2007 International Symposium on Nonlinear Theory and itsApplications
NOLTA'07, Vancouver, Canada, September 16-19, 2007

Synchronization Phenomena in van der Pol Oscillators
Coupled by Fifth-Power Nonlinear Resistor
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— In this study, synchronization phenomena observed in van der Pol oscillators coupled by afifth-power nonlinear resistor are investigated. By carrying out computer simulations, interesting synchronization phenomena can be confirmed to be generated
in this system. Namely, the synchronization stateschange according to the coupling strength and nonlinearity of the coupling resistor.
1. Introduction
Synchronization phenomena in complex systems arevery interesting to describe various higher-dimensional
nonlinear phenomena in the field of natural science.
Studies on synchronization phenomena of coupled oscillators are extensively carried out in various fields,
physics [1]-[4], biology [5], [6] engineering [7]-[12] andso on. Because many researchers suggest that synchronization phenomena of coupled oscillators have somerelations to information processing in the brain. We
consider that it is very important to investigate thesynchronization phenomena of coupled oscillators torealize a brain computer for the future engineering 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 discussions
about coupled oscillators coupling by a time-varyingresistor.
In our previous research, we have investigated synchronization phenomena in van der Pol oscillators coupled by a time-varying resistor. We realized the timevarying resistor by switching a positive and a negative
resistors periodically. By changing the duty ratio p, weconfirmed that the characteristics of the synchronization phenomena changed as follows. First, for smallerp, the two coupled oscillators are synchronized onlyin anti-phase. Second, for intermediate p, the coexistence of the in-phase and the anti-phase synchroniza-tions can be observed. Finally, for larger p, only thein-phase synchronization can be confirmed.
Coexistence of attractors in coupled oscillatory systems is important for modeling of information processing mechanisms of the brain and applying them tonovel parallel information processing systems. Generally, in two oscillators coupled by a resistor, only
one synchronization state becomes stable even if plural states exist. In this study, two van der Pol oscillators coupled by a fifth-power nonlinear resistor
are investigated. By carrying out computer calculations, interesting synchronization phenomena can beconfirmed to be generated in this system. First, thecoexistence of in-phase and anti-phase synchronization
are observed. Next, we investigate the influence of thecoupling strength and nonlinearity of the van der Poloscillator.
2. Circuit Model
Figure 1 shows the circuit model. In this circuit, two
identical van der Pol oscillators are coupled by a fifthpower nonlinear resistor. Synchronization phenomena
for the case that the coupling resistor is a simple resistor have been investigated in [7],[8]. Namely, thein-phase synchronization is stable for a negative coupling resistor, while the anti-phase synchronization isstable for a positive coupling resistor.
LI 1 vR I 2

Li R2i R1v1CCv2

Fifth-power
nonlinear resistor

Third-power
nonlinear resistor

Figure 1: Circuit model.
In this study, we consider the fifth-power nonlinear
resistor as coupling resistor. The characteristics of the- 232 -fifth-power nonlinear resistor is defined as follows,x1y2

y1vR = g1 (I1 + I2 ) − g3 (I1 + I2 )3 + g5 (I1 + I2 )5 . (1)
x1x2

x1

Next, the vk - iRk characteristics of the third-power
nonlinear resistor are defined as following equation.

(a)

(b)

(c)

x1

iRk = −g1 vk + g3 vk3(2)x2

By changing the variables and parameters,
rr r
√
g1 L
g1, ik =, t = LCτ ,vk =
g5 C
g5r
r
L
g3 g1
α = g1, β=,C
g1 g5rCL
ε = g1, ζ= ,LC
the normalized circuit equations are given as dxk3 dτ = −yk + ε(xk − βζxk )τ(d)Figure 3: Anti-phase synchronization. (a) 1st circuit attractor(3)(x1 vs y1 ). (b) 2st circuit attractor (x2 vs y2 ). (c) Phase difference (x1 vs x2 ). (d) Time wave form (τ vs x1 and x2 ). ε = 0.4,
α = 0.002, β = 4.0 and ζ = 4.0.

Phase difference

phase difference when the two parameters β and ζ arechanged. This simulated results are shown in Fig. 5.
In this figure, the black color means the in-phase state
and the yellow color means the anti-phase state. We
can also observe the anti-phase state in these parameter area. Namely, the in-phase state area means thedzk35= xk − α{(y1 + y2 ) − β(y1 + y2 ) + (y1 + y2 ) } coexistence area. The coexistence area can be con dτfirmed when β are large values and ζ are small values.
(k = 1, 2)(4)200
where α is the coupling factor and ε is the strength180
nonlinearity.
150
3. Synchronization Phenomena
3.1. In-Phase and Anti-Phase Synchronization
We observed that the two coupled oscillators aresynchronized in in-phase and anti-phase as shown inFigs. 2 and 3. These two synchronization states canbe obtained by giving different initial conditions. Theparameters of the van der Pol oscillator are fixed as
ε = 0.4, α = 0.002, β = 4.0, ζ = 4.0.
x1y2

y1(a)

(b)

x1(c)

x1x2(d)100

coexistence

5002345678910βFigure 4: Phase difference for ε = 0.4, α = 0.002 and ζ = 5.0.

3.2. Synchronization in dependence on εx2x1

anti-phaseτFigure 2: In-phase synchronization. (a) 1st circuit attractor(x1 vs y1 ). (b) 2st circuit attractor (x2 vs y2 ). (c) Phase difference (x1 vs x2 ). (d) Time wave form (τ vs x1 and x2 ). ε = 0.4,
α = 0.002, β = 4.0 and ζ = 4.0.

Figure 4 shows the one parameter diagram of thephase difference. We can confirm the coexistence ofin-phase and anti-phase synchronization around for
5.0 < β < 10.0. Furthermore, we investigate theWe investigate how the coexistence area depends onnonlinearity ε of the van der Pol oscillator. The antiphase state is confirmed everywhere in these parameterarea. So, the in-phase state area is corresponding to
the coexistence area.
The computer simulation when ε is fixed to 1.2 and
2.0 are shown in Fig. 6 (a), (b). The horizontal axis isβ, the vertical axis is ζ and the black and yellow color
denotes the the in-phase and the anti-phase states between the two coupled oscillators. In the case of weak
nonlinearity ε = 1.2, the in-phase state area can beoccurred in ζ < 5.0 (Fig. 6 (a)). On the other hand,the case of ε = 2.0, the in-phase state area is observedin ζ < 7.0 (Fig. 6 (b)). Figure 7 shows the area ratio of in-phase state when the nonlinear parameter εis changed. From this figure, we confirm that the in-- 233 -

110
10090

Phase difference
180160140120
100
80
60
40
20080

[%]

180160
140
120
10080
60
40
20070
α=0.002
60
50
40

10

α=0.01

9830

7

2

34667

beta

0

0.5

1

1.5

2

48

92.5

ε

zeta

55

3

3.5

43
10 2Figure 7: Area ratio of in-phase by changing ε.

(a) 3D plot.
10180
160
140
120
10080604020098zeta

7654322345678910beta(b) 2D plot.

Figure 5: Phase difference for ε = 0.4 and α = 0.002.

phase (coexistence) area becomes large when the nonlinearity ε of the van der Pol oscillator becomes large.
3.3. Synchronization in dependence on αNext, we investigate the coexistence area depends
on the coupling strength α. The computer simulation when α is fixed to 0.001 and 0.01 are shown inFig. 8 (a), (b). In the case of small coupling strength,
the in-phase state area can be occurred in ζ < 6.0 andβ > 6.0 (Fig. 8 (a)). On the other hand, the case of
ε = 2.0, the in-phase state area is observed in ζ < 3.0(Fig. 8 (b)). Figure 9 shows the area ratio of in-phasestate when the parameter α is changed. From this figure, we confirm that the in-phase (coexistence) area
becomes small when the coupling strength α becomeslarge.

10180
160
140120
100
8060
40
200

98zeta

76180
160
140
120
10080
60
40
200

987zeta

51046534223456789310beta

22345678910beta(a) ε = 1.2.
(a) α = 0.001.
1010180160
140120100806040
2008zeta

765180160
140120100806040
200987zeta

965443322223456789345678910beta

10beta(b) α = 0.01.
(b) ε = 2.0.

Figure 8: Phase difference for ε = 0.4.

Figure 6: Phase difference for α = 0.002.

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References

80757065[%]605550ε=1.24540ε=0.4

3530
0.001

0.002

0.003

0.004

0.005

α0.006

0.007

0.008

0.009

0.01

Figure 9: Area ratio of in-phase by changing α.
3.4. Amplitude of van der Pol Oscillator
Finally, we investigate the amplitude of van der
Pol oscillator when the two parameters β and ζ arechanged. The simulated result is shown in Fig. 10.
We confirm that the amplitude of van der Pol oscillator becomes small when the both parameters β and ζincrease.

100.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1

98zeta7654322345678910

betaFigure 10: Amplitude of van der Pol oscillator for ε = 0.4 andα = 0.002.

4. Conclusions
In this study, two van der Pol oscillators coupled by
a fifth-power nonlinear resistor were investigated. By
carrying out computer calculations, interesting synchronization phenomena could be confirmed to be generated in this system. First, the coexistence of inphase and anti-phase synchronization were observed.
Furthermore, we have investigated the influence of thecoupling strength and nonlinearity of the van der Poloscillator.

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Acknowledgments
This work was partly supported by Yazaki Memorial
Foundation for Science and Technology.

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