International Journal of Bifurcation and Chaos, Vol. 17, No. 10 (2007) 3565–3569
c World Scientiﬁc Publishing Company
SYNCHRONIZATION PHENOMENA IN VAN DERPOL OSCILLATORS COUPLED BY
A TIME-VARYING RESISTOR
YOKO UWATE∗ and YOSHIFUMI NISHIO†
Department of Electrical and Electronics Engineering,
Tokushima University, 2-1 Minami-Josanjima,
Tokushima 770-8506, Japan∗uwate@ee.tokushima-u.ac.jp†nishio@ee.tokushima-u.ac.jp
Received October 10, 2005; Revised March 9, 2006
In this study, synchronization phenomena observed in van der Pol oscillators coupled by a timevarying resistor are investigated. We realize the time-varying resistor by switching a positive anda negative resistor periodically. By carrying out circuit experiments and computer calculations,interesting synchronization phenomena can be conﬁrmed to be generated in this system. Namely,
the synchronization states change according to the switching frequency of the time-varyingresistor.
Keywords: Coupled oscillators; synchronization; time-varying resistor.

1. Introduction
Synchronization phenomena in complex systems
are very interesting to describe various higherdimensional nonlinear phenomena in the ﬁeld ofnatural science. Studies on synchronization phenomena of coupled oscillators are extensively carried out in various ﬁelds, such as in physics [Bonillaet al., 1998; Sherratt, 1998; Abramson et al., 2002;
Belykh, 2005], biology [Gray, 1994; Cosp et al.,2004] and engineering [Suezaki & Mori, 1965;
Kimura & Mano, 1965; Endo & Mori, 1976a, 1976b,1978; Datardina & Linkens, 1978; Nishio & Mori,1992; Yamauchi et al., 1999; Lin et al., 2005].
Furthermore many researchers suggest that synchronization phenomena of coupled oscillators havesome relations to information processing in thebrain. We consider that it is very important toinvestigate the synchronization phenomena of coupled oscillators to realize a brain computer forfuture engineering applications.
On the other hand, there are some systems
whose dissipation factors vary with time, forexample, under the time-variation of the ambient

temperature, an object moving in a space with somefriction and a circuit with a resistor whose temperature coeﬃcient is sensitive such as thermistor. However, oscillators coupled by a time-varying resistor
have been barely studied.
In this study, synchronization phenomenaobserved in van der Pol oscillators coupled by atime-varying resistor are investigated. We realize
the time-varying resistor by switching a positive anda negative resistor periodically. By carrying out circuit experiments and computer calculations, interesting synchronization phenomena can be conﬁrmed
to be generated in this system. Further, we investigate the inﬂuence of the duty ratio of the switching,namely we vary the ratio of time intervals connecting to the positive and the negative resistors. Various interesting features of the coupled system canbe clariﬁed.

2. Circuit Model
Figure 1 shows the circuit model. In this circuit,
two identical van der Pol oscillators are coupled by3565

3566

Y. Uwate & Y. Nishio

LI2I1

Γ(τ)

L

γiR2iR1v1

TVR

C

2πCv2

ωτ

0-γ
Fig. 1.

2πpCircuit model (TVR is a time-varying resistor).
Fig. 3.

a Time-Varying Resistor (TVR). Synchronization
phenomena for the case that the coupling resistor is a simple time-invariant resistor have been
investigated in [Suezaki & Mori, 1965; Kimura &Mano, 1965]. Namely, the in-phase synchronizationis stable for a positive coupling resistor, while theanti-phase synchronization is stable for a negative
coupling resistor. In this study, we consider the casethat the coupling resistance R(t) varies with time.
The characteristics of the TVR are shown in Fig. 2.
In this study, we consider the case that the function
representing the variation of the VTR is a squarewave with the angular frequency ωt and the duty
ratio p.
Firstly, the vk − iRk characteristics of the nonlinear resistor are deﬁned as follows,
iRk = −g1 vk + g3 vk3 .

(1)

By means of the following change of variables,


√
g1
g1
Cyk , t = LCτ ,xk , ik =vk =
g3
3g3 L
1LC
ωt ,, γ=r, ω=√ε = g1
CLLCR(t)the normalized circuit equations are given bydxk= (1 − x2k ) − yk

 dτ

(k = 1, 2)
2

dyk


=x−
Γ(τ)
yjk


 dτ

(2)

j=1

where Γ(τ ) corresponds to the normalized characteristics of the time-varying resistor and is shown inFig. 3.

3. Synchronization Phenomena3.1. In-phase and anti-phasesynchronization
The two coupled oscillators can be synchronized inin-phase and anti-phase as shown in Fig. 4. Thesetwo synchronization states can be obtained startingat diﬀerent initial conditions.
We have conﬁrmed the two synchronization
states in-phase and anti-phase in circuit experiments, as shown in Fig. 5. In the circuit experiments, the TVR is realized by using an analog
switch as shown in Fig. 6 [Nishio & Mori, 1993].

3.2. Synchronization in dependenceon ω2πr
ωtt

0-r
2πpFig. 2.

Normalized characteristics of the TVR.

Characteristics of the TVR.

We investigate how the synchronization state
depends on ω. The computer simulation is shownin Fig. 7. The horizontal axis is ω and the verticalaxis is the phase diﬀerence between the two coupledoscillators. We set the parameters of the circuits as
ε = 2.0, γ = 0.1 and p = 0.5.
From Fig. 7, we can conﬁrm the coexistence ofin-phase and anti-phase synchronizations for 1.44 <ω < 1.58. We also conﬁrmed that the range of thecoexistence region shifts toward smaller ω when εincreases.

Synchronization Phenomena in Van Der Pol Oscillators Coupled by a Time-Varying Resistor

x1x1x2x23567

ττ(a) In-phase.

(b) Anti-phase.

Fig. 4. Time waveform of two synchronization states (computer simulation). ε = 2.0, γ = 0.1, ω = 1.5, p = 0.5. (a) In-phasesynchronization (the initial states: x1 = 1.0, y1 = 1.0, x2 = 1.1, y2 = 1.1). (b) Anti-phase synchronization (the initial states:
x1 = 1.0, y1 = 1.0, x2 = −1.1, y2 = −1.1).

v1v1v2v2tt(a) In-phase.

(b) Anti-phase.

Fig. 5. Time waveform of two synchronization states (circuit experimental results). L = 10 mH, C = 33 nF, r = 152 Ω.
(a) In-phase synchronization, (b) Anti-phase synchronization.

200
180

Anti-phase+

150

TVR+-Phase difference

-

100

50
In-phase
01.2

1.3

1.4
1.5
Omega

1.6

1.7

Circuit realization of the TVR [Nishio & Mori, 1993].

Fig. 7. Synchronization in dependence on ω ( = 2.0, γ =0.1, p = 0.5).

It is also interesting to see that the antiphase synchronization is not completely anti-phase.
The solution oscillates around 150◦ and oscillation amplitude becomes larger as ω increases.

This means that another anti-phase synchronization mode coexists around 210◦ because of thesymmetry of the coupled circuits. Anti-phase synchronization becomes unstable around ω = 1.58,Fig. 6.

Y. Uwate & Y. Nishio

3568

4. Conclusions

200180Phase difference

150

Anti-phase

10050In-phase
00.46

0.48

0.5
p0.52

0.54

Fig. 8. Synchronization in dependence on p ( = 2.0, γ =0.1, ω = 2.0).

where two anti-phase synchronizations seem to collide with each other. We consider that the detailed
study on this bifurcation would help to clarify themechanism of the coexistence of synchronization
modes in this circuits.

3.3. Synchronization in dependence
on pNext, we investigate the inﬂuence of the duty ratioof the switching, namely we vary the ratio of timeintervals connecting to the positive and the negative resistors. Figure 8 shows the phase diﬀerence
between both oscillators in dependence on the duty
ratio p obtained from Eq. (2). We set the parameters as  = 2.0, γ = 0.1 and ω = 2.0.
From Fig. 8, we can see the hysteresis of thesynchronization phenomena. First, for p smaller
than 0.462, the two coupled oscillators are synchronized only in anti-phase. Second, when the p isbetween 0.462 and 0.490, the coexistence of the inphase and the anti-phase synchronizations can beobserved. Finally, for p larger than 0.490, only thein-phase synchronization can be conﬁrmed.
It should be noted that the in-phase synchronization is not completely in-phase in the coexistence region. The phase diﬀerence becomes larger
as p decreases.
The width of the coexistence region depends onthe values of the parameter , which is responsible
for the nonlinearity of the oscillators. We conﬁrmedthat the width of the coexistence region becomesnarrower, when  increases.

In this study, we investigated synchronization phenomena of two van der Pol oscillators coupled bya time-varying resistor in dependence on ω and p.
We realized the time-varying resistor by switchinga positive and a negative resistor periodically. By
carrying out circuit experiments and computer calculations, we conﬁrmed that the range of the coexistence region shifts toward smaller ω, and the widthof the coexistence region becomes narrower, when increases. A detailed study of the found bifurcations
will be the subject of a forthcoming paper.

References
Abramson, G., Kenkre, V. M. & Bishop, A. R. [2002]“Analytic solutions for nonlinear waves in coupled
reacting systems,” Physica A: Statist. Mech. Appl.
305, 427–436.
Belykh, I., Hasler, M., Lauret, M. & Nijmeijer, H. [2005]“Synchronization and graph topology,” Int. J. Bifurcation and Chaos 15, 3423–3433.
Bonilla, L. L., Perez Vicente, C. J. & Spigler, R. [1998]“Time-periodic phases in populations of nonlinearly
coupled oscillators with bimodal frequency distributions,” Physica D: Nonlin. Phen. 113, 79–97.
Cosp, J., Madrenas, J., Alarcon, E., Vidal, E. & Villar, G. [2004] “Synchronization of nonlinear electronicoscillators for neural computation,” IEEE Trans.
Neural Networks 15, 1315–1327.
Datardina, S. P. & Linkens, D. A. [1978] “Multimodeoscillations in mutually coupled van der Pol typeoscillators with ﬁfth-power nonlinear characteristics,”
IEEE Trans. Circuits Syst. CAS-25, 308–315.
Endo, T. & Mori, S. [1976a] “Mode analysis of a multimode ladder oscillator,” IEEE Trans. Circuits Syst.
CAS-23, 100–113.
Endo, T. & Mori, S. [1976b] “Mode analysis of a twodimensional low-pass multimode oscillator,” IEEETrans. Circuits Syst. CAS-23, 517–530.
Endo, T. & Mori, S. [1978] “Mode analysis of a ring of alarge number of mutually coupled van der Pol oscillators,” IEEE Trans. Circuits Syst. CAS-25, 7–18.
Gray, C. M. [1994] “Synchronous oscillations in neural systems: Mechanisms and functions,” J. Comput.
Neurosci. 1, 11–38.
Kimura, H. & Mano, K. [1965] “Some properties ofmutually synchronized oscillators coupled by resistance,” Trans. IECE 48, 1647–1656.
Liu, B., Liu, X., Chen, G. & Wang, H. [2005] “Robustimpulsive synchronization of uncertain dynamical networks,” IEEE Trans. Circuits Syst.-I 52, 1431–1441.
Nishio, Y. & Mori, S. [1992] “Mutually coupled oscillators with an extremely large number of steady states,”
Proc. ISCAS’92 2, 819–822.

Synchronization Phenomena in Van Der Pol Oscillators Coupled by a Time-Varying Resistor

Nishio, Y. & Mori, S. [1993] “Chaotic phenomena in
nonlinear circuits with time-varying resistors,” IEICE
Trans. Fund. E76-A, 467–475.
Sherratt, J. A. [1998] “Invading wave fronts and theiroscillatory wakes are linked by a modulated travelingphase resetting wave,” Physica D: Nonlin. Phen. 117,145–166.

3569

Suezaki, T. & Mori, S [1965] “Mutual synchronization oftwo oscillators,” Trans. IECE 48, 1551–1557.
Yamauchi, M., Wada, M., Nishio, Y. & Ushida, A. [1999]“Wave propagation phenomena of phase states inoscillators coupled by inductors as a ladder,” IEICE
Trans. Fund. E82-A, 2592–2598.