Two van der Pol Oscillators Coupled by Chaotically Varying Resistor
Yoko Uwate and Yoshifumi Nishio
Tokushima University, JAPANe-mail: {uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, two van der Pol oscillators
coupled by a time-varying resistor are investigated.
We assume that the time-varying resistor is realized
by switching a positive and a negative resistors alternately with a fixed frequency and chaotically varying
duty ratio. By carrying out computer simulations, weconfirm that interesting change of phase difference between the two oscillators is observed.
I. I NTRODUCTION
Synchronization phenomena in complex systems
are very interesting to describe various higherdimensional nonlinear phenomena in the field of natural science. Studies on synchronization phenomena ofcoupled oscillators are extensively carried out in various fields, physics [1]-[3], biology [4], engineering[5]-[12] and so on. Because many researchers suggestthat synchronization phenomena of coupled oscillators have some relations to information processing in
the brain. We consider that it is very important to investigate the synchronization phenomena of coupledoscillators to realize a brain computer for the futureengineering application.
On the other hand, there are some systems whose
dissipation factors vary with time, for example, under the time-variation of the ambient temperature, anequation describing an object moving in a space withsome friction and an equation governing a circuit witha resistor whose temperature coefficient is sensitive
such as thermistor. However, there are few discussion
about coupled oscillators coupling by a time-varying
resistor.
In our previous research, we have investigated synchronization phenomena in van der Pol oscillators
coupled by a time-varying resistor [13]. We realized
the time-varying resistor by switching a positive and anegative resistors periodically [14]. By changing theduty ratio p, we can confirmed that the characteristicsof the synchronization phenomena as follows. First,
for smaller p, the two coupled oscillators are synchronized only in anti-phase. Second, for intermediate p,
the coexistence of the in-phase and the anti-phase synchronizations can be observed. Finally, for larger p,

only the in-phase synchronization can be confirmed.
What happen in the two coupled oscillators? if theduty ratio changes to between smaller and larger values. This question is very interesting to make clear themechanism of coupled oscillators with time-varyingresistor.
In this study, we investigate two van der Pol oscillators coupled by a time-varying resistor when theswitching duty ratio of the time-varying resistor is determined by the value of the logistic map as an external signal. First, we investigate the phenomena when
the switching duty ratio is changed as one-period,two-period, four-period, and chaos. Next, we investigate the phenomena when the switching duty ratio isaround the intermittency chaos near the three-periodic
window. By carrying out computer calculations, wecan confirm that the phase difference between the two
coupled oscillators changes according to the switching duty ratio.
II. C IRCUIT M ODEL
Figure 1 shows the circuit model. In this circuit,
two identical van der Pol oscillators are coupled bya Time-Varying Resistor (TVR). We have known thesynchronization phenomena for the case that the coupling resistor is a simple time-invariant resistor [5][6].
Namely, the in-phase synchronization is stable for apositive coupling resistor, while the anti-phase synchronization is stable for a negative coupling resistor.
In this study, we consider the case that the coupling resistance R(t) of the TVR varies with time. The characteristics of the TVR is shown in Fig. 2. In this study,
we consider the case that the function representing thevariation of the TVR is the square wave 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 vk 3 .

(1)

By changing the variables and the parameters,vk =rg1
xk , ik =
g3r

g1
g3s√
Cyk , t = LCτ ,L

L
iR1
CI2

I1

L

x1

iR2

v1

TVRx2

Cv2

τ

(a) In-phase.

x1x2

Fig. 1. Circuit model (TVR is a Time-Varying Resistor).

τ(b) Anti-phase.

R(t)2πFig. 3. Time waveform of two synchronization states(computer calculated results). ε = 2.0, γ = 0.1, p =0.5, ω = 1.5. (a) In-phase synchronization. (b) Antiphase synchronization.

rωtt

0200-r1802πpPhase difference

150

Fig. 2. Characteristics of the TVR.

ε = g1sL, γ=rC

s100

1C, ω=√
ωt ,LLCdyk= xk ± γ(τ ) dτ

2X(k = 1, 2)

50In-phase0

the normalized circuit equations are given asdxk= ε(1 − xk 2 ) − yk

dτ



Anti-phase0.46(2)

0.48

0.5
p0.52

0.54

Fig. 4. Synchronization in dependence on p (ε = 2.0,γ = 0.1, ω = 2.0).

yjj=1

where the sign of the coupling term changes according
to the value of the time-varying resistor.

0.46 and 0.492, the coexistence of the in-phase andthe anti-phase synchronizations can be observed. Finally, for p larger than 0.492, only the in-phase synchronization can be confirmed.
IV. S WITCHING D UTY R ATIO BY L OGISTIC M AP

III. S YNCHRONIZATION P HENOMENA [13]
We observed that the two coupled oscillators aresynchronized in in-phase and anti-phase as shown inFig. 3. These two synchronization states can be obtained by giving different initial conditions.
We have investigated the influence of the duty ratioof the switching, namely we vary the ratio of time intervals connecting to the positive and the negative resistors. Figure 4 shows the phase difference between
both oscillators in dependence on the duty ratio p. We
set the parameters as ε = 2.0, γ = 0.1 and ω = 2.0.
From this figure, we can see the hysteresis of thesynchronization phenomena. First, for p smaller than0.46, the two coupled oscillators are synchronized
only in anti-phase. Second, when the p is between

We investigate the phase difference between the oscillators when the duty ratio of the TVR changes between smaller and larger values. We assume that theswitching duty ratio p is determined by the value ofthe logistic map as an external signal. The logistic
map is defined by the following equation.
zn+1 = αzn (1.0 − zn ).

(3)

Varying parameter α, Eq. (3) behaves chaotically via
a periodic-doubling cascade. We transform the valueof the logistic map into the switching duty ratio p by
the following equation:
1pn = (zn + 0.85).
3(4)

200

200

150

150
phase difference

phase difference

Further, we assume that the duty ratio pn is fixed during N periods.
In the following sections, we set the parameters ofthe circuit as ε = 2.0, γ = 0.1 and ω = 2.0.

100

500
4000

V. P ERIODIC AND C HAOTIC S WITCHING

VII. C ONCLUSIONSIn this study, we have investigated phase differences in two van der Pol oscillators coupled by atime-varying resistor when the duty ratio of the timevarying resistor varies chaotically. We realized thetime-varying resistor by switching a positive and anegative resistors using the value of the logistic map.

4400

4600

4800

5000

4000

4200

4400

4600

4800

5000

time

(b) N=400.

200

200

150

150
phase difference

phase difference

Fig. 5. Phase differences for 1-periodic duty ratio (α =2.9).

1005004000

1005004200

4400

4600

4800

5000

4000

4200

4400

time

4600

4800

5000

time

(a) N=200.

(b) N=400.

Fig. 6. Phase differences for 2-periodic duty ratio (α =3.2).
200150150
phase difference

200

100

500
4000

1005004200

4400

4600

4800

5000

4000

4200

4400time

4600

4800

5000

time

(a) N=200.

(b) N=400.

Fig. 7. Phase differences for 4-periodic duty ratio (α =3.5).
200200150150phase difference

Next, we compare the phenomena for the cases thatthe logistic map is three-periodic, intermittency chaos
near the three periodic window and fully-developed
chaos. We consider the three types of intermittencychaos near the three periodic window. Namely, thelaminar part of the intermittency chaos is long, middleand short. The simulated results are shown in Fig. 9.
From these figures, we can see the different behaviors of the phase differences by changing the switching duty ratio. First, for the case that the laminar partis long (Fig. 9(b)), the behavior is similar to the threeperiodic (Fig. 9(a)). Secondly, for the case that thelaminar part is short (Fig. 9(d)), the behavior is similar to the fully-developed chaos (Fig. 9(e)). Finally,
for the case that the laminar part is middle (Fig. 9(c)),
the behavior includes both the three periodic and thefully-developed chaos. In that case, the changing of
the phase difference is very complicated.

04200(a) N=200.

phase difference

VI. I NTERMITTENCY AND F ULLY-D EVELOPEDC HAOTIC S WITCHING

50

time

phase difference

Figure 5 shows the phase differences when the logistic map is one-periodic. The phase difference isconstant. This is because the duty ratio, which isdetermined by the value of the logistic map, is alsoconstant. Figures 6 and 7 show the phase differences when the logistic map is two-periodic and fourperiodic, respectively. From these figures, we canconfirm that the phase difference oscillates as almost
one or two periodic according to the change of theduty ratio. Further, when the period N becomes large,
the amplitude of the oscillation of the phase difference
becomes large, and the changing speed of the phasedifference becomes slow. Further, the phase differences when the logistic map is chaotic are shown inFig. 8. We can see that the phase difference between
the two coupled oscillators behaves chaotically.

100

100

5004000

100

5004200

4400

4600
time

(a) N=200.

4800

5000

4000

4200

4400

4600

4800

5000

time

(b) N=400.

Fig. 8. Phase differences for chaotic duty ratio (α = 3.68).

By carrying out computer calculations, we confirmedthat the synchronization phenomena of the two coupled oscillators depended on the switching duty ratio.
Namely, for the case of the intermittency chaos, thesynchronization had coexistence of the three period
and the fully-developed chaos. The varying phase difference was very complex.

R EFERENCES

200

phase difference

150

100

5004000

4200

4400

4600

4800

5000

time

(a) 3 periodic (α = 3.835).
200

phase difference

1501005004000

4200

4400

46004800

5000

time

(b) Intermittency chaos: laminar part is long.
(α = 3.8284).
200

phase difference

150

10050

04000

4200

4400

4600

4800

5000

time

(c) Intermittency chaos: laminar part is middle.
(α = 3.8260).
200

phase difference

150

100

50

04000

4200

4400

4600

4800

5000

time

(d) Intermittency chaos: laminar part is short.
(α = 3.8250).
200

phase difference

150

100

5004000

4200

4400

4600

4800

5000

time

(e) Fully-developed chaos (α = 3.99).
Fig. 9. Comparison of intermittency chaos and fullydeveloped chaos.

[1] L.L. Bonilla, C.J. Perez Vicente and R. Spigler, “Timeperiodic phases in populations of nonlinearly coupled oscillators with bimodal frequency distributions,” Physica D: Nonlinear Phenomena, vol.113, no.1, pp.79-97, Feb. 1998.
[2] J.A. Sherratt, “Invading wave fronts and their oscillatory
wakes are linked by a modulated traveling phase resettingwave,” Physica D: Nonlinear Phenomena, vol.117, no.1-4,pp.145-166, June 1998.
[3] G. Abramson, V.M. Kenkre and A.R. Bishop, “Analytic solutions for nonlinear waves in coupled reacting systems,” Physica A: Statistical Mechanics and its Applications, vol.305,no.3-4, pp.427-436, Mar. 2002.
[4] C.M. Gray, “Synchronous oscillations in neural systems:mechanisms and functions,” J. Computational Neuroscience,
vol.1, pp.11-38, 1994.
[5] T. Suezaki and S. Mori, “Mutual synchronization of two oscillators,” Trans. IECE, vol.48, no.9, pp.1551-1557, Sep. 1965.
[6] H. Kimura and K. Mano, “Some properties of mutually synchronized oscillators coupled by resistance,” Trans. IECE,
vol.48, no.10, pp.1647-1656, Oct. 1965.
[7] T. Endo and S. Mori, “Mode analysis of a multimode ladder oscillator,” IEEE Trans. Circuits Syst., vol.CAS-23, no.2,
pp100-113, Feb. 1976.
[8] T. Endo and S. Mori, “Mode analysis of a two-dimensional
low-pass multimode oscillator,” IEEE Trans. Circuits Syst.,
vol.CAS-23, no.9, pp517-530, Sep. 1976.
[9] S.P. Datardina and D.A. Linkens, “Multimode oscillations
in mutually coupled van der Pol type oscillators with fifthpower nonlinear characteristics,” IEEE Trans. Circuits Syst.,
vol.CAS-25, no.5, pp.308-315, May 1978.
[10] T. Endo and S. Mori, “Mode analysis of a ring of a large
number of mutually coupled van der Pol oscillators,” IEEETrans. Circuits Syst., vol.CAS-25, no.1, pp.7-18, Sep. 1978.
[11] 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.
[12] M. Yamauchi, M. Wada, Y. Nishio and A. Ushida, “Wavepropagation phenomena of phase states in oscillators coupled by inductors as a ladder,” IEICE Trans. Fundamentals,
vol.E82-A, no.11, pp.2592-2598, Nov. 1999.
[13] Y. Uwate and Y. Nishio, “Synchronization phenomena invan der Pol oscillators coupled by a time-varying resistor,” International Journal of Bifurcation and Chaos (to appear).
[14] 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.