoko Uwate† and Yoshifumi Nishio††Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minamijosanjima, Tokushima, 770-8506, Japan
Email: {ooi, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract— Investigation of synchronization phenomena in chaotic oscillator is expected to application tophysics, biology, medical science, and engineering equivalent. In this study, we focus on chaotic oscillation by applying parametrically excite to van der Pol oscillator, and apply small mismatch. We investigate synchronization phenomena in random network with hub by using computer
simulation. In addition, we investigate more detailed effects of small mismatch added in hub. As a result, the small
mismatch to hub has eﬀect on synchronization phenomena
in network.
1. Introduction
Synchronization is one of the fundamental phenomena
in nature 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 isimportant to investigate the synchronization phenomena ofcoupled oscillators for the future engineering application.
The coupled van der Pol oscillator is one of coupled oscillators, and synchronization generated in the system canmodel certain synchronization of natural rhythm phenomena. The van der Pol oscillator is studied well because itis expressed in simple circuit. Parametric excitation circuit
is one of resonant 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 thatthe almost periodic oscillation occurs if nonlinear inductor
has saturation characteristic. Additionally the occurrenceof chaos is referenced in Refs. [4] and [5].
In our research group, we have investigated synchronization of parametrically excited van der Pol oscillators [6].
By carrying out computer calculations for two or threesubcircuits case, we have confirmed that various kinds ofsynchronization phenomena of chaos are observed. In thecase of two subcircuits, the anti-phase synchronization isobserved. In the case of three subcircuits, self-switching
phenomenon of synchronization states is observed.
However, we have investigated the only simple networkmodels. It is important to investigate more complex net-work for the broad-ranging future engineering applications.
In our previous study, we investigated synchronization phenomena in random network by using parametrically excited
van der Pol oscillators with small mismatch [7]. First, twooscillators are combined by resisters in one-dimensional
coordinate system. We have investigated synchronization
between two oscillators by changing the value of couplingstrength. We also investigated synchronization phenomena of complex network by applying this circuit model toten coupled oscillators as random network model with hub
which include small mismatch in amplitude of parametrically excitation to each of node. We focus on eﬀect of small
mismatch of hub in random network through this previous
study by reason of we expect this mismatch make an eﬀect
to synchronization phenomena in random network.
In this study, we investigate full synchronization in random network by adding several types of small mismatch
to hub of random network. We confirm that the full synchronous state depends on the small mismatch in hub.
2. Circuit Model
Figure 1 shows the parametrically excited van der Poloscillator.
iidnCv

L

Figure 1: Parametrically excited van der Pol oscillator.

This circuit includes a time-varying inductor L whose characteristic is given as the following equation.
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 andω, respectively. The α expresses the amplitude of function- 439 -921+
( )

hub

70/

2 /

Figure 3: A random network model with a hub.

relating to parametrically excitation. The v − i characteristics of the nonlinear resistor are approximated by the following equation.

(2)

The normalized circuit equations are given by the followingequations.
==ε(xn − xn3 ) − yn + δ1xnγ(τ)∑(xk − xn )

k∈S n

R

5Figure 2: Function relating to parametrically excitation.

id = −g1 vk + g3 vk .

64101-dxn

 dτ

dyn


dτ

8

1

31(3)

(n = 1, 2, 3, · · · , 10))S n is the set of nodes which are directly connected to thenode n.
3. Random Network with Hub
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 whichis corresponding to the amplitude of the function relating
to parametrically excitation. We investigate eﬀect of small
mismatch of hub in random network by considering threecases (Mismatch of hub: zero, random and constant). The
small mismatch is generated by random and the range ofthe mismatch is set to [-0.01:0.01]. Each table (Table 1,Table 2 and Table 3) shows small mismatch pattern which
is used in this computer simulations. In these Tables, w expresses the pattern of small mismatch. In Table 1, the valueof small mismatch in hub (α1 ) is zero. In Table 2, the valueof α1 is random. In Table 3, the value of α1 is fixed with0.01.
Figure 4 shows attractors and lissajous of each node andinternode when w=1, α = 0.958 and δ = 1.3. This figure
expresses a part of example of full synchronization. Figure 5 shows the simulation results of are mechanism ofsynchronization. The horizontal axis denotes the couplingstrength δ, and the vertical axis denotes the parameter of α
expresses the threshold of full synchronization in network.

In this graph, the lower area of each line denotes full synchronous area and the upper area of each line denotes unsynchronous area. By increasing the value of the couplingstrength, the full synchronous area becomes large. Namely,
several types of full synchronization states can be observed
from Fig. 5. We can see that the spread of full synchronous
area is determined by the small mismatch pattern.
Next, we compare process of full synchronization bycontriving ways to add small mismatch. We add small mismatch to all nodes include hub or except hub. Figure 6
shows an average of mechanism spreading full synchronization of three types. In this graph, the lower area of
each line denotes full synchronous area and the upper areaof each line denotes unsynchronous area in common with
Fig. 5. In Fig. 6. (b) and Fig. 6. (c), become full synchronization in smaller value of α than Fig. 6. (a), andwe confirm the diﬀerence of process of spreading full synchronization. In these results, we have confirmed that thefull synchronous state depends on the pattern of small mismatch.

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Table 1: Mismatch of hub: zero.
small mismatch
α1
α2
α3
α4
α5
α6
α7
α8
α9
α10

w=1
0
-0.009-0.006-0.001
0.009-0.0020.003
0.007-0.0010.005

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
0.009-0.009w=40-0.01
0.009-0.0030.001-0.002
0.007-0.006-0.01-0.009w=50-0.009
0.01-0.007-0.001
0.004
0.009
0.007
0.005
-0.001

Table 2: Mismatch of hub: random.
small mismatch
α1
α2
α3
α4
α5
α6
α7
α8
α9
α10

w=1
-0.005-0.009-0.006-0.001
0.009-0.0020.003
0.007-0.0010.005

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
0.009-0.009w=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

x3x2x1y2

y1x5x4y4

y3

y6

y5x8x7x6

y7x9x10y9y8y10(a) Attractors of each node.

x2x3x5x4x1x1x4x6x3x7x1
x10x1x6x4x1x7x4x9x8x1x10x1x3
x1x4x2x2x9x5x6x7(b) Lissajous of each internode.

Figure 4: Attractors and lissajous of each node and internode when w=1.

Table 3: Mismatch of hub: constant.
small mismatch
α1
α2
α3
α4
α5
α6
α7
α8
α9
α10

w=1
0.01
-0.009-0.006-0.001
0.009-0.0020.003
0.007-0.001
0.005

w=2
0.01
0.008
-0.006-0.0050.007-0.01-0.001-0.004
-0.01
0.009

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

w=4
0.01
-0.01
0.009
-0.0030.001-0.0020.007-0.006-0.01-0.009w=5
0.01-0.009
0.01-0.007-0.001
0.004
0.009
0.007
0.005
-0.001

4. Conclusions
In this study, we have investigated synchronization phenomena of parametrically excited van der Pol oscillatorsin random network with hub by adding several types small
mismatch to hub of random network in order to investigate
more detailed eﬀects of small mismatch of hub in random
network. In this result, we have confirmed that the full synchronous state depends on the small mismatch in hub. For
the future work, we would like to consider the influenceof the small mismatch for synchronization state of anothertype and more large scale network structure.

[2] J. Cosp, J. Madrenas, E. Alarcon, E. Vidal and G. Villar, “Synchronization of 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 of mapping 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 Bifurcationof the Almost Periodic Oscillation and the Generationof Chaos in a Parametric Excitation Circuit,” Trans. ofIEICE, vol. J68-A, no. 7, pp. 621-626, 1985.
[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 andClustering in Coupled Parametrically Excited Oscillators with Small Mismatch,” Proc. ISCAS’15, pp. 910913, May, 2015.

References[1] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer,“Synchronization and graph topology,” Int. J. Bifurcation and Chaos, vol.15, no.11, pp.3423-3433, Nov.
2005.

- 441 -0.958
0.958

0.956
0.956

0.954
0.954

0.952
0.950

0.950

αα

0.952

w=10.948

w=20.946

0.948
0.946
w=30.944

0.944
w=40.942

0.942

w=50.940

0.940

0.938

0.9380.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0.5

1.3

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.1

1.2

1.3

1.2

1.3

δ

δ

(a) Mismatch of hub: zero.

(a) Mismatch of hub: zero.

0.958

0.958

0.956

0.956

0.954

0.954

0.952

0.952

0.950

α

α

0.950

w=1
0.948

0.948

w=2
0.946

0.946

w=3
0.944
0.942

w=4

0.944

w=5

0.942

0.940

0.940

0.938
0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0.938

1.3

0.5

δ

0.6

0.7

0.8

0.9

1.0

δ

(b) Mismatch of hub: random.

(b) Mismatch of hub: random.

0.958

0.958

0.956

0.956

0.954

0.954

0.952

0.952

0.950

αα

0.950

w=10.948

0.948

w=2
0.946

0.946

w=3
0.944

0.944

w=4
0.942

0.942

w=5
0.940

0.940

0.938

0.938
0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

0.5

δ

0.6

0.7

0.8

0.9

1.0

1.1

δ

(c) Mismatch of hub: constant.

(c) Mismatch of hub: constant.

Figure 5: Mechanism of synchronization when α1 = 0.

Figure 6: Average of mechanism of synchronization.

- 442 -