Synchronization Phenomena in Star-Coupled
van der Pol Oscillators
by Adding Different Frequency Oscillators
Minh Hai Tran, Kosuke Oi, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima,Tokushima 770–8506, Japan
Email: {minhhai, ooi, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract—In many cases, mutual synchronization systems
consisting of a large number of oscillators are used for practicalmodel. In this work, we investigate synchronization phenomena
observed by adding different frequency van der Pol oscillators
coupled with star combination. By computer simulations, weconfirm some of oscillators in the system are synchronized at
anti-phase.

I. I NTRODUCTION
Synchronization has grown to a considerable research fieldin recent years. Synchronization phenomena in large populations of interacting elements are the subject of intense research
efforts in physical, biological, chemical, and social systems.
The synchronization in the research is one of the basic phenomena of nature and it is observed over a wide range of fields
for example: human applause, digital telephony, video, digitalaudio, frogs, etc. In addition, in the organism, synchronization
phenomena has been observed in the activity of the brain andthe operation of the heart. It is considered to have become
an important role. In the future, analysis of brain activity andanalysis of heart activity are very important. Its application toengineering fields such as the realization of brain computer isexpected.
Now, synchronization phenomena have been studied bymany researchers in various field: two van der Pol oscillators
coupled by chaotically varying resistor [1], synchronization incoupled van der Pol oscillators involving periodically forced
capacitors [2], comparing two-layer CNN with van der Pol
oscillators coupled by inductors [3], group synchronization ofvan der Pol oscillators with different frequencies [4], stochastic
bifurcations in a bistable Duffing−van der Pol oscillator withcolored noise [5], synchronization and anti-synchronization ofchaos in an extended Bonhoffer−van der Pol oscillator using
active control [6].
In coupled oscillators, synchronization phenomena dependon the type of coupling. When two identical oscillators arecoupled, there are two basic possibilities of synchronization:
in-phase synchronization and anti-phase synchronization.
In this study, we investigate the effect to three star-coupledoscillators by adding another oscillator with different fre-978-1-5090-1570-2/16/$31.00 ©2016 IEEEquency. In this work, we confirm the case of four oscillatorsand the case of five oscillators in the system. First, wechange the parameter α of the fourth oscillator and investigate
the effect to the star-coupled circuits with three oscillators.
Next, we add the fifth oscillator and investigate the effect tothe star-coupled oscillators. We consider that many unknown
phenomena remain in such systems. Therefore, it is very
important to investigate such systems.

II. C IRCUIT M ODEL WITH F OUR O SCILLATORS  

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Fig. 1. Circuit model.

The circuit model used in the first pattern is shown inFig. 1. Three van der Pol oscillators are connected as thestar combination. In addition, we add a different frequency
oscillator to the star-coupled van der Pol oscillators. Wechange the frequency of the 4th oscillator and investigate theinfluence of the 4th oscillator to the overall star circuit.
Firstly, the v − iRk characteristics of the nonlinear resistors
are defined as follows:629

iRk = −g1 vk + g3 vk3 .

(1)

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By changing the variables and the parameters:


g1
Cg1
1
vk =xk , ik =
yk , α = 2 , t = L1 Cτ (2)
3g3
3Lg3
ω
and defining:CCLδ 0 = R0, δ1 = R1, ε = g1.
(3)
LLCThe⎧ normalized circuit equations are represented as follows:dxk⎪⎪= ε(xk − 13 x3k ) − yk − zk (k = 1, 2, 3)
⎪⎪ dτ
⎪
⎪⎪dyk
⎪
⎪
= 12 xk − 12 δ0 (y1 + y2 + y3 ) (k = 1, 2, 3)
⎪⎪
⎪dτ⎪
⎪
⎪dzk
⎪
⎪
= 12 xk (k = 1, 2, 4)
⎨dτ
(4)
dz3
⎪
1
⎪
x−
δ(z+y)=⎪31342⎪dτ
⎪
⎪
⎪
⎪
dx4
⎪
⎪
= ω 2 (ε(x4 − 13 x34 ) − y4 − z4 )
⎪⎪dτ⎪
⎪
⎪
dy4⎪⎪⎩= 12 x4 − 12 δ1 (z3 + y4 )dτ



Fig. 2. Simulation result (ω=1).

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We use Runge-Kutta method to calculate the values of x,y and their phase difference. We investigate synchronization
phenomena of the oscillators by using computer simulation
with ε=0.03, δ0 =0.1 and δ1 =0.3. We investigate the change ofvarying ω. Figures 2 to 4 show the simulation results.
In Fig. 2, in the case of ω =1, the all four oscillators
oscillated. Only between the 3rd oscillator and the 4th oscillator are synchronized at anti-phase. Consequently, we did
not see the effects of ω to the star-coupled circuit. And then,as ω increases to 1.2, the oscillations of the 3rd and the 4thoscillators stop, namely oscillation death appears as Fig. 3.
In that moment, the 1st oscillator and the 2nd oscillator
synchronize at anti-phase. By these result, we can see theeffect of α to the star-coupled oscillators.
In Fig. 4, the 4th oscillator oscillates again and the amplitude increase by increasing ω. When the value of ω above 1.2,frequency of the 4th oscillator becomes faster.
The amplitudes of the oscillator are shown in the Figs. 5
and 6. In Fig. 5, ω is changed inside a large range value[1.0,2.0]. As the result, the amplitudes of the 1st and the 2ndoscillator unchanged. Therefore, frequency is not effect to thestar-coupled oscillators.
In Fig. 6, when the ω increase from 1 to 1.2, the amplitudesof the 3rd and the 4th oscillator become gradually smaller.
Next, when the ω is increased to 1.4, the amplitudes of the3rd and the 4th oscillator are perfect stop. When ω above
1.4, the amplitude of the 4th oscillator goes up. However, theamplitudes of the 3rd oscillator are almost unchanged.

where ε is the nonlinear intensity.
III. S IMULATION R ESULT



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Fig. 3. Simulation result (ω=1.2).

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Fig. 4. Simulation result (ω=2).

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In the case of ω1 =1 and ω2 =1, from before result we
can imagine no effect from ω1 and ω2 to the star-coupledoscillators and that all of the five oscillators oscillate. Between
the 3rd oscillator and the 4th oscillator, the 2nd oscillator andthe 5th oscillator, the phase difference is synchronized at antiphase.
In Fig. 8, the phase difference between the 3rd oscillator
and the 4th oscillator, the 2nd oscillator and the 5th oscillator
are quasi synchronized at anti-phase. The phase difference
become asynchronous between the 1st oscillator and the 3rdoscillator, the 2nd oscillator and the 5th.
In the next case, we increase only ω1 to 1.1 and ω2
remains 1. Now, we observe the effect of ω to the star-coupledoscillators. Oscillations of the 3rd oscillator and 4th oscillator
become small.
From the result in Fig. 10, the 1st, the 2nd and the 5th
oscillators oscillate and the 3rd and the 4th oscillators stop,
namely oscillation death appear. The phase difference between
the 1st oscillator and the 2nd oscillator, the 2nd oscillator
and the 5th oscillator become synchronized at anti-phasecompletely.

Fig. 5. Amplitude of oscillator 1 and oscillator 2.

Fig. 6. Amplitude of oscillator 3 and oscillator 4.

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IV. F IVE O SCILLATORS C ASE

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(5)We investigate synchronization phenomena with ε = 0.03,
δ0 =0.1, δ1 =0.3 and δ2 =0.3 as varying ω1 and ω2 . The simulation results are shown in Fig. 8 to Fig. 12.

978-1-5090-1570-2/16/$31.00 ©2016 IEEE

Fig. 9. Simulation result (ω1 =1.1, ω2 =1).

Next, we use a circuit model in Fig. 7. This system has fiveoscillators. We add the 5th oscillator in this system. We change
the frequency of the 4th and the 5th oscillators and investigate
their influence to the overall star-coupled oscillators.
In this system, the values of frequency α1 and α2 arenormalized as Eq. (5)(i = 1, 2)

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Fig. 7. Circuit model with five oscillators.

1ωi2

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αi =



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Fig. 8. Simulation result (ω1 =1, ω2 =1).

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Fig. 10. Simulation result (ω1 =1.2, ω2 =1).

In Fig. 11, we increase not only ω1 , but also ω2 to 1.2.
Frequency is given a strong influence on circuit. Only the 1stoscillator still oscillate and the remaining oscillators stop.

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In Fig. 12, if we still increase ω2 to 2.5, oscillation of the5th oscillator also starts again. Frequency of the 5th oscillator
becomes faster when ω2 is 1.2 and higher.

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confirmed oscillation of the oscillator stop in some rangeof the frequency. When the frequency of the 4th and 5thoscillator is varied, oscillation of some oscillator stop, namelyoscillation death appear. By increasing the frequencies, theattractor becomes asynchronous states.
For the future work, we would like to increase the numberof the oscillators to six and to change the star combination
to the ring combination. We also would like to do real circuit
experiments to confirm the results.

R EFERENCES
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

[1] Y. Uwate, Y. Nishio, “Two van der Pol Oscillators Coupledby Chaotically Varying Resistor,” Proceedings of International Workshop on Nonlinear Dynamics of Electronic
Systems (NDES’06), pp. 189-192, Jun. 2006.





[2] H. Kumeno, Y. Nishio, “Synchronization in Coupled vander Pol Oscillators Involving Periodically Forced Capasitors,” Proceedings of International Symposium on Nonlinear Theory and its Applications (NOLTA’09), pp. 226229, Oct. 2009.



[3] K. Urabe, Y. Nishio, “Comparing Two-Layer CNN withvan der Pol Oscillators Coupled by Inductors,” Proceedingsof International Symposium on Nonlinear Theory and its
Applications (NOLTA’06), pp. 723-726, Sep. 2006.

Fig. 11. Simulation result(ω1 =1.2, ω2 =1.2).

[4] Y. Uwate, Y. Nishio, R. Stoop, “Group Synchronizationof van der Pol Oscillators with Different Frequencies,”
Proceedings of International Symposium on Nonlinear
Theory and its Applications (NOLTA’08), pp. 612-615,
Sep. 2008.
[5] R. S. Barbosa, J.A. T. Machado, B.M. Vinagre, “Initialconditions-independent limit cycles of a fractional-ordervan der Pol oscillator,” Journal of Vibration and Control,
May 2016, 22: 2135-2146, first published on July 15, 2015.

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[6] Y. Xu, R. Gu, H. Zhang, W. Xu, J. Duan, “Stochasticbifurcations in a bistable Duffing−van der Pol oscillator
with colored noise,” Phys. Rev. E 83, 056215, Published
26 May 2011.



[7] A. Njah, U.E. Vincent, “Synchronization and antisynchronization of chaos in an extended Bonhoffer-van der
Pol oscillator using active control,” Journal of Sound andVibration, Volume 319, Issues 1-2, 9 January 2009, Pages
41-49.



Fig. 12. Simulation result (ω1 =2.36, ω2 =1.6).

V. C ONCLUSION
In this study, we have investigated synchronization phenomena and oscillators of four and five oscillators with different frequencies. By carrying out computer simulations, we

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