2015 International Symposium on Nonlinear Theory and its Applications
NOLTA2015, Kowloon, Hong Kong, China, December 1-4, 2015

Synchronization in Two van der Pol Oscillators
Coupled by Transmission Line Model
Yoko Uwate† , Yuichi Tanji‡ 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‡Dept. of Electronics and Information Engineering,
Kagawa University,Takamatsu, 761-0396 Japan.
Email: tanji@eng.kagawa-u.ac.jp
Abstract—The internal wiring of high-speed VLSI isconsidered for transmission line, because its structure isvery complex and high density.
In this study, we investigate synchronization phenomena
in two van der Pol oscillators coupled by transmission line
model. By using the computer simulations, we confirm
several types of synchronization states such as coexistencein-phase and anti-phase by changing the circuit parameter.
Figure 1: Two Chua’s circuits with transmission lines.
1. Introduction
Recently, the synchronization observed from coupled oscillators and chaotic circuits systems have been studied actively [1], [2]. It is important to investigate basic synchronization observed in coupled oscillatory systems for
future engineering applications such as chaotic communication and chaotic cryptography. In high-speed VLSI, theinternal wiring is considered for transmission line because
the structure of high-speed VLSI is complex and high density. However, there are not many discussions about effectcaused by transmission line model for coupled oscillatorysystems.
In our previous study, the synchronization of two chaotic
circuits with transmission line coupled by the cross talkhave been investigated [3]-[7]. We have observed in-phaseand anti-phase synchronization phenomena from these circuits in computer simulations (see Figs. 1, 2) [6]. In thissystem model, the part of inductor and capacitor of chaotic
circuit is modeled by transmission line. The synchronization of oscillators coupled by transmission line using theladder circuit of inductor and capacitor is not really investigated.
This paper presents synchronization phenomena in two
van der Pol oscillators coupled by transmission line model.
We model transmission line using ladder circuits of inductor and capacitor as lossless transmission line. By using
the computer simulations, several types of synchronization
states such as coexistence in-phase and anti-phase are confirmed by changing the circuit parameter.

(a) In-phase (coupling capacitors).

(b) Anti-phase (mutal inductors).

Figure 2: Synchronization states.
2. Two van der Pol Oscillators Coupled by Transmission Line Model
Figure 3 shows the conceptual circuit model of thisstudy. Two van der Pol Oscillators are coupled by transmission line as shown in Fig. 4.
Next, we develop the expression for the circuit equationsof the circuit model as shown in Fig. 4. The vk − iRk characteristics of the nonlinear resistor are approximated by thefollowing third order polynomial equation,- 109 -iRk = −g1 vk + g3 vk 3 (g1 , g3 > 0), (k = 1, 2).

(1)

TML model

iR1

iR2

i1
v1

C0

i2
v2

C0

L0

van der Pol -1

L0

van der Pol -2

Figure 3: Conceptual circuit model.
L1

i3
iR1

i5 L1

i6 L1

i7 L1

i8 L1

i9 L1

i10 L1

i11 L1i12 L1i13 L1

L1

i1
C0v1

L0C1

C1v3v5v6

C1

C1

C1

C1v7

C1v9v8

C1v10C1v11C1v12i4 L1
iR2v4van der Pol -1

i2C0v2L0

van der Pol -2

Figure 4: Two van der Pol Oscillators coupled by transmission line model.
3. Synchronization Phenomena
The normalized circuit equations governing the circuit
are expressed as [van der Pol Oscillator]
s



√dxk
1 21
= ε 1 − xk xk − yk − αyn


dτ
3β


(2)
dyk


= ykdτ(k = 1, 2).
[Transmission Line Model]
√ pdxk= α β(yk − yn )dτ
 dy√ pk= α β(xk − xn )

dτ


(k = 3...13).

(3)

3.1. Attractor
Figures 5-7 show the attractors and the lissajous when
the parameters α and β are changed.
In the case of α=β=1.0, coexistence of in-phase and antiphase states of periodic solutions are observed as shown inFig. 5. Figure 6 shows the simulation results when α andβ are set to 1.9. In this case, we also confirm coexistenceof in-phase and anti-phase state. We confirm that the torus
attractor can be obtained when two oscillators are synchronized at in-phase, while the periodic attractor is generated
when two oscillators are synchronized at anti-phase.

y1, 3y2, 4

X1

X2

X2

where

t=

p

L0C0 τ, vk =r

g1
xk
3g3

(k = 1...12),

r
g1
C0
ik =
yk
(k = 1, 2),
3g3
L0
r
r
g1
C1yk
(k = 3...13),
ik =3g3L1L0C0α=, β=,
L1
C1
r

X1(a) In-phase.

In this equations, ε denotes nonlinearity of the oscillators.
α and β denote the ratio of inductor and capacitor of theoscillators and transmission line, respectively.
For the computer simulations, we calculate Eq. (3) using
the fourth-order Runge-Kutta method with the step size h =0.005. The parameter ε is fixed with 0.1.

- 110 -y1, 3y2, 4

X1

X2(b) Anti-phase.

Figure 5: Attractor (α = 1.0).

X2

X1

Figure 7 shows the results when α and β are set to 2.4
and 3.0. In the case of α=β=2.4, torus attractor can be observed. In the case of α=β=3.0, two oscillators generate theperiodic attractor. In the case of α=β=5.0, we observe coexistence of in-phase and anti-phase states again as shownin Fig. 8.
By increasing the value of α and β, only in-phase state
can be observed (see. Fig. 9). Namely coexistence of inphase and anti-phase disappears if the ratio of α and β is set
to large values.

y2, 4y1, 3y2, 4

X1

X2

X1

X2

X2

X1

(a) In-phase.

y1, 3y2, 4

X1

X2

X2

X1(b) Anti-phase.

X1Figure 8: Attractor (α=5.0).

(a) In-phase.

y1, 3y2, 4X2

X2

X1y1, 3

X2

X1(b) Anti-phase.

y1, 3y2, 4

X1

X2X2

X1

Figure 9: Attractor (α=8.0).

Figure 6: Attractor (α = 1.9).

y1, 3y2, 4

X1

X2

and anti-phase mode, respectively. We can observe coexistence with different solutions when α, β is smaller than
2.3. By increasing the parameter α and β, only one solution exists. Furthermore, we observe periodic window
around α=β=2.35 (see. Fig. 11).

X2

X1
3.3. Phase Difference(a) α=2.4.

y1, 3y2, 4X1X2X2X1(b) α=3.0.

Figure 7: Attractor (α=2.4, 3.0).

3.2. Bifurcation diagram
Next, we consider one parameter bifurcation when theparameters α and β are changed. Figure 10 shows thesimulation results when initial condition is set to in-phase

In this subsection, we calculate the phase difference between two oscillators. The simulation result of the phasedifference is shown in Fig. 12. We can observe coexistencewith different solutions when α, β is smaller than 2.3 asknown from the bifurcation diagram. When the parameters
α and β is larger than 2.3, in-phase synchronization disappear and only anti-phase synchronization is observed. After that, coexistence of in-phase and anti-phase states canbe observed around α=β=5.0. Finally, by increasing thevalue of α and β, only in-phase state can be observed.
In order to confirm the switching several synchronization
states, Fig. 13 shows the expanded graph of Fig. 12. From
this figure, we can see that synchronization state is classified in in-phase, anti-phase, coexistence and asynchronous
states depending on the parameter. Investigation of synchronization state with small step size of the parameter is
one of our future work.

- 111 -180

phase difference

160
140
120
100
806040(a) Initial condition: In-phase mode.

200
12

34

56

78alpha, beta

Figure 12: Phase difference.
180
160
140
phase difference(b) Initial condition: Anti-phase mode.

Figure 10: Bifurcation diagram.

120
1008060
40-1.220-1.4

0
1.56

-1.6

1.58

1.6

1.62

1.64

1.66

1.68

1.7

1.72

1.74

x1alpha, beta
-1.8-2In-phase area

Coexistence area

Anti-phase area

Asynchronous area

-2.2

Figure 13: Phase difference.
-2.4
2.25

2.3

2.35

2.4

2.45

2.5

alpha, beta

References
Figure 11: Periodic solution in bifurcation.

[1] S. Boccaletti, J. Kurths, G. Osipov, D. Valladares and C.
Zhou, “The Synchronization of Chaotic Systems” Physics
Reports, 366, pp. 1-101, 2002.

4. Conclusions
We have investigated synchronization phenomena in twovan der Pol oscillators coupled by transmission line model.
By using the computer simulations, several types of synchronization states such as coexistence in-phase and antiphase were confirmed by changing the circuit parameter.
In our future works, we would like to investigate effectof the length of the transmission line and apply this model
to more complex networks such as smart grid network andsocial network.

[2] A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno andC. Zhou, “Phase Synchronization of Chaotic Oscillators”
Physics Reports, 469, pp. 93-153, 2008.
[3] J. Kawata, Y. Nishio and A. Ushida, “Analysis of Chua’s
Circuits with Transmission Line” IEEE Transaction on Circuits and Systems I, vol. 44, no. 6, pp. 556-558, 1997.
[4] J. Kawata, Y. Nishio and A. Ushida, “Chaos Synchronization by Transmission Line Coupling” Proc. of NOLTA’96,pp. 109-112, 1996.
[5] Y. Nakaaji and Y. Nishio, “Synchronization of Chaotic Circuits with Transmission Line” Proc. of NCSP’06, pp. 353356, 2006.

Acknowledgment[6] Y. Nakaaji and Y. Nishio, “Synchronization of Chaotic Circuits Linked by Cross Talk” Proc. of NOLTA’06, pp. 43-46,
2008.

This work was partly supported by JSPS Grant-in-Aidfor Challenging Exploratory Research 26540127.

[7] N. Kondo, Y. Uwate and Y. Nishio, “Synchronization ofTwo Chaotic Circuits with Transmission Line Coupled byResistors” Proc. of NCN’13, pp. 17-19, 2013.

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