2009 International Symposium on Nonlinear Theory and its Applications
NOLTA'09, Sapporo, Japan, October 18-21, 2009

Synchronization of Distant Oscillators in a Ring Network
Yoko Uwate† , Yoshifumi Nishio‡ and Ruedi Stoop††Institute of Neuroinfomatics, University / ETH Zurich,
Winterthurerstrasse 190, 8057 Zurich, Switzerland,
email: {yu001, ruedi}@ini.phys.ethz.ch‡Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—High-dimensional nonlinear phenomena in
the field of natural sciences can often be explained as theresult of a synchronization phenomenon of coupled oscillatory systems. In this study, we investigate how distantvan der Pol oscillators coupled by resistors in a ring topology may synchronize. Using computer simulations, wefind that distant oscillators can synchronize in-phase, antiphase, and at N/2-phase (where N is the number of oscillators), in dependence of the oscillation frequency of adjacent oscillators.

pendence of the diﬀerent frequencies, we observed several
interesting synchronization phenomena [12]. So far, themechanisms of this type of group synchronization have,
however, remained only partially understood.
In the present study, we focus on synchronization phenomena between distant van der Pol oscillators with diﬀerent frequencies that are coupled by means of resistors ina ring topology. Using computer simulations, we find thatthe distant neighbors oscillators are synchronized in-phase,anti-phase or at N/2-phase, dependent on the oscillation
frequency of the adjacent oscillators.

1. Introduction
2. Circuit Model
Synchronization phenomena can be observed in a widerange of fields such as physics, chemistry, biology, neuroscience, engineering, and so on. Synchronization may
prove crucial in understanding - and mimicking in applications - decentralized information processing mechanisms,
as found, in particular, in the living nature. Therefore,
mathematical modeling studies of synchronization in oscillatory networks have become a topical issue [1]-[3]. As
particularly nice examples, Endo et al. have presented thedetails of a theoretical analysis and corresponding circuit
experiments on electrical circuits oscillators arranged in aladder [4], a ring [5] and in a two-dimensional array topology [6]. Moreover, coupled oscillatory systems can also
produce interesting phase patterns, including wave propagation, clustering, and complex phase patterns [7]-[10].
Seto et al. [11] have observed interesting synchronization phenomena when van der Pol oscillators with diﬀerent frequencies are coupled by means of a resistor, in a star
topology. This phenomenon is, however, restricted to small
networks: Oscillators coupled in a star topology fail to synchronize if their number exceeds 4.
In our search for synchronization phenomena present
in large-scale networks, we have therefore concentrated
on van der Pol oscillators arranged in a ring topology,
where each oscillator has its own oscillation frequency. For
this arrangement, we found interesting nonlinear phenomena such as oscillation death, independent oscillations anddouble-mode oscillations [12]. Moreover, we have studied the interaction of van der Pol oscillators drawn fromtwo sets of distinct oscillatory frequencies, where, in de-The basic circuit model and the topology according towhich the oscillators are arranged is shown in Fig. 1. Inorder to study the synchronization on this topology, we detune the frequency of each second oscillator by means ofchanging its capacitance, proceeding cyclically through thering.
2L
2L i a(k+1) ib(k+1) 2L
rm

2L i aki bk
rm

rm

rm

i RNiRki Rk
R

2L2Lv(k+1)
αC

R(k+1)-th oscillator

Cvk

k-th oscillator

R...

αCvN

N-th oscillator

Figure 1: Ring of van der Pol oscillators with diﬀerent frequencies.
We assume that the vk − iRk characteristics of the nonlinear resistor in each oscillator is given by the following thirdorder polynomial equation
iRk = −g1 vk + g3 vk 3 .
By changing the variables and the parameters,
√√
√
√
g1
g1
C
vk =xk , ik =yk , t = LCτ,
3g3
3g3 L- 439 -(1)√ε = g1L, γ=rC√

C1, α = 2 , η = rmLω√

anti-phase, whereas oscillator 2 does not synchronize withoscillator 4 at all.

C,
L

the normalized circuit equations of the ring of oscillators
finally read
dxk= ω2k εxk (1 − xN 2 ) − ω2k (yaN + ybN )dτ
dyak 1= xk − ηyak − γ(yak) + yb(k+1) )
dτ2



dybk 1= xk − ηybk − γ(ya(k−1) + ybk) )


dτ2




(k = 1, 2, · · · , N),(2)

Next, we calculated the phase diﬀerence between oscillator 1 and oscillator 3 if the frequencies α of the oscillators 2 and 4 are increased from 0.1 to 3.5. The results
shown in Fig. 5 demonstrate that when α is smaller than0.5, non-synchronization is observed. Around α ' 1.0, inphase synchronization is obtained. By further increasing
the value of α, anti-phase synchronization can be triggered.
y1

y2

y3

y4

Attractor

whereya0 = yaN ,yb(N+1) = yb1 .

x1

(3)

In this system of equations, ωk denotes the frequency of thekth oscillator, γ corresponds to the coupling strength and εcaptures the nonlinearity of the oscillators. For the simulations, we evaluated Eq. (2) using a fourth-order RungeKutta method. For this setting, we obtained results asshown and discussed in the following paragraph.

x3x2x3x4x5x4x6Phase
differencex1x2x3x4x1x2x3x4Time
wave
form

Figure 3: Synchronization results for N = 4 (α = 0.64).

3. Synchronization phenomena when each second oscillator is detuned
3.1. N=4y1

ω=α3rd osc.
ω=1

1st osc.

y3

y4

Attractor

In this case, the two types of oscillators with detunablefrequencies are placed alternately, as is shown in Fig. 2,
where α denotes the from the standard frequency ω = 1.0
detuned frequency. We now focus on the synchronization
between oscillator 1 and oscillator 3, which have the standard frequency (ω = 1.0), when the frequencies oscillator
2 and oscillator 4 are periodically changed to ω = α .
4th osc.

y2x1x3x2x3x4x5x4x6

Phase
differencex1x2x3x4x1x2x3x4

Time
wave
form

Figure 4: Synchronization results for N = 4 (α = 3.47).

ω=12nd osc.
ω=α180phase difference (average)
phase difference (max)phase difference (min)160

phase difference

140

Figure 2: Model circuit for N = 4.

120
100
8060

The obtained results are collected in Figs. 3 and 4. In
these figures, the top rows show the attractors obtained
from oscillator 1 and oscillator 3, the middle row displays
the phase diﬀerence between the ith and the (i + 2)th oscillator, whereas the bottom row shows the time wave-forms.
Figure 3 has been obtained for α = 0.64. In this case, theoscillators 1 and 3 are almost synchronized in-phase. The
result obtained for α = 3.47 is shown in Fig. 4. Now, oscillator 1 and oscillator 3 oscillators are almost synchronized

4020000.5
non-synchronization

11.5

α2

2.5

33.5

Figure 5: Phase diﬀerence between x1 and x3 , in dependence of the frequencies of x2 and x4 .

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13.2. N=6x1

6th osc.
ω=αx2

0.5x1, x2, x3

Next, we consider a circuit model which is composedof six oscillators, as shown in Fig. 6. The observed synchronization phenomena for α = 0.64 are shown in Fig. 7.
From this figure, we infer that almost the same synchronization phenomena as those observed for N = 4 emerge.
Namely, that any ith oscillator is almost in-phase synchronized with the (i + 2)th oscillators, at the two oscillators’
inherent frequency.This synchronization pattern changes at
α = 3.47. In this case, the three oscillators with the standard frequency are synchronized at 120 degree phase difference (see Figs. 8,9).

x3

0-0.5

-1
3.99e+06

Figure 9:(α=3.47).

1st osc.
ω=1

3.9905e+06

3.991e+06

3.9915e+06

τp3.992e+06

Synchronization at 120 degree diﬀerence

180
phase difference (average)
phase difference (max)phase difference (min)160
140
phase difference

2nd osc.

5th osc.
ω=1

ω=α

120
100
80
60

4th osc.

ω=α

ω=140

3rd osc.

20
0

Figure 6: Circuit model for N = 6.

0

0.5
non-synchronization

1

1.5

α2

2.5

3

3.5

Figure 10: Phase diﬀerence between x1 and x3 in dependence of the frequencies of x2 , x4 and x6 .
y1

y2

y3

y4y5y6

Attractor
x1x3x2x4x3x5x4x6x5x1x6x2Phase
differencex1Timewave
form

x2x3x4x5x6x1x2x3x4x5x6

Figure 7: Synchronization results for N = 6 (at α=0.64).

y1y2y3y4y5y6

Attractor
x1x3x2x4x3x5x4x6x5x1x6x2

8 and 10, respectively. In the both cases, if we choose
α = 0.64, we observed that, as before for N = 4, 6, theith and the (i + 2) oscillators synchronize in-phase. Whenfrequency α is fixed to α = 3.47, however, 2 pairs of antiphase synchronized distant neighbors emerge for N = 8(see Fig. 11), and 5-phase synchronization emerges for
N = 10 (see Fig. 12).
From these results, we infer that N/2 pairs of anti-phasesynchronized oscillators could be observed if half of thetotal number of coupled oscillator is even. If the numberof pairs N/2 is an odd number, we, however, expect N/2phase synchronization to occur (at least for a not too highnumber N of oscillators).

Phase
differencex1
Timewaveform

x2x3x4x5x64. Synchronization phenomena when two out of threeoscillators are detuned

x1x2x3x4x5x6

Figure 8: Synchronization results for N = 6 (at α=3.47).

3.3. N=8, 10Furthermore, we investigated what synchronization patterns emerge if the number of coupled oscillators is set toThe conceptual circuit model corresponding to this situation is shown in Fig. 13. Figure 14 shows the simulation results if α is set to α = 0.64. In this case, we observe non-synchronization between oscillator 1 and oscillator 4, although they have the same standard oscillation
frequency. Instead, neighboring oscillators that have thechanging frequencies (2nd-3rd, and 5th-6th oscillators) are
synchronized anti-phase, as is shown in Fig. 14. At variance with the results presented in the previous section, the- 441 -1

1st Osc.

x2

0.8
0.6

x8x2, x8

0.4

2nd Osc.

y2

y1

3rd Osc.

4th Osc.

y4y3

5th Osc.

6th Osc.

y6

y5

Attractor

0.2
0x1

-0.2

x3x2x4

x6

x5

-0.4
-0.6
-0.8

Phase
difference-12.49e+06

2.4905e+06

2.491e+06

2.4915e+06

τp2.492e+06x6x3

10.8x6

0.6x2x4x5
Anti-synchronization

x4, x60.4
0.2
0-0.2x4
Phase
difference(x1-x4)

-0.4
-0.6
-0.8

-1
2.49e+06

2.4905e+06

2.491e+06
p2.4915e+06

τ

Non-synchronization
x1

2.492e+06

Figure 14:
Anti-phase synchronization and nonsynchronization (α = 0.64).

Figure 11: 2 pair anti-phase synchronization.
1x2

x3x5

x2x4

References

x1, x2, x3, x4, x5

0.5[1] M. G. Rosenblum, A. S. Pikovsky and J. Kurths, “Fromphase to lag synchronization in coupled chaotic oscillators,”
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0-0.5

-1
4.99e+06

4.9905e+06

4.991e+06τp4.9915e+06[2] Z. Zheng, G. Hu and B. Hu, “Phase slips and phase synchronization of coupled oscillators,” Phys. Rev. Let., vol.81,
no.24, pp.5318-5321, Dec. 1998.

4.992e+06[3] C. Zhou, J. Kurths, I. Z. Kiss and J. L. Hudson, “Noiseenhanced phase synchronization of chaotic oscillators,”
Phys. Rev. Let., vol.89, no.1, (014101), Jul. 2002.

Figure 12: 5-phase synchronization.

oscillators with ω = 1 can now oscillate even at the lowervalue α = 0.64, since seemingly they can benefit from theneighbor that is in the same situation. Together, they canmake it.
6th osc.
ω=α

4th osc.

[7] Y. Nishio and S. Mori, “Mutually coupled oscillators with anextremely large number of steady states,” Proc. of ISCAS’92,
vol.2, pp.819-822, May 1992.

2nd osc.
ω=α

ω=1

ω=α

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

1st osc.
ω=1

5th osc.
ω=α[4] T. Endo and S. Mori, “Mode analysis of a multimode ladderoscillator,” IEEE Trans. Circuits Syst., vol.23, pp.100-113,
Feb. 1976.

[8] L. Pivka, “Autowaves and spatio-temporal chaos in CNNs
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3rd osc.

[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.

Figure 13: Conceptual circuit model (N=6).

[10] Y. Uwate and Y. Nishio, “Complex Phase Synchronizationin an Array of Oscillators Coupled by Time-Varying Resistor,” Porc. of IJCNN’06, pp. 8345-8350, Jul. 2006.

5. Conclusions
In this study, we have investigated synchronization phenomena of distant van der Pol oscillators on a ring topology, where we cycle through the topology, detuning oscillators in a regular fashion. Using computer simulations, wehave confirmed that the distant oscillators can synchronize
in-phase, anti-phase, and N/2-phase, in dependence of theoscillation frequency of the adjacent oscillators.

[11] Y. Setou, Y. Nishio and A. Ushida, “Synchronization Phenomena in Resistively Coupled Oscillators with Diﬀerent
Frequencies,” IEICE Trans. Fundamentals, vol.E79-A,
no.10, pp.1575-1580, Oct. 1996.
[12] Y. Uwate, Y. Nishio and R. Stoop, “Group Synchronizationof van der Pol Oscillators with Diﬀerent Frequencies,” Porc.
of NOLTA’08, pp. 612-615, Sep. 2008.

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