2008 年 電子情報通信学会総合大会

A-2-7

Synchronization in a Ring of van der Pol
Oscillators with Different Frequencies
Yoko Uwate(Tokushima University)
Introduction

In this study, we investigate synchronization phenomena
in a ring of van der Pol oscillators with different frequencies.
Namely, only the N th oscillator has different frequency. By
carrying out computer simulations, we observe that oscillation of the N th oscillator stops in some range of the frequency
ratio and that others are synchronized as if the N th oscillator
does not exist.

2.

rmC

2L2L ia(k+1) ib(k+1) 2L

rmvk

k-th oscillator

x1x1x2x2x3

τ x3(a) Three-phase synchronization (ω=1.0).

iRk

rm

rmC2L

iRNvNvk(k+1)th oscillator

αC...

x1x2

τ(d) Double-mode oscillation (ω=1.09).

Fig. 2: Computer simulation results (ε=0.2, γ=0.2).
Next, we calculate the relationship between the amplitudeof the N th oscillator and the frequency ω. The simulated
results for N = 3, 4 is shown in Fig. 3.
Figure 4 shows the frequency at the break down of the N phase synchronization by changing the number of oscillators.
From this results, we confirm that the odd number couplingis more stable than even number coupling.
1.4

1.4

1.2

1.2

1

amplitude

The normalized circuit equations of the array of oscillators
are given as= xk − ηyak − γ(yak) + yb(k+1) )

dτ2



1dybk


=xk − ηybk − γ(ya(k−1) + ybk) )
dτ2



1
ε=0.1
ε=0.2
ε=0.3

0.8
0.6
0.4

ε=0.1
ε=0.2
ε=0.3

0.8
0.6
0.40.20dxk= ε(1 − xk 2 ) − (yak + ybk )dτdxN
= ω 2 ε(1 − xN 2 ) − ω 2 (yaN + ybN )
 dτ
1x3

τ
(b) Anti-phase and oscillation death (ω=2.47).

N-th oscillator

Fig. 1: Circuit model.

dyakτ(c) Anti-phase and independent oscillations(ω=0.64).

x1x3

The circuit model is shown in Fig. 1. In the system only
the N th oscillator has different oscillation frequency.

iRkoscillates alone and the others keep anti-phase synchronization. Furthermore, we observe the double-mode oscillation
as shown in Fig. 2(d).

x2Circuit Model

2L iak ibk 2L

Ruedi Stoop(UNI / ETH Zurich)

amplitude

1.

Yoshifumi Nishio(Tokushima University)

0.2

0

0.5

1

1.5

ω

2

2.5

3

0

3.5

0

0.5

1

(a) N=3.

1.5

ω

2

2.5

3

3.5

(b) N=4.

Fig. 3: Relation between amplitude and ω .
(1)

1.12
1.1
1.08
1.06
1.04

ω

(k = 1, 2, · · · , N )

1.02
1

where

0.98

(2)

0.96

It should be noted that γ corresponds to the couplingstrength and that ε corresponds to the nonlinearity of oscillators.

0.92

ya0 = yaN ,yb(N +1) = yb1 .

3. Synchronization Phenomena
Figure 2 shows observed phenomena for N = 3. For thecase that all of these oscillations have the same frequency, wecan observe that 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 not synchronized to the others. Namely, the 3rd oscillator

2008/3/18 〜 21

北九州市

0.94

0.92

46810121416N

Fig. 4: Frequency for break down of N -phase synchronization (ε=0.2, γ=0.2).
5. Conclusions
In this study, we have investigated synchronization phenomena in a ring of van der Pol oscillators with different
frequencies. By carrying out computer simulations, we observed that oscillation of the N th oscillator stops in somerange of the frequency ratio and that others are synchronized
as if the N th oscillator does not exist.

46（基礎・境界講演論文集）

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