IEEE Workshop on Nonlinear Circuit Networks
December 11-12, 2015

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

Abstract— In this work, we investigated synchronization phenomena observed in van der Pol oscillators coupled with star
combination. By computer simulations, we confirmed some ofoscillators in the system is synchronized at anti-phase. The nonlinear phenomena can make synchronization such as oscillation
death.

I. I NTRODUCTION
Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts inphysical, biological, chemical, and social systems.
Different kinds of synchronization are important in fields
such as digital telephony, video, digital audio, frogs, humanapplause, etc. Now, synchronization phenomena has beenstudied by many researchers in various field: synchronization
phenomena in van der Pol oscillators coupled by a timevarying resistor [1], synchronization phenomena observed
from coupled van der Pol oscillators involving periodicallyforced capacitors [2], investigation on the similarity between
the 2-layer CNN and the van der Pol oscillators coupled byinductors [3], consider a ring consisting of van der Pol oscillators with different oscillation frequencies for investigation oflarge-scale circuits [4].
The van der Pol oscillator is an oscillator with nonlinear
damping governed by the second-order differential equation.
It was originally proposed by the Dutch electrical engineer
and physicist Balthasar van der Pol while he was working atPhilips. The coupled van der Pol oscillator is one of coupledoscillators. It is very simple circuit how to study nonlinear
system and important to investigate various nonlinear phenomena of the parametric excitation circuits for future engineering
applications.
In this study, we investigate the effect to three-coupledcentral star circuits by adding another oscillator with different frequency. In this time, we confirmed the case of four
oscillators and the case of five oscillators in the system issynchronized at anti-phase. We concentrate from the simplest
case of van der Pol oscillators that oscillation frequency isvaried. Fzirst, we are changing the parameter α of fourth
oscillator and investigating the effect to the central star circuitswith three oscillators. The case that all of oscillators have thesame oscillation. Next, we are increasing fifth oscillators withparameter and investigating the effect to the three oscillators.
We consider that many unknown phenomena remain in such
systems. Therefore, it is very important to investigate such
systems.
II. C IRCUIT M ODELIn this study, the circuit model used in the first pattern isshown in Fig. 1. Three van der Pol oscillators connected as
the star combination. In addition, we also add the 4th van der
Pol oscillator on the three-couple star circuits. So we change
the frequency of 4th oscillator and investigate the influence of4th oscillator for the overall star circuit.
2L
2L

I2B

iR2

v2

(2)

2L

iR1

2L

I1A

I2A

I3A 2L

R0C

(1)

v1

iR3

C

I3B 2LC2L

v3(3)

Fig. 1.

R1

I4AI4B 2LiR4

Cv4(4)Circuit model.

The normalized equations are represented as follows: dxk
dτdykdτ
 dzk

dτdz3dτ
dx4dτ
dy4dτ

=ε(xk − 13 x3k ) − yk − zk=

1
12 xk − 2 β0 (y1 + y2 + y3 )(k = 1, 2, 3)
12 xk (k = 1, 2, 4)
12 x3 − β1 (z3 + y4 )
ω 2 (ε(x4 − 13 x34 ) − y4 − z4 )1
12 x4 − 2 β1 (z3 + y4 )====(k = 1, 2, 3)

where ε is the non-linear intensity.
Firstly, the v − iRk characteristics of the nonlinear resistor
are defined as follows,-1-iRk = −g1 vk + g3 vk3(1)

By changing the variables and the parameters,
√
√
g1
Cg1
vk =xk ; ik =yk
3g3
3Lg3
√
1
α = 2 ; t = L1 Cτ.
ωand defining√√
CLβ=R; ε = g1
LC

(2)
(3)

(4)

III. S IMULATION R ESULTS
We investigate synchronization phenomena and oscillationof the oscillators by using computer simulation with β0 =0.1
and β1 =0.3. We investigate the change of varrying ω (ω=1 toω=1.8). Figures 2, 3 and 4 show the simulation results.

y1x2x1xx3

2x1x1y1

Fig. 4.

Simulation result (ω=1.5).

Fig. 7.

Simulation result (ω=2.1).

x4

x4xx

3

3

Simulation result (ω=1.2).

y4

y3x2x1

Fig. 6.

3x3
2x3

2

Simulation result (ω=1).

y4

xy2x1

Fig. 5.

x3

y3x1

Fig. 3.

4

Simulation result (ω=1).

x32x4xxx2x1xx3
2y2y1y4xx1
Fig. 2.

xy3

y2

2nd oscillator become anti-phase synchronization. From here
we can see effect of α to star circuit.
In Fig. 4, as ω increases from 1.2 to 1.8, 1st oscillatorand 2nd oscillator still are anti-phase synchronization, however
the fourth oscillator oscillates again and frequency of the 4thoscillator become faster than before from above 1.8 and higher.
Next, we increased ε to 0.15 and we investigate synchronization phenomena and oscillation of the oscillators by using
computer simulation with β0 from 0.1 and to 2.1.
In Fig. 5, in the case of ω =1, ε = 0.15, all the four oscillators
oscillated and do not have any oscillators is synchronization
at anti-phase. Consequently, we did not see the effects ofomega to star circuit. And then, as the ω increases to 1.5,
only oscillation of 3rd oscillator stop, namely oscillator deathappears as shown in Fig. 5. In Fig. 7, four oscillators alway
oscillate from ω = 2.1 and higher.

x4

2x1

x4x3xx

3x3

Simulation result (ω=1.8).

In Fig. 2, in the case of ω =1, all the four oscillators
oscillated. Only between 4th oscillator and 1st oscillator aresynchronized at anti-phase. Consequently, we did not see theeffects of omega to star circuit. And then, as the ω increases
to 1.2, the oscillation of 4th oscillator and the oscillation of
3rd oscillator stop, namely oscillation death appears as shownin Fig. 3. At the same time, between the 1st oscillator and theIV. C IRCUIT M ODEL W ITH F IVE O SCILLATORS
Next, the circuit model used in the second pattern is shownin Fig. 8. This pattern have five oscillators in circuit. In
addition, we also add the 4th and 5th van der Pol oscillator
on the three-couple star circuits. So we change the frequency
of 4th and 5th oscillator and investigate the influence of 5thoscillator for the overall star circuit.

-2-

2L

2L I2B

2L I5B

I5A

iR2

2L

2L

I1A

I2A

I3A

v2

C

R0
v1

C

(1)

R2

2L

iR3

R1

v3

C

v5

C

I4A

I4B 2LiR4

CFrom Fig. 9 to Fig. 11, we investigated 5 oscillators coupled
by a resistor with different frequency oscillators. In the caseof ω=1 , from before result we can imagining no have affectfrom ω to central circuit and all of the five oscillators will beoscillation. In Fig. 9, the result that was as expected. All of
the five oscillators oscillated. Between 4th oscillator and 3rdoscillator are synchronized at anti-phase. It s the same result
between 3rd oscillator and 2nd oscillator.
In the next case, we increased only ω to 1.2 was used thatlike that. Now, we can saw the effect from ω to circuit. Fromresult, have only 1st, 2nd, 5th oscillator oscillated and 3rd,4th oscillation had been stopped. Oscillators 1 and 2 generate
double-mode oscillation and its envelopes are synchronized atanti-phase. It s the same result with oscillator 4 and 3.
In Fig. 11, we still increase ω to 2.36. The circuit was bestrongly affect. 4th oscillator oscillate and only 3rd oscillator
stop. We could not confirm the phase between 2nd oscillator
and 5th oscillator.
In Fig. 7 with 4 van der Pol and in Fig 11 with 5 vander Pol, we saw different phase with 2 simulation result. In
Fig. 7 we can only confirm oscillation from 1st oscillator to
4th oscillator but in Fig 11 not only oscillator but also phasebetween 2 oscillators except 2nd and 5th oscillator.
In the second case with 5 van der Pol, oscillation star again
earlier than 4 van der Pol when we still increase ω.

2L(3)Fig. 8.

iR2

I3B
2L

iR1

with ε = 0.15, β0 =0.1, β1 =0.3 and β2 =0.3. We investigate thechange of varrying ω . Figures 9, 10, 11 show the simulation
results.

(5)

(2)

v4

(4)

Circuit Model with five oscillators.

The normalized equations are represented as follows:

 dxk
dτ
 dykdτ dzkdτdz3dτdxkdτdykdτ




= ε(xk − 13 x3k ) − yk − zk(k = 1, 2, 3)

1
12 xk − 2 β0 (y1 + y2 + y3 )(k = 1, 2, 3)
12 xk (k = 1, 4, 5)
12 x3 − β1 (z3 + y4 )
ω 2 (ε(xk − 13 x3k ) − yk − zk )(k = 4, 5)112 xk − 2 βz (z3 + yk )(k = 4, 5; z = 1, 2)=====V. S IMULATION R ESULTS W ITH F IVE O SCILLATORS

y1y3y2x2x1xx3x32y5y4xx

542x5x4x

VI. C ONCLUSIONSx1Fig. 9.

y1

y3x2

23y5y4x3x32xx3Simulation result (ω=1).

y2x1xxx1xx542x5x4xIn this study, we have investigated synchronization phenomena and oscillators of 4 and 5 oscillators with different
frequencies. By carrying out computer simulations, we confirmed oscillation of the oscillator stop in some range of thefrequency. When the frequency of the 4th and 5th oscillator isvaried, oscillation of some oscillator stop, namely oscillation
death appear of start again. By increasing the frequencies, theattractor becomes asynchronous states.
R EFERENCES

x1xx1
Fig. 10.

y1y3x2xx3

2x1x3x5

4xx5x4x2

Fig. 11.

y5y4xx1[1] Y. Uwate、Y. Nishio Two van der Pol Oscillators Coupled by 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 van der Pol
Oscillators Involving Periodically Forced Capasitors Proceedings ofInternational Symposium on Nonlinear Theory and its Applications(NOLTA’09), pp. 226-229, Oct. 2009.
[3] K. Urabe, Y. Nishio
Comparing Two-Layer CNN with van der
Pol Oscillators Coupled by Inductors Proceedings of International
Symposium on Nonlinear Theory and its Applications (NOLTA’06), pp.
723-726, Sep. 2006.
[4] Y. Uwate, Y. Nishio, R. Stoop Group Synchronization of van der PolOscillators with Different Frequencies Proceedings of International
Symposium on Nonlinear Theory and its Applications (NOLTA’08), pp.
612-615, Sep. 2008.

23Simulation result (ω=1.2).

y2x1xx

33x

3x

2

Simulation result (ω=2.36).

In this case, we investigate synchronization phenomena andoscillation of the oscillators by using computer simulation-3-