Wave Propagation in Oscillators Coupled byTime-Varying Resistor with Timing Mismatch
Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp

Abstract— Synchronization phenomena in coupled oscillatory
systems are very important model to describe various higherdimensional nonlinear phenomena in the ﬁeld of natural science.
In this study, we investigate synchronization phenomena in vander Pol oscillators coupled by time-varying resistors with timing
mismatch as a ring. By carrying out computer simulations, we
conﬁrm the various interesting phenomena (wave propagation,clustering, complex phase pattern) which cannot be observed insimple oscillatory systems coupled by resistors.

I. I NTRODUCTION
In recent years, many people have been trying to develop
some applications to information processing by exploiting
synchronization phenomena in coupled oscillatory systems.
Such oscillatory systems can produce some kinds of phasepatterns, and they may be utilized modeling of associative
memory and learning process. Endo et al. have reported detailsof theoretical analysis and circuit experiments about somecoupled oscillators as a ladder, a ring and a two-dimensional
array [1]. Yamauchi et al. have discovered very interesting
wave propagation phenomena of phase states between two
adjacent oscillators in an array of van der Pol oscillators
coupled by inductors [2].
On the other hand, there are some systems whose dissipation
factors vary with time, for example, under the time-variationof the ambient temperature, an equation describing an object
moving in a space with some friction and an equation governing a circuit with a resistor whose temperature coefﬁcient issensitive such as thermistor. However, there are few discussion
about coupled oscillators coupling by a time-varying resistor.
In our previous research, we have investigated the synchronization phenomena in van der Pol oscillators coupled by timevarying resistor as a ring [3]. We conﬁrmed the coexistence ofin-phase and anti-phase synchronization and wave propagation
phenomena. However, it is not make clear the mechanism ofwave propagation in detail.
In this study, we investigate the wave propagation phenomena when timing mismatch is added to time-varying
resistors. We realize the time-varying resistor by switchinga 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, the caseof even number coupling, the coexistence of in-phase andanti-phase states are observed. In contrast, the case of odd
number coupling, we can conﬁrm the coexistence between
in-phase and n-phase states. Second, the coexistence areawith changing nonlinearity of oscillator is investigated. 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 coupled system is978-1-4244-1684-4/08/$25.00 ©2008 IEEE

investigated. We conﬁrm that the different synchronization
phenomena from the original coupled oscillatory system. For
example, wave propagation disappear and wave extinctionchanges to wave propagation.
II. C OUPLED O SCILLATORS M ODELIn this study, we consider a ring of oscillators as shownin Fig. 1. In this circuit adjacent two oscillators are coupledby one time-varying resistor (TVR). We realize the TVR byswitching a positive and a negative resistors periodically as
shown in Fig. 2. We introduce small resistors rm in computer
simulations to avoid L-loop.
2L iak ibk 2L

rm

iR1
TVR

rm

v1

C

Fig. 1.

R(t)TVR

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

iR2

rm

C

v2

rm
TVR

Coupled oscillators model.

2π

r
ωtt

0

-r
2πp

Fig. 2.

Characteristics of the TVR.

First, the vk − iRk characteristics of the nonlinear resistor
are deﬁned as follows,
iRk = −g1 vk + g3 vk 3 .

(1)

By changing the variables and the parameters,



√
g1
g1
C
xk , ik =vk =yk , t = LCτ ,
g3
3g3 L



L
C
C
1
ε = g1
ωt , η = rm
, γ=r
, ω= √
,
C
L
L
LC

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the normalized circuit equations of the array of oscillators are
given as⎧dxk⎪⎪
= ε(1 − xk 2 ) − (yak + ybk )
⎪
⎪
dτ
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ dyak = 1 xk − ηyak − γ(τ )(yak) + yb(k+1) )
dτ
2
(2)
⎪
⎪
⎪
dybk
1
⎪
⎪
= xk − ηybk − γ(τ )(ya(k−1) + ybk) )
⎪
⎪
⎪
dτ
2
⎪
⎪
⎪
⎩
(k = 1, 2, · · · , N )wherein not around π. Namely, in this case 15-phase synchronization
are observed.
Attractor
Phase
difference
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
x15

τ

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

(a) In-phase synchronization.

(3)
Attractor

It should be noted that γ(τ ) corresponds to the couplingstrength and that ε corresponds to the nonlinearity of oscillators. Eq. (2) is calculated by using the fourth-order RungeKutta method.

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

III. S YNCHRONIZATION P HENOMENA
A. Even Number Coupling: N = 14
Figure 3 shows the computer simulated result for the caseof 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.

τ(b) 15-phase synchronization.
Fig. 4. Computer simulated result for N = 15. ε = 2.6, ω = 1.37, γ =(0.2 or −0.2), η = 0.01. Upper ﬁgures: xk vs yak +ybk . Middle ﬁgures: xk
vs x(k+1) . Lower ﬁgures: τ vs xk . k = 1, 2, 3, . . . , 15.

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

C. Coexistence Area
Next, we investigate coexistence area between in-phase andanti-phase or N -phase states when the nonlinearity parameter
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
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.
We can conﬁrm that the special section around ε = 4.0, thecoexistence range is narrow extremely.

τ(a) In-phase synchronization.
Attractor
Phase
difference
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14

2.2

A

2
1.8

τ

1.6

ω

(b) Anti-phase synchronization.

D

1.2

Fig. 3. Computer simulated result for N = 14. ε = 2.6, ω = 1.37, γ =(0.2 or −0.2), η = 0.01. Upper ﬁgures: xk vs yak +ybk . Middle ﬁgures: xk
vs x(k+1) . Lower ﬁgures: τ vs xk . k = 1, 2, 3, . . . , 14.

B. Odd Number Coupling: N = 15
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)). However, theadjacent oscillators are almost synchronized with anti-phase as
shown in Fig. 4(b). Because, the boundary condition is the ring
structure, the phase difference between the adjacent oscillators

C

B

1.4

1
0.8
0.6
0.4
0.0

1.0

2.0

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.

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D. Wave Propagation Phenomena
In this section, we pay our attention to synchronization
phenomena on outside of coexistence area. The simulation
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 thevoltages of adjacent oscillators and the horizontal axis is time.
White regions in the diagram correspond to the states that thesum of the voltages is close to zero, namely the adjacent two
oscillators are synchronized at anti-phase. While, black regions
correspond to the state the sum of the voltages has large
amplitude. We can see that the adjacent two oscillators aresynchronized at in-phase in the black regions form Fig. 6. We
carried out computer simulation by setting initial condition at
random. In the region-A, wave extinction can be observed andthe synchronization states becomes stable at in-phase in wholeoscillators ﬁnally (Fig. 6 (a)). In the region-B, the coupledoscillators produce complex pattern, however the oscillators
do not synchronize stably (Fig. 6 (b)). In the region-C, we
can conﬁrm the wave propagation phenomena continue toexist (Fig. 6 (c)). In the region-D, the clustering phenomena
can be observed (Fig. 6 (d)). We consider that it becomes
possible to generate interesting synchronization phenomena bycoupling TVR.
x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1(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(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(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+x1(d) Region-D: ε=3.8, ω=1.31, γ=(0.2 or −0.2) and η=0.01.
Fig. 6.

Synchronization phenomena.

τ

τ

τ

τ

IV. TVR WITH T IMING M ISMATCH
We consider the synchronization phenomena when timing
mismatch is added to the switching of TVR. The characteristicof TVR with timing mismatch is shown in Fig. 7. From thisﬁgure, Ts means the one period of TVR and Tε denotes the
maximum value of timing mismatch. We produce the timing
mismatch range of 0 to Tε by uniform random.
0.3

Ts : period of TVR
Tε : maximum of timing mismatch
0.2

0.1

γ (τ) 0
-0.1

-0.2

Tε

Tε
Ts
-0.3

ωτ

Fig. 7.

TVR with timing mismatch (N=15).

Figure 8 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. 8 (a)).
In the region-B, the coupled oscillators are synchronized inanti-phase state (Fig. 8 (b)). In the region-C, wave extinction
can be observed and the synchronization states becomes inphase (Fig. 8 (c)). In the region-D, the clustering phenomena
can be observed (Fig. 8 (d)).
From these results of Fig. 6 and 8, we compare the synchronization phenomena between the original TVR and theTVR with timing mismatch. By adding the timing mismatch
to the switching of TVR, the different type of synchronization
phenomena can be obtained. Namely, in the region-A, thewave extinction change to wave propagation phenomena byadding timing mismatch. In the region-B, the complex patterndisappear and the synchronization state becomes anti-phase.
In the region-C, the wave propagation phenomena changes tothe wave extinction. Finally, in the region-D, we can conﬁrm
the same synchronization as clustering.
Next, we investigate the synchronization phenomena inRegion-C when the value of timing mismatch is changed
from 0 to 0.25. The simulated result is shown in Fig. 9.
Fist, for smaller Tε /Ts , the coupled oscillators produce wave
propagation. Second, for intermediate Tε /Ts , only in-phasesynchronization state can be observed. Finally, for larger
Tε /Ts , we could not conﬁrm synchronization phenomena,
namely the coupled oscillators do not synchronous.
Furthermore, we show synchronization phenomena when
the timing mismatch becomes large (Tε /Ts =0.3) and one example of simulated result is shown in Fig. 10. From this result,
we conﬁrm that the coupled oscillators are not synchronizedstably. We consider that timing of TVR switching is importantrole for synchronization state of whole oscillatory systems.
V. C YCLE S WITCHING OF TVR
Finally, we propose a cycle switching method of TVR as
follows. In this method, the switching of TVR is shifted by
2π/N regularly as shown in Fig. 11. Figure 12 shows thesimulation result. We can conﬁrm that the two types of wave
propagation repeat with irregularity. The wave which forms a115

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

0.204
0.203
0.202

Wave
propagation

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

γτIn-phase state

0.2
0.199

non-synchronization

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

0.198
0.197
0.196

0

0.05

0.1

0.15

0.2

0.25

Timing Mismatch(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(c) Region-C: ε=3.8, ω=1.43, γ=(0.2 or −0.2) and η=0.01.

τ

0.3

Τε / ΤsFig. 9. Synchronization types in 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+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

Fig. 10. Synchronization phenomena for TVR with parameter mismatch(Tε /Ts =0.3). (Region-C: ε=3.8, ω=1.43, γ=(0.2 or −0.2) and η=0.01).
0.3(d) Region-D: ε=3.8, ω=1.31, γ=(0.2 or −0.2) and η=0.01.

τ

0.2

0.1

Fig. 8.

Synchronization phenomena for TVR with timing mismatch.
γ

diamond propagates and the waveform changes to one wave
propagation.

-0.1

-0.2

VI. C ONCLUSIONSIn this study, we have investigated synchronization phenomena in van der Pol oscillators coupled by a time-varying
resistor as a ring. By carrying out computer simulations, we
conﬁrm the various interesting phenomena (wave propagation,clustering, complex phase pattern) which cannot be observedin simple oscillatory systems coupled by a resistor.

2π/Ν
-0.3

ωτ

Fig. 11.

TVR with cycle switching (N=15).
τ

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1R EFERENCES[1] T. Endo and S. Mori, “Mode analysis of a ring of a large number ofmutually 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 a ladder ofoscillators” IEICE Trans. Fundamentals, vol.E86-A, no.4, pp.891-899,
Apr. 2003.
[3] Y. Uwate and Y. Nishio, “Complex Phase Synchronization in an Array ofOscillators Coupled by Time-Varying Resistor,” Porc. of IJCNN’06, pp.
8345-8350, Jul. 2006.
[4] Y. Nishio and S. Mori, “Mutually coupled oscillators with an 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.

0Type-A

Type-B

Diamond wave propagation

One wave propagation

Fig. 12. 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).

116