平成 27 年度電気関係学会四国支部連合大会 講演論文集 (2015 高知工科大学)
2015 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (KOCHI)

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Synchronization Phenomena of Coupled Parametrically Excited Oscillators
Containing Small Mismatch in Random Network
Kosuke OI

Yoko UWATE
Yoshifumi NISHIO(Tokushima University)

1. Introduction

The method of applying α1 is expressed in Table 1.
The value of α10 from α2 is fixed as shown in Table 2.
Figure 2 shows the simulation results of mechanism of fullsynchronization. The horizontal axis denotes the couplingstrength δ, and the vertical axis denotes the parameter ofα express the threshold of full synchronization in network.
In this graph, the lower area of each line denotes fullsynchronous area and the upper area of each line denotes
unsynchronous area. In Fig. 2, there are two patterns offull synchronization when the value of α1 is random orzero. When δ is 0.5, there is a diﬀerence in α between α1for random and zero. By increasing the value of the coupling strength, the full synchronous state becomes large.
We can see that the spread of full synchronous state isdetermined by the small mismatch pattern.

In this study, we investigate the synchronization phenomena in random network model by using parametricallyexcited van der Pol oscillators containing small mismatch.
We apply this circuit model to ten coupled oscillators as
random network. We observe wherein become full synchronization in random network by using computer simulation.

2. Circuit model
The circuit model of parametrically excited van der
Pol oscillators is shown in Fig. 1 (a).

iidnv

CLTable 1: Small mismatch of α1 .

(a) Circuit model.

α1

w=1

w=2

w=3

w=4

w=5

Zero
Random0-0.00900.00800.0080-0.010-0.00992
hub

Table 2: Small mismatch of α10 from α2 .
8

α10 from α2

w=1

w=2

w=3

w=4

w=5

α2
α3
α4
α5
α6
α7
α8
α9
α10

-0.009-0.006-0.001
0.009-0.0020.003
0.007-0.001
0.005

0.008-0.006-0.0050.007-0.01-0.001-0.004
-0.01
0.009

0.008
0.009
-0.01
-0.01
-0.004-0.0030.005
0.009-0.009-0.01
0.009-0.003
0.001-0.0020.007-0.006-0.01-0.009-0.0090.01-0.007-0.001
0.004
0.009
0.007
0.005
-0.0011

37
106

45R(b) A random network model.

Figure 1: Circuit model and network model.
In Fig. 1, n express the circuit number of the network isshown in Fig. 2. The normalized circuit equations of thiscircuit equations are given by the following equations.
 dx
∑
n3

(xk − xn ) dτ = ε(xn − xn ) − yn + δk∈Sn(1)1 dyn =xndτ
γ(τ )

0.958
0.956
0.954
0.952
0.950
0.948
0.946
1=random

1=00.944

where n = 1, 2, 3, ..., 10. Sn is the set of nodes which are
directly connected to the node n. We apply the circuit
model to ten coupled oscillators as random network modelwith hub which is shown in Fig. 1 (b). In this study, thevalue of ε is fixed as 1.0.

0.942
0.940
0.938
0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

δ

3. Simulation results

Figure 2: Mechanism of full synchronization when α1 set toWe add the small mismatch to α which is corresponding to the amplitude of the function relating to parametrically excitation. The small mismatch is generated by random and the range of the mismatch is set to [-0.01:0.01].
In this study, first, we investigate an eﬀect of small mismatch of hub in random network by researching two cases
which the values of small mismatch are zero or random.
Next, we compare two patterns of full synchronization innetwork.

random and zero.

4. Conclusions
In this study, we have investigate synchronization phenomena of parametrically excited van der Pol oscillatorsin random network with hub by adding two types small
mismatch to hub of random network. In this result, wehave confirmed that the full synchronous state depends on
the small mismatch in hub.

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