2010 International Workshop on Nonlinear Circuits,Communication and Signal Processing
NCSP'10, Waikiki, Hawaii, March 3-5, 2010

Synchronization in Three Coupled van der Pol Oscillatorswith Different Coupling Strength
Yoko Uwate† , Yosifhimi Nishio‡ and Ruedi Stoop††Institute of Neuroinfomatics, UZH/ETH Zurich,
Winterthurerstrasse 190,
CH-8057 Zurich, Switzerland,
Email: yu001@ini.phys.ethz.ch‡Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470
Email: nishio@ee.tokushima-u.ac.jp

1st oscillator

1. Introduction

2LCv1

βR

2LrmRib32L

rm

ia32L

The circuit model of three coupled van der Pol oscillator as
ring topology is shown in Fig. 1. In this figure, β denotes theamplitude for changing the coupling strength between firstand second oscillators.
In the computer simulations, we assume that the vk − iRkcharacteristics of the nonlinear resistor in each oscillator isi R1rm

2. Circuit Modelia12L

High-dimensional nonlinear phenomena in the field of natural sciences can often be explained as the result of a synchronization phenomenon of coupled oscillatory systems. Therefore, studies of synchronization in oscillatory networks havebeen extensively reported in the various fields such as physical [1], biological [2] and electrical [3] systems. Endo etal. have presented the details of a theoretical analysis andcorresponding circuit experiments on electrical circuits oscillators arranged in a ladder, a ring and in a two-dimensional
array topology [4]-[6]. Moreover, coupled oscillatory systems can also produce interesting phase patterns, including
wave propagation, clustering and complex patterns [7]-[10].
Setou et al. have observed interesting synchronization phenomena (oscillation death, independent oscillation and double mode oscillation) when van der Pol oscillators with different frequencies are coupled by means of a resistor, in a star
topology [11]. We assume that any fluctuations such as different frequencies have one possibility to induce interesting
synchronization phenomena in oscillatory networks.
In this study, we discuss the synchronization phenomenain three coupled van der Pol oscillators as a ring topology
with different coupling strength. As a first step, we consider
the simplest case that only one coupling strength is varied.
Namely, the coupling strength between the first and the second oscillators has the different coupling strength to the others. By carrying out computer simulations, we confirm thatthe phase differences of the three oscillators depend on thecoupling strength between the first and the second oscillators. These phenomena are also analyzed theoretically by using the averaging method. Furthermore, we investigate thepower consumption in the whole system.

rm

ib1

2L
rm

ib2

ia2
rm

i R3iR2v3
CR

3rd oscillator

Cv2

2nd oscillator

Figure 1: Three coupled van der Pol oscillators as a ring
topology.

given by the following third order polynomial equation
iRk = −g1 vk + g3 vk 3 (g1 , g3 > 0),(1)(k = 1, 2, 3).
The normalized circuit equations governing the circuit inFig. 1 are expressed as[First oscillator]
dx11 2=ε1−x1 x1 − (ya1 + yb1 )dτ3
 dya1 1(2)x1 − ηya1 − βγ(ya1 + yb2 )=dτ2

dyb1 1 
x1 − ηyb1 − γ(ya3 + yb1 )=dτ
2[Second oscillator]
dx21 2=
ε1−x2 x2 − (ya2 + yb2 )


dτ3






 dya2 1(3)x−ηy−γ(y+y)=2a2a2b3

dτ2






dyb2 1 
x2 − ηyb2 − βγ(ya1 + yb2 )=dτ
2- 109 -180[Third oscillator]

1 2
dx3


=
ε1−x3 x3 − (ya3 + yb3 )dτ3
 dya3 1(4)x3 − ηya3 − γ(ya3 + yb1 )=dτ2
dyb3 1 
x3 − ηyb3 − γ(ya2 + yb3 )=dτ2wherer√
g1
t = LCτ, vk =xk ,3g3r
r
r
r
g1
C
g1
C
iak =
yak , ibk =ybk ,3g3 L
3g3 Lrr
r
LCC
ε = g1, γ=R, η = rm,CLL(k = 1, 2, 3).
In this equations, γ is the coupling strength and ε denotes
the nonlinearity of the oscillators.
3. Synchronization Phenomena

ϕ12phase difference160
140120ϕ23, ϕ31100806040
20000.5

11.5

β22.5

3Figure 2: Phase differences.

3.2. Power Consumption of Resistors
In this section, we focus on the power consumption of thecoupling resistors between the coupled oscillators. The average power P is calculated as follows,
Tp

For the computer simulations, we calculate Eqs. (2)-(4)using a fourth-order Runge-Kutta method with the step size
h = 0.005. The parameters of this circuit model are settingfor ε = 0.1, γ = 0.02 and η = 0.001.

1 Xβγ(ya1 + yb2 )2T p τ=1
Tp
1 X
P23 =γ(ya2 + yb3 )2
T p τ=1
Tp
1 Xγ(ya3 + yb1 )2
P31 =T p τ=1PAll = P12 + P23 + P31

P12 =3.1. Phase Difference(6)

We carry out computer simulations passing one hundred
thousand iterations of the Poincaré section. Figure 2 shows
the computer simulation result of the phase difference. Whenβ is smaller than 0.5, first and second oscillators are synchronized at in-phase and other oscillators are synchronized atanti-phase. By increasing β, phase difference between thefirst and the second oscillators close to anti-phase synchronization and other oscillators synchronize with 90◦ shift.

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power consumption

First, in order to calculate the phase difference between
oscillators, let us define the Poincaré section as xk < 0 andyak + ybk = 0 (k = 1, 2, 3). Next, we introduce the following
independent variables from the discrete data of xkl (n), yakl (n) where T p is fixed for one thundered thousand as long period.
Figure 3 shows the power consumption of the coupling reand ybkl (n) on the Poincaé section. The phase difference besistors.
From this figure, we confirm that the energy of coutween k-th and l-th oscillators is defined as following equaplingstrength
between the first and the second oscillators P12tion.
has the peak around β = 0.5. When β is smaller than 0.5, thepower consumption of P23 and P31 shows zero. Because, in−1 yakl (n) + ybkl (n),π−tan
this case, these oscillators are synchronized at the anti-phase.
xkl (n)
By increasing the value of β, the energy of whole systemxkl (n) ≥ 0(PAll ) increases.
yakl (n) + ybkl (n),−tan−10.035
xkl (n)
0.03(5)ϕkl (n) = xkl (n) < 0 and yakl + ybkl (n) ≥ 0




PAll


0.025



−1 yakl (n) + ybkl (n)
2π − tan,
0.02xkl (n)P23, P31
xkl (n) < 0 and yakl + ybkl (n) < 0
0.015





0.01
(k, l = 1, 2, 3)

P120.005
000.5

11.5

β2

Figure 3: Power consumption.

2.5

3

By using Eq. (13), we obtain the solution of Eq. (7).

4. Theoretical Analysisρ˙k4.1. Case for β = 1.0
Consider the case of that the three coupling strength has thesame value (β = 1.0) and the tiny resistors to avoid L-loop forcomputer simulations is set to zero (η = 0.0). The normalized
circuit equations are given as follows.
 dx

1 2
k

=ε1
−xk xk − (yak + ybk )


dτ3






 dyak = 1 x − γ(y+y)
ka(k−1)
bk


dτ2






dybk 1=x−γ(y+y)kakb(k+1) dτ2

θ˙k(k = 1, 2, 3)ε2 ρk 2(ρk − 4)8εγ 2ρk + γρk−1 sin(θk − θk−1 )− 28(γ + 1)
−ρk−1 cos(θk − θk−1 ) + γρk+1 sin(θk − θk+1 )−ρk+1 cos(θk − θk+1 )(14)

εγ2γρk + γρk−1 cos(θk − θk−1 )8ρk (γ2 + 1)
−ρk−1 sin(θk − θk−1 ) + γρk+1 cos(θk − θk+1 )−ρk+1 sin(θk − θk+1 )

(15)

=−=−

(k = 1, 2, 3).

(7)

where

In the steady state,ya0 = ya3 , yb4 = yb1(8)

θ1 − θ2 = θ2 − θ3 = θ3 − θ1 =

(10)

where
Y ≡ yak + ybk + ya(k−1) + yb(k+1) .
Equation (10) is first order linear differential equation. Thesolution is given as following equation.

andθ˙k = 0(16)In this section, we analyze the synchronization phenomena
by using the averaging method. Equation (6) is combined to must be satisfied. We obtain the solutions as follows. For thegive the second order nonliner differential equation as follows amplitude:rγ
ρk = 4 −(17)2 + 1)2ε(γ1d xk
2 dxk+ γY ≡ Fk+ xk = ε(1 − xk )dτ2dτ2(9)(k = 1, 2, 3).
(k = 1, 2, 3)
For the phase difference:
11dY+ γY = xk + xk−1 + xk+1dτ
22(k = 1, 2, 3)

ρ˙k = 02eγτ xk +1
1xk−1 + xk+1 + Ce−γτ
22

Theoretical result
Computer simulation result

1.8(11)

amplitude

Y = e−γτ(C: const.)

1.7
1.6
1.5
1.4

In the steady state, the second term of Eq. (11) becomes tozero. Let us assume the solutions of Eq. (7) isxk (τ) = ρk sin(τ + θk ).

(18)

In order to confirm the credibility of theoretical analysis,
we compare the amplitude of oscillator and the phase difference of the theoretical result with the computer simulation
result (see. Fig. 4 and Tab. 1). From these results, we confirm
that they match very well.

1.9Z

2π.
3(12)

(k=1, 2, 3)

1.3

0

0.05

0.1

γ

0.15

0.2

Figure 4: Comparison between theoretical and computer simulation results for the amplitude of oscillator.

We pay attention to treat the non-resonance system and apply
for the averaging method to Eq. (7). The average method isdefined as following equations.
Table 1: Comparison between theoretical and computer simulation results for the phase difference (θ1 − θ2 ).
Z TεFk cos(τ + θk )dτρ˙k = lim

T →∞θ˙k = lim

T →∞0Z T0

εFk sin(τ + θk )dτρk(k = 1, 2, 3)(13)- 111 -

Phase difference

Theory120◦

Simulation
122◦5. Conclusions

4.2. Case for β , 1.0

phase difference: θ1−θ2We have investigated the synchronization phenomena in a
Next we consider the case of that the coupling strength between the first and the second oscillators are changed (β , ring van der Pol oscillators when the symmetric property of1.0). The solutions of phase differences including the β factor network is destroyed. By computer simulations, we have confirmed that three coupled oscillators tend to synchronize to beread tominimum energy of the whole system. Furthermore, we have
−1 = βcos(θ1 − θ2 ) + cos(θ2 − θ3 )applied the theoretical method to these synchronization phe
−1 = βcos(θ1 − θ2 ) + cos(θ3 − θ2 )(19) nomena and the similar solutions for the amplitude and the −1 = cos(θ − θ ) + cos(θ − θ )phase difference were obtained.
2332In our future works, we discuss the stability of solutions
Namely each difference phase shift is obtained.
and explain the synchronization mechanism of a large-scale
network by using this basic circuit model.
 1
θ1 − θ2 = arccos −2βReferences(20)
θ2 − θ3 = −1 − cos(θ1 − θ2 )[1] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synθ3 − θ1 = −1 − cos(θ1 − θ2 )chronization and graph topology,” Int. J. Bifurcation andChaos, vol.15, no.11, pp.3423-3433, Nov. 2005.
Finally, we compare the theoretical method and the computer simulations for the phase differences when the cou- [2] R. Stoop and C. Wagner, “Neocortex’s architecture optimizes computation, information transfer and synchropling strength between the first and the second oscillator (β)
nizability, at given total connection length,” Internais changed from 0.02 to 3.0 (see. Figs. 5, 6). From Fig. 5 wetional Journal of Bifurcation and Chaos, vol.17, no.7,
can see that the phase difference between the theoretical andpp.2257-2279, 2007.
the computer simulation results has small difference around10 degree. In the case of Fig. 6, they match perfectly.
[3] X.F. Wang and G. Chen, “Synchronization in scale-free
dynamical networks: robustness and fragility,” IEEE
180
Trans. Circuit and Systems-I, vol.49, no.1, pp.54-62,
Computer simulation result
160
2002.
140

[4] T. Endo and S. Mori, “Mode analysis of a multimodeladder oscillator,” IEEE Trans. Circuits Syst., vol.23,
pp.100-113, Feb. 1976.

Theoretical result

120
10080604020000.511.5

β22.5

3

Figure 5: Phase difference (θ1 − θ2 ).

[7] Y. Nishio and S. Mori, “Mutually coupled oscillatorswith an extremely large number of steady states,” Proc.
of ISCAS’92, vol.2, pp.819-822, May 1992.
[8] L. Pivka, “Autowaves and spatio-temporal chaos in
CNNs Part I: A tutorial,” IEEE Trans. Circuits Syst. I,vol.42, pp.638-649, Oct. 1995.

phase difference: θ2−θ3, θ3−θ1180
160
140

Theoretical result120[9] M. Yamauchi, Y. Nishio and A. Ushida, ”Phase-waves
in a ladder of oscillators” IEICE Trans. Fundamentals,
vol.E86-A, no.4, pp.891-899, Apr. 2003.

100

Computer simulation result

806040200[5] T. Endo and S. Mori, “Mode analysis of twodimensional low-pass multimode oscillator,” IEEE
Trans. Circuits Syst., vol.23, pp.517-530, Sep. 1976.
[6] T. Endo and S. Mori, “Mode analysis of a ring of a large
number of mutually coupled van der Pol oscillators,”
IEEE Trans. Circuits Syst., vol.25, no.1, pp.7-18, Jan.
1978.

00.5

11.5

β22.5Figure 6: Phase difference (θ2 − θ3 , θ3 − θ1 ).

3[10] Y. Uwate and Y. Nishio, “Complex Phase Synchronization in an Array of Oscillators Coupled by TimeVarying Resistor,” Porc. of IJCNN’06, pp. 8345-8350,Jul. 2006.
[11] Y. Setou, Y. Nishio and A. Ushida, “Synchronization Phenomena in Resistively Coupled Oscillators withDifferent Frequencies,” IEICE Trans. Fundamentals,vol.E79-A, no.10, pp.1575-1580, Oct. 1996.

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