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THE INSTITUTE OF ELECTRONICS,
INFORMATION AND COMMUNICATION ENGINEERS

信学技報
TECHNICAL REPORT OF IEICE.

タイミングミスマッチを伴う時変抵抗で結合された発振器で観測される同期現象について
上手 洋子†

西尾 芳文†† 徳島大学大学院ソシオテクノサイエンス研究部〒 770–8506 徳島市南常三島 2–1
E-mail: †{uwate,nishio}@ee.tokushima-u.ac.jp
あらまし

結合発振器の同期現象は自然界で観測される高次元な非線形現象を記述するのに優れたモデルである。本

研究では、タイミングミスマッチを伴う時変抵抗で結合された van der Pol 発振器で観測される同期現象について調査を行う．コンピュータシミュレーションの結果，位相伝播波，クラスタリング，複雑位相パターンなどの様々な興味深い同期現象を観察することができた。これらの現象はシンプルな抵抗で結合された発振器では観測できない現象
である。
キーワード

結合発振器，伝播波，時変抵抗

Synchronization Phenomena in Oscillators Coupled by Switching
Resistors with Timing Mismatch
Yoko UWATE† and Yoshifumi NISHIO†† Tokushima University 2–1 Minami Josanjima, Tokushima, 770–8506 Japan
E-mail: †{uwate,nishio}@ee.tokushima-u.ac.jp
Abstract Synchronization phenomena in coupled oscillatory systems are very important model to describe various
higher-dimensional nonlinear phenomena in the ﬁeld of natural science. In this study, we investigate synchronization
phenomena in van der Pol oscillators coupled by time-varying resistors with timing mismatch as a ring. By carryingout computer simulations, we conﬁrm the various interesting phenomena (wave propagation, clustering, complex
phase pattern) which cannot be observed in simple oscillatory systems coupled by resistors.
Key words coupled oscillators, wave propagation, time-varying resistor

1. Introduction

pation factors vary with time, for example, under the timevariation of the ambient temperature, an equation describing

In recent years, many people have been trying to develop

an object moving in a space with some friction and an equa-some applications to information processing by exploitingtion governing a circuit with a resistor whose temperaturesynchronization phenomena in coupled oscillatory systems.

coeﬃcient is sensitive such as thermistor. However, there

Such oscillatory systems can produce some kinds of phase

are few discussion about coupled oscillators coupling by apatterns, and they may be utilized modeling of associa-

time-varying resistor.

tive memory and learning process. Endo et al. have re-In our previous research, we have investigated the synchro-ported details of theoretical analysis and circuit experiments

nization phenomena in van der Pol oscillators coupled byabout some coupled oscillators as a ladder, a ring and a two-time-varying resistor as a ring [3]. We conﬁrmed the coexis-

dimensional array [1]. Yamauchi et al. have discovered very

tence of in-phase and anti-phase synchronization and wave

interesting wave propagation phenomena of phase states be-

propagation phenomena. However, it is not make clear thetween two adjacent oscillators in an array of van der Pol

mechanism of wave propagation in detail.

oscillators coupled by inductors [2].
On the other hand, there are some systems whose dissi-

In this study, we investigate the wave propagation phenomena when timing mismatch is added to time-varying re—1—

2L iak ibk 2L

rm

rm

iR1
TVR

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

v1

C

iR2

rm

rm
C

TVR

v2

TVR

Fig. 1 Coupled oscillators model.

sistors. We realize the time-varying resistor by switching a
positive and a negative resistor periodically. This proposed
oscillatory system is novelty, this is because the coupling
term has dynamics by switching value of resistor. First, theare deﬁned as follows,
iRk = −g1 vk + g3 vk 3 .
By changing the variables and the parameters,√case of even number coupling, the coexistence of in-phaseand anti-phase states are observed. In contrast, the case ofvk =odd number coupling, we can conﬁrm the coexistence be-

By carrying out computer simulations, we conﬁrm the various interesting phenomena (wave propagation, clustering,
complex phase pattern) which cannot be observed in simple
oscillatory systems coupled by a resistor. Next, we carry out
computer simulations when the timing mismatch is added
to the TVR. The inﬂuence of small error for whole coupledsystem is investigated. We conﬁrm that the diﬀerent synchronization phenomena from the original coupled oscillatory system. For example, wave propagation disappear andwave extinction changes to wave propagation.

g1
xk ,
g3

√

tween in-phase and n-phase states. Second, the coexistencearea with changing nonlinearity of oscillator is investigated.

(1)

ε = g1

L
,
C

√
ik =

√
γ=r

g1
3g3

C
,
L

√

C
yk ,
L

ω= √

t=

√

1
ωt ,
LC

LCτ ,

√
η = rmC,
L

the normalized circuit equations of the array of oscillators
are given as

 dx
k

= εxk (1 − xk 2 ) − (yak + ybk )


dτ






dyak
1


 dτ = 2 xk − ηyak − γ(τ )(yak) + yb(k+1) )


dybk
1


= xk − ηybk − γ(τ )(ya(k−1) + ybk) )


dτ
2







(2)

(k = 1, 2, · · · , N )2. Coupled Oscillators Model

where

In this study, we consider a ring of oscillators as shown in

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

(3)Fig. 1. In this circuit adjacent two oscillators are coupledby one time-varying resistor (TVR). We realize the TVR by

It should be noted that γ(τ ) corresponds to the couplingswitching a positive and a negative resistors periodically as

strength and that ε corresponds to the nonlinearity of oscil-shown in Fig. 2. We introduce small resistors rm in computer

lators. Eq. (2) is calculated by using the fourth-order Runge-

simulations to avoid L-loop.

Kutta method.

R(t)3. Synchronization Phenomena
2π

3. 1 Even Number Coupling: N = 14r

Figure 3 shows the computer simulated result for the case

ω tt

0

of N = 14. N denotes the number of coupled oscillators.
We can see that the array of oscillators coupled by TVR are
synchronized at in-phase or at anti-phase.

-r

3. 2 Odd Number Coupling: N = 152πpFig. 2 Characteristics of the TVR.

Figure 4 shows the computer simulated result for the caseof N = 15. We can see that the array of oscillators coupled
by TVR are synchronized with in-phase (Fig. 4(a)). How-First, the vk − iRk characteristics of the nonlinear resistor

ever, the adjacent oscillators are almost synchronized with—2—

3. 3 Coexistence Area

Attractor

Next, we investigate coexistence area between in-phase andPhase
differenceanti-phase or N -phase states when the nonlinearity param-

x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14

eter ε is changed from 0.4 to 5.0. We carried out computer
calculations for the number of the coupling oscillators is set
to N = 15. The simulated result is shown in Fig. 5. The
horizontal axis is ε and the vertical axis is the frequency
τ(a) In-phase synchronization.

of TVR. From this ﬁgure, the coexistence range becomeswidely with increasing the value of ε. The average value ofthe coexistence range decreases gradually when ε increases.

Attractor

We can conﬁrm that the special section around ε = 4.0, thePhasedifference

coexistence range is narrow extremely.

x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14

2.2

A

2
1.8
1.6

ω

τ
(b) Anti-phase synchronization.

Fig. 3 Computer simulated result for N = 14.

ε = 2.6, ω =

C

B

1.4

D

1.2

1.37, γ = (0.2 or −0.2), η = 0.01. Upper ﬁgures: xk vs1

yak + ybk . Middle ﬁgures: xk vs x(k+1) . Lower ﬁgures: τ

0.8

vs xk . k = 1, 2, 3, . . . , 14.

0.6
0.4
0.0

1.0

2.0

anti-phase as shown in Fig. 4(b). Because, the boundary
condition is the ring structure, the phase diﬀerence between
the adjacent oscillators in not around π. Namely, in this case

ε

3.0

4.0

5.0

Fig. 5 Coexistence area of in-phase and anti-phase synchronizations. γ = (0.2 or −0.2), η = 0.01.

15-phase synchronization are observed.
3. 4 Wave Propagation Phenomena
Attractor

In this section, we pay our attention to synchronization

Phase
difference

phenomena on outside of coexistence area. The simulation

x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
x15

A °,
B °,
C °)
D from Fig. 5 are shown
results of four-regions (°,in Fig. 6.

In this ﬁgure, the vertical axis is the sum of

the voltages of adjacent oscillators and the horizontal axis istime. White regions in the diagram correspond to the statesτthat the sum of the voltages is close to zero, namely the adjacent two oscillators are synchronized at anti-phase. While,(a) In-phase synchronization.

black regions correspond to the state the sum of the voltages
Attractor

has large amplitude. We can see that the adjacent two oscil-Phase
difference

lators are synchronized at in-phase in the black regions form

x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
x15

Fig. 6. We carried out computer simulation by setting initialcondition at random. In the region-A, wave extinction can beobserved and the synchronization states becomes stable at inphase in whole oscillators ﬁnally (Fig. 6 (a)). In the regionτ

B, the coupled oscillators produce complex pattern, however
the oscillators do not synchronize stably (Fig. 6 (b)). In the(b) 15-phase synchronization.

ε = 2.6, ω =region-C, we can conﬁrm the wave propagation phenomena

1.37, γ = (0.2 or −0.2), η = 0.01. Upper ﬁgures: xk vscontinue to exist (Fig. 6 (c)). In the region-D, the clusteringyak + ybk . Middle ﬁgures: xk vs x(k+1) . Lower ﬁgures: τphenomena can be observed (Fig. 6 (d)). We consider thatvs xk . k = 1, 2, 3, . . . , 15.

it becomes possible to generate interesting synchronization

Fig. 4 Computer simulated result for N = 15.

phenomena by coupling TVR.
—3—

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1

τ

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1

τ

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1

τ

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1ττ(c) Region-C: ε=3.8, ω=1.46, γ=(0.2 or −0.2) and η=0.01.

(c) Region-C: ε=3.8, ω=1.43, γ=(0.2 or −0.2) and η=0.01.
x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1τ(b) Region-B: ε=1.8, ω=1.27, γ=(0.2 or −0.2) and η=0.01.

(b) Region-B: ε=1.8, ω=1.47, γ=(0.2 or −0.2) and η=0.01.
x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1

τ(a) Region-A: ε=1.8, ω=2.15, γ=(0.2 or −0.2) and η=0.01.

(a) Region-A: ε=1.8, ω=1.99, γ=(0.2 or −0.2) and η=0.01.

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1(d) Region-D: ε=3.8, ω=1.31, γ=(0.2 or −0.2) and η=0.01.

(d) Region-D: ε=3.8, ω=1.21, γ=(0.2 or −0.2) and η=0.01.

Fig. 6 Synchronization phenomena.

Fig. 7 Synchronization phenomena.

τ

0.3

Ts : period of TVR
Tε : maximum of timing mismatch

3. 5 Synchronization in dependence on ω
Next, we investigate the synchronization phenomena when

0.2

the frequency of TVR ω is changed. The simulated results ofA °,B °C and °
D are shown in Fig. 7. From this reRigions °,
A °
B °
D are
sults, we conﬁrm that the phenomena of Rigions °0.1

γ (τ) 0almost similar to Fig. 6. However, the phenomena of Rigion
C is diﬀerent which means the wave propagation disappear°

by changing ω.

-0.1

-0.2

Tε

Tε

4. TVR with Timing Mismatch
We consider the synchronization phenomena when timing
mismatch is added to the switching of TVR. The character-

Ts
-0.3

ωτ

Fig. 8 TVR with timing mismatch (N=15).

istic of TVR with timing mismatch is shown in Fig. 8. From
this ﬁgure, Ts means the one period of TVR and Tε denotes
the maximum value of timing mismatch. We produce thetiming mismatch range of 0 to Tε by uniform random.
Figure 9 shows the simulation results when the parametersA °,B °C and °
D from Fig. 5. In the regionare set to region °,A, wave propagation phenomena can be observed (Fig. 9 (a)).
In the region-B, the coupled oscillators are synchronized inanti-phase state (Fig. 9 (b)). In the region-C, wave extinc-tion can be observed and the synchronization states becomesin-phase (Fig. 9 (c)). In the region-D, the clustering phenomena can be observed (Fig. 9 (d)).
From these results of Fig. 6 and 8, we compare the synchronization phenomena between the original TVR and the TVRwith timing mismatch. By adding the timing mismatch tothe switching of TVR, the diﬀerent type of synchronization
phenomena can be obtained. Namely, in the region-A, the—4—wave extinction change to wave propagation phenomena by

is important role for synchronization state of whole oscilla-

adding timing mismatch. In the region-B, the complex pat-

tory systems.

tern disappear and the synchronization state becomes anti0.204

phase. In the region-C, the wave propagation phenomenachanges to the wave extinction. Finally, in the region-D, we

0.203

can conﬁrm the same synchronization as clustering.

0.202

Wave
propagation

0.201
x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1γIn-phase state

0.2
0.199

non-synchronization
0.198
0.197

τ
(a) Region-A: ε=1.8, ω=1.99, γ=(0.2 or −0.2) and η=0.01.

0.196

0

0.05

0.1

0.15

0.2

Timing Mismatch

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x10.25

0.3

Τε / ΤsFig. 10 Synchronization types in Region-C. ε=3.8, ω=1.43,γ=(0.2 or −0.2) and η=0.01.

τ(b) Region-B: ε=1.8, ω=1.47, γ=(0.2 or −0.2) and η=0.01.

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1ττ(c) Region-C: ε=3.8, ω=1.43, γ=(0.2 or −0.2) and η=0.01.

Fig. 11 Synchronization phenomena for TVR with parameter mismatch (Tε /Ts =0.3). (Region-C: ε=3.8, ω=1.43,

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1γ=(0.2 or −0.2) and η=0.01).

5. Cycle Switching of TVRτFinally, we propose a cycle switching method of TVR as(d) Region-D: ε=3.8, ω=1.31, γ=(0.2 or −0.2) and η=0.01.

follows. In this method, the switching of TVR is shifted by

Fig. 9 Synchronization phenomena for TVR with timing mis-

2π/N regularly as shown in Fig. 12. Figure 13 shows the sim-

match.

ulation result. We can conﬁrm that the two types of wave
propagation repeat with irregularity. The wave which forms

Next, we investigate the synchronization phenomena inRegion-C when the value of timing mismatch is changed froma diamond propagates and the waveform changes to one wave
propagation.

0 to 0.25. The simulated result is shown in Fig. 10. Fist, forFigre 14 shows the examples of wave propagation whensmaller Tε /Ts , the coupled oscillators produce wave propaga-

the number of coupled oscillators is changed for TVR withtion. Second, for intermediate Tε /Ts , only in-phase synchro-cycle switching.

nization state can be observed. Finally, for larger Tε /Ts , wecould not conﬁrm synchronization phenomena, namely the
coupled oscillators do not synchronous.

6. Conclusions
In this study, we have investigated synchronization phe-

Furthermore, we show synchronization phenomena whennomena in van der Pol oscillators coupled by a time-varying

the timing mismatch becomes large (Tε /Ts =0.3) and one ex-resistor as a ring. By carrying out computer simulations,

ample of simulated result is shown in Fig. 11. From thiswe conﬁrm the various interesting phenomena (wave propa-

result, we conﬁrm that the coupled oscillators are not syn-gation, clustering, complex phase pattern) which cannot bechronized stably. We consider that timing of TVR switchingobserved in simple oscillatory systems coupled by a resistor.

—5—

0.3

0.2

0.1

γ

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x1

0

-0.1

-0.2

τ
(a) N=6.

2π/Ν

-0.3

ωτ

x1+x2
x2+x3
x3+x4
x4+x5

Fig. 12 TVR with cycle switching (N=15).

x5+x6
x6+x7
x7+x1

τ

τ
x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1(b) N=7.
x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x1

Type-A

Type-B

Diamond wave propagation

One wave propagation

τ(c) N=11.

Fig. 13 Two types of wave propagation for TVR with cycle
switching. (Region-C: ε=3.8, ω=1.43, γ=(0.2 or -0.2)
and η=0.01).

References[1] 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.
[2] M. Yamauchi, Y. Nishio and A. Ushida, ”Phase-waves in aladder of oscillators” IEICE Trans. Fundamentals, vol.E86A, no.4, pp.891-899, Apr. 2003.
[3] 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.
[4] Y. Nishio and S. Mori, “Mutually coupled oscillators withan extremely large number of steady states,” Proc. of ISCAS’92, vol.2, pp.819-822, May 1992.
[5] Y. Nishio and S. Mori, “Chaotic phenomena in nonlinear
circuits with time-varying resistors,” IEICE Trans. Fundamentals, vol.E76-A, no.3, pp.467-475, Mar. 1993.

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x1

τ(d) N=12.

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x1τ(e) N=13.

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15

τ(f) N=14.

Fig. 14 Examples of wave propagation by changing numberof coupled oscillators for TVR with cycle switching.
(Region-C: ε=3.8, ω=1.43, γ=(0.2 or -0.2) and η=0.01).

—6—