2008 International Symposium on Nonlinear Theory and its Applications
NOLTA'08, Budapest, Hungary, September 7-10, 2008

Group Synchronization of van der Pol Oscillators with Different Frequencies
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—In this study, synchronization phenomena ofvan der Pol oscillators coupled by resistors in a ring are investigated, when the frequency error are added to some oscillators. Using computer simulations, we observe that thevan der Pol oscillators with frequency error can produce thenonlinear phenomena related with synchronization such as
oscillation death, independent oscillation and double-modeoscillation.

has different oscillation frequency from the others. We realize the frequency error of the Nth van der Pol oscillator
by setting the different value of the capacitor of this oscillator in comparison to the others.
2L
2L i akibk2Lrm

2L i a(k+1) i b(k+1) 2L
rmrm

1. Introduction
Various synchronization phenomena observed in coupled oscillatory systems have the possibility to model highfunctional information processing of a human brain. Suchoscillatory systems can produce the interesting phase pattern including: wave propagation, clustering, complex
phase pattern [1]–[4]. Setou et al. have observed interesting synchronization phenomena when van der Pol oscillators with different oscillation frequencies are coupled bya resistor as a star structure [5]. However, it is difficult toinvestigate synchronization phenomena of large-scale circuits, because, van der Pol oscillators coupled in a star
topology do not synchronize when the number of couplingoscillators is larger than 4.
In this study, we consider a ring consisting of van der
Pol oscillators with different oscillation frequencies for investigation of large-scale circuits. It is possible to add thefrequency error to the van der Pol oscillators by setting thedifferent value of a capacitor of oscillators from the others.
Using computer simulations, we observe that the van derPol oscillators with frequency error can produce synchronization phenomena such as oscillation death, independentoscillation and double-mode oscillation for large-scale oscillatory systems. Furthermore, we investigate four typesof arrangements of the van der Pol oscillators with frequency error and investigate the synchronization phenomena for each coupling method.

v(k+1)

RC(k+1)-th oscillator

i RNC

vk

k-th oscillator

αCR...

vN

N-th oscillator(a) Ring of van der Pol oscillators.
with different frequency
ω=?

ω=1

ω=1

ω=1

ω=1

ω=1(b) Conceptual circuit model.

Figure 1: Circuit model and concept of circuit system.

First, we assume that the vk − iRk characteristics of thenonlinear resistor in each oscillator is represented by thefollowing third order polynomial equation.
iRk = −g1 vk + g3 vk 3

(1)

By changing the variables and the parameters,
rrr
√
g1
g1
Cvk =
xk , ik =yk , t = LCτ,
3g3
3g3 L

2. Circuit Modelε = g1The basic circuit model and the concept of circuit system
are shown in Fig. 1. In the system, only the Nth oscillator

rmiRki Rk
R2Lr

L, γ=rC

r

1C, α = 2 , η = rm
L
ωrC,
L

the normalized circuit equations of the ring of oscillators- 612 -

10.5

are given as

x1 0-0.5-1where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)
ya0 = yaN ,yb(N+1) = yb1 .

10.5

x2 0-0.5-110.5(2)x3 0-0.5-1
199800199840199880199920199960200000τ(a) Three-phase synchronization (ω1 =ω3 =ω3 =1.0).
10.5

x1 0-0.5
-1(3)

1
0.5

x2 0-0.5-1

It should be noted that ωk denotes the frequency of kth oscillators, γ corresponds to the coupling strength and that
ε corresponds to the nonlinearity of oscillators. Eq. (2) iscalculated by using the fourth-order Runge-Kutta method.

10.5

x3 0-0.5-1
199800199840199880199920199960200000τ(b) Anti-phase and oscillation death (ω1 =1.0, ω2 =1.0, ω3 =0.64).

3. Synchronization Phenomena

10.5

x1 0

Figure 2 shows the observed phenomena for coupling
oscillator number N=3. For the case that all of these oscillations have the same frequency, we can observe thatthe system is synchronized at the three-phase (Fig. 2(a)).
When the frequency of the 3rd oscillator is varied, we confirm that oscillation of the 3rd oscillator stops, namely oscillation death appears (Fig. 2(b)). As increasing the frequency of the 3rd oscillator, oscillation of the 3rd oscillator
starts again (Fig. 2(c)). However, the 3rd oscillator is notsynchronized to the others. Namely, the 3rd oscillator oscillates alone and the others keep anti-phase synchronization. Furthermore, we observe the double-mode oscillation
as shown in Fig. 2(d).
Next, we calculate the relationship between the amplitude of the Nth oscillator and the frequency ωN when thenonlinearity parameter are changed from ε=0.06 to 0.2.
The simulated results for N=3, 4, 9, 10, 15 and 16 areshown in Fig. 3. The graph form of amplitude is different between odd and even number of coupling oscillators.
In the case of odd number, the amplitude takes the peak on
ωN =1.0 and around ωN =1.2 the oscillation death appears
when the nonlinearity parameter εN is set to 0.06. While, inthe case of even number, although the amplitude takes thepeak on ω=1.0, the oscillation death can not be observed.
Figure 4 shows the frequency at the break down of theanti-phase or N-phase synchronization by changing thenumber of oscillators. From these results, we confirm thatthe even number coupling is more stable than the odd number coupling systems for the frequency error.
4. Group Coupling
In this section, we investigate four types of arrangementsof the van der Pol oscillators with frequency error and investigate the synchronization phenomena for each coupling
method.

-0.5
-1
10.5

x2 0-0.5-1
10.5

x3 0-0.5-1199800199840199880199920199960200000τ(c) Anti-phase and independent oscillation (ω1 =1.0, ω2 =1.0, ω3 =3.47).
10.5

x1 0-0.5-1
10.5

x2 0-0.5-1
10.5

x3 0-0.5-1195000196000197000198000199000200000τ(d) Double-mode oscillation (ω1 =1.0, ω2 =1.0, ω3 =1.07).

Figure 2: Computer calculation for N=3 (ε = 0.2, γ = 0.2,
η = 0.01).

4.1. Two groups
First, we consider that the coupled oscillatory systems
are composed of two groups with different oscillatory frequencies as shown in Fig. 5. In the Type-A, the two typesof oscillators with different frequencies are set to half andhalf. And, in the case of Type-B, we place the two types ofoscillators alternately. From Fig. 5, α denotes the different
frequency to the standard oscillators with ω=1.0. As a firststep, we consider the case of that number of coupling oscillators are set to even number to distinguish two groups
clearly.
The simulated results of Type-A are shown in Fig. 6.
From this figure, we confirm that each group are synchronized in anti-phase with own frequency and two groups do- 613 -

1.2

1

1

0.8

ε=0.06ε=0.08
ε=0.1
ε=0.2

0.6

amplitude

amplitude1.2

ε=0.06ε=0.08
ε=0.1
ε=0.2

0.6
0.4

0.4

0.2

0.2
00.8

00.5

1

1.5

ωΝ2

2.5

3

0

3.5

00.5

1(a) N=3.

ε=0.06ε=0.08
ε=0.1
ε=0.2

0.6

amplitude

amplitude

2.5

3

3.5

1

0.8

0.4

0.8

ε=0.06ε=0.08
ε=0.1
ε=0.2

0.6
0.4

0.2

0.2
00.5

1

1.5

ωΝ2

2.5

30

3.5

00.51(c) N=9.

1.5

ωΝ2

2.5

33.5(d) N=10.

1.2

1.2

1

10.8

ε=0.06ε=0.08
ε=0.1
ε=0.2

0.6

amplitude

amplitude

21.2

10.4

ω=α0.8ε=0.06ε=0.08
ε=0.1
ε=0.2

0.6

ω=1

ω=α

ω=α0.40.20ωΝ(b) N=4.

1.2

0

1.5

not synchronize. Figure 7 shows the observed phenomenaof coupled oscillators as Type-B. In this case, every oscillators are not synchronized. When the value of α equal to0.64, the oscillation death can be observed (Fig. 7(a)).
Next, we compare the break down frequency fromsynchronization to non-synchronization between Type-A
and Type-B coupling methods by changing the number of coupling oscillators from 4 to 14. Where ”nonsynchronization” denotes the break down of group synchronization. Figure 8 shows the frequency for break downof synchronization. In the case of Type-A, the synchronization area is decreasing with number of coupling oscillators. While in the case of Type-B, the frequency at break
down of synchronization is constant even if the number ofcoupling oscillators is changed. We confirm that Type-B
is more stable for synchronization by increasing the frequency error than Type-A coupling.
ω=1

ω=1

ω=1

ω=α0.2
00.5

11.5

ωΝ

2

2.5

3

0

3.5

00.5

1(e) N=15.

1.5

ωΝ

2

2.5

3

ω=α3.5ω=α

ω=1(a) Type-A(f) N=16.

ω=1(b) Type-B

Figure 5: Coupling model for two groups.

Figure 3: Relation between amplitude and ωN .
1.2

Independent oscillation

1.15

y1

y2

y3

y4y5

y6

Attractor

1.1x1

ωΝ

1.05

Phase
differencex2

Anti-phase synchronization

1x2x3x1x3x4x2x4x5x3x5x6x4x6x1x5x6x1x2

0.95

x3

0.9x4x50.85
0.8

Oscillation death46810x612τ

14(a) α=0.64.

Number of oscillators(a) Break down for anti-phase synchronization.

y1

y2

y3

y4

y5

y6

Attractor

1.2

x1x2

Independent oscillation

1.15x11.1

ωΝ

x3x4x4x5x5x6x6x1x2x3x4x5x6x1x2

1.05

x3x4

N-phase synchronization

1x5x6

0.95
0.9

τ(b) α=3.47.
Oscillation death0.85
0.8x2x3

Phase
difference

35791113

Figure 6: Computer simulation results of Type-A (N=6).

15

Number of oscillators(b) Break down for N-phase synchronization.

4.2. Three groups
Figure 4: Frequency for break down of synchronization(ε = 0.2, γ = 0.2, η = 0.01).

Next, we consider that the coupled oscillatory systems
are composed of three groups with different oscillatory fre-- 614 -y1y2

y3y4y5

ω=βy6

ω=1

ω=β

ω=1

Attractorx1

Phase
differencex2x2x3x3x1x4x4x5x5x2x6x3x6

ω=βx1x4x5

ω=α

ω=1

ω=αx6

ω=α

ω=αx1

ω=β

ω=1x2

(a) Type-Cx3(b) Type-Dx4x5x6

Figure 9: Coupling model for three groups.

τ(a) α=0.64.
y1

y2

y3

y4

y5

y6

y1

Attractor

y2

y3

y4

y5

y6

Attractor
x1

Phase
difference

x2

x2x3

x3x4x4x5x5x6x6x1x1

Phase
differencex1

x2x3x4x5x6x2x3

x1

x1x1x2x2x3x3x4x4x5x5x6x2

x3x4x2x4x5x3x5x6x4x6x1x5x6x6

τ

τ(b) α=3.47.

(a) Type-A.

Figure 7: Computer simulation results of Type-B (N=6).

y1y2y3y4y5y6

Attractor
x1

1.15

1.15

αα

Synchronization

10.9
46810
Number of oscillators(a) Type-A.

1214

0.85

x3x4x2x4x5x3x5x6x4x6x1x5x6x2

Synchronization

1x3x4x5

0.9

Non-synchronizationx2x3x1

0.95

0.95

x2x1

1.05

1.05

0.85

Non-synchronization

1.1

Non-synchronization

1.1

Phase
differencex6

Non-synchronization
46810Number of oscillators

12τ

14(b) Type-B.

(b) Type-B.

Figure 8: Frequency for break down of synchronization(ε = 0.2, γ = 0.2, η = 0.01).

quencies as shown in Fig. 9. In the Type-C, the three typesof oscillators with different frequencies are set as shownin Fig. 9(a). And, in the case of Type-D, we place thethree types of oscillators alternately. From Fig. 9, α and β
denote the different frequencies to the standard oscillatorswith ω=1.0.
The simulated results are shown in Fig. 10. From thisfigure, in the case of Type-C, we confirm that each groupare synchronized in anti-phase with own frequency and twogroups do not synchronize. In the Type-D, any adjacentoscillators are not synchronized.
5. Conclusions
In this study, we have investigated a ring coupling vander Pol oscillators with different oscillation frequencies.
By computer simulations, we observe that the van der Pol

Figure 10: Computer simulation results of Type-C and
Type-D (N=6, α=0.64, β=3.47).

oscillators with frequency error can produce the interesting
nonlinear phenomena such as oscillation death, independent oscillation and double-mode oscillation.
References[1] T. Endo and S. Mori, “Mode analysis of a ring of a large numberof mutually coupled van der Pol oscillators,” IEEE Trans. Circuits
Syst., vol.25, no.1, pp.7-18, Jan. 1978.
[2] 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.
[3] M. Yamauchi, Y. Nishio and A. Ushida, ”Phase-waves in a ladderof oscillators” IEICE Trans. Fundamentals, vol.E86-A, no.4,pp.891-899, Apr. 2003.
[4] Y. Uwate and Y. Nishio, “Complex Phase Synchronization in an
Array of Oscillators Coupled by Time-Varying Resistor,” Porc. ofIJCNN’06, pp. 8345-8350, Jul. 2006.
[5] 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.

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