Synchronization and Clusteringin Coupled Parametrically Excited Oscillatorswith Small Mismatch
Kosuke Oi, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {ooi,uwate,nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, we investigate synchronization ofparametrically excited van der Pol oscillators with small mismatch. In the case of two subcircuits, we confirm that the twosubcircuits are synchronized at in-phase state when the adding
mismatch is small. By increasing the small mismatch, we observe
unsynchronous phenomena. Furthermore, we apply this circuit
model to ten coupled oscillators as random network model withhub.

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 of
the parametric excitation circuits for future engineering applications. In simple oscillator including parametric excitation,Ref. [3] reports that the almost periodic oscillation occurs ifnonlinear inductor has saturation characteristic. Additionallythe occurrence of chaos is referenced in Refs. [4] and [5].
In our previous study, 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 twosubcircuits, the anti-phase synchronization is observed. In
the case of three subcircuits, self-switching phenomenon ofsynchronization states is observed.
However, in the previous study, we have investigated theonly simple network models. It is important to investigate more
complex network for the broad-ranging future engineering
application. In this study, we challenge to investigate more
complex network using parametrically excited van der Pol
oscillators with small mismatch.

978-1-4799-8391-9/15/$31.00 ©2015 IEEEFirst, we investigate synchronization of parametrically excited van der Pol oscillators with small mismatch. The small
mismatch is added to the amplitude of the function relating
to parametrically excitation. In the case of two subcircuits,
we confirm 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 apply this circuit model to ten coupledoscillators as random network model with hub.
II. VAN DER P OL OSCILLATORUNDER PARAMETRIC EXCITATIONThe circuit model of van der Pol oscillator under parametric
excitation is shown in Fig. 1.

910

iidnCv

L(a) Parametrically excited van der Pol oscillator(b) Time-varying inductor
Fig. 1. Circuit model.

γ(τ)

1+α

1Fig. 3. Two coupled oscillators.

1-αFigure 4 shows the computer calculated results when theparameter 𝛼 is changed. In the case of two parametrically
excitations are same, two oscillators are fully synchronized atin-phase state as shown in Fig. 4(a). If the second oscillator
has little bit different value of 𝛼 with the first oscillator, fully
synchronization state changes to quasi synchronization state
as shown in Fig. 4(b). Figure 4(c) shows that the state of twooscillators becomes to unsynchronus state with large difference
between 𝛼 of the first and the second oscillators.

0π/ω

2π/ω

τFig. 2. Function relating to parametrically 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)Y1𝛾(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)

Y2

X1

By changing the variables and the parameters,
⎧√
√
𝑔1

𝐿𝐶𝜏,𝑣=𝑥𝑛𝑡=
0𝑛
𝑔3




√


𝜔 = 𝜔 0 𝐿0 𝐶⎨
√ √(3)
𝑔1 𝐶
𝑖𝑛 =𝑦𝑛
𝑔 3 𝐿0




√
√




⎩ 𝜀 = 𝑔 1 𝐿0 , 𝛿 = 1 𝐿
𝐶𝑅 𝐶
The normalized circuit equations are given by the followingequations.
⎧∑𝑑𝑥𝑛= 𝜀(𝑥𝑛 − 𝑥3𝑛 ) − 𝑦𝑛 + 𝛿(𝑥𝑘 − 𝑥𝑛 )⎨ 𝑑𝜏𝑘∈𝑆𝑛(4)

1𝑑𝑦𝑛
⎩
=𝑥𝑛𝑑𝜏
𝛾(𝜏 )

X2

X2

X1(a) 𝛼1 = 0.950 and 𝛼2 = 0.950

Y1Y2

X1X2X2

X1(b) 𝛼1 = 0.950 and 𝛼2 = 0.946

Y1

where 𝑛 = 1, 2, 3, ..., 10. 𝑆𝑛 is the set of nodes which aredirectly connected to the node 𝑛.

Y2

X1

X2

X2

X1(c) 𝛼1 = 0.950 and 𝛼2 = 0.936

III. T WO S UBCIRCUIT C ASE
In this section, we consider the case of 𝑛 = 2. Only twoparametrically excited van der Pol oscillators are coupled by
one resistor (see Fig. 3). We investigate synchronization state
when two oscillators have different parametrically excitations.
The parameter 𝛼 of the first oscillator is fixed with 0.95, andthe value of 𝛼 of the second oscillator is changed. For thecomputer simulations, we fix the circuit parameter as follows;𝜀 = 1.00, 𝛼 = 0.95, 𝜔 = 1.00 and 𝛿 = 0.50.

Fig. 4. Attractor and phase differences.

Next, we investigate synchronization area when the couplingstrength 𝛿 is changed. In this simulation, 𝛼 of the firstoscillator is fixed with 0.95. The computer simulated results
are shown in Fig. 5. The horizontal axis denotes the couplingstrength 𝛿, and the vertical axis denotes 𝛼 of the second

911oscillator. In this figure, synchronous area is shown between
the upper and the lower lines. We can see that synchronous
area becomes bigger by increasing the coupling strength.

TABLE ISMALL MISMATCH

small mismatch
𝛼1𝛼2
𝛼3
𝛼4
𝛼5𝛼6𝛼7𝛼8𝛼9𝛼10

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=30
0.008
0.009
-0.01
-0.01
-0.004-0.0030.005
0.009-0.009

w=40
-0.01
0.009
-0.0030.001-0.0020.007-0.006-0.01-0.009w=50-0.0090.01-0.007-0.001
0.004
0.009
0.007
0.005
-0.001

Fig. 5. Synchronization area by changing 𝛿.

IV. R ANDOM N ETWORK C ASE
Next, we apply the previous circuit model to ten coupled oscillators as random network model. The ten coupled oscillator
model is shown in Fig. 6. 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.

hub

Finally, we focus on clustering phenomena of ten coupledoscillators as random network with hub. Figure 8 shows theclustering phenomena (how do the oscillators change the fully
synchronization state) when the coupling strength is changed.
The horizontal axis is the coupling strength, and the verticalaxis denotes the node number between the oscillators.
At the beginning, first, two oscillators becomes synchronousstate. By increasing the coupling strength, the number ofsynchronized oscillators becomes large.

81310V. C LUSTERING P HENOMENA

927645Fig. 7. Synchronous state with 𝛿.

RFig. 6. A random network model with a hub.

VI. C ONCLUSIONS

Figure 7 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 upper area of eachline denotes synchronous area and the lower 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.

In this study, we have investigated 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.
For the future work, we would like to consider the influenceof the small mismatch for synchronization state of more large912

(a) 𝛿 = 0.01(b) 𝛿 = 0.1

92hub131087645(c) 𝛿 = 0.25

:synchronous state(d) 𝛿 = 0.80
Fig. 8. Circuit model.

scale networks.

[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).

510[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).

Circuit number

2364[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.

98[6] H. Kumeno, Y. Nishio, “Synchronization Phenomena in Coupled Parametrically Excited van der Pol Oscillators,” Proc. NOLTA’08, pp. 128-131,
Sep. 2008.

0.60

0.38

0.02
0.07
0.09
0.24

10.30

7

δ

Fig. 9. 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.

913