RISP

NCSP’16

International Workshop on
Nonlinear Circuits, Communications and Signal Processing

Honolulu, Hawaii, USA, March 6-9, 2016

Synchronization in Oscillators Coupled by Transmission Line Modelwith Different Length
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
This paper presents synchronization phenomena in two vander Pol oscillators coupled by transmission line model withdifferent length. By using the computer simulations, several
types of synchronization states such as coexistence in-phaseand anti-phase are confirmed by changing the circuit parameters.
1. Introduction
Recently, the synchronization observed from coupled oscillators and chaotic circuits systems have been studied actively in various fields [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,
the internal 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 effect causedby transmission line model for coupled oscillatory systems.
In our research group, the synchronization of two chaotic
circuits with transmission line coupled by the cross talk havebeen investigated [3]-[7]. We have observed in-phase andanti-phase synchronization phenomena from these circuits incomputer simulations [6]. In this system model, the part ofinductor and capacitor of chaotic circuit is modeled by transmission line. The synchronization of oscillators coupled bytransmission line using the ladder circuit of inductor and capacitor has not really investigated.
In our previous study, we have investigated synchronization phenomena in two van der Pol oscillators coupled bytransmission line model. We modeled transmission line usingladder circuits of inductor and capacitor as lossless transmission line. By using the computer simulations, several types ofsynchronization states such as coexistence in-phase and antiphase have been confirmed by changing the circuit parameters.
However, we have investigated only basic synchronization
state by using only one transmission line model. The transmission line length is determined by the number of inductors and capacitors. Previously, the number of inductors andcapacitors of transmission line model is fixed to 11 and 10,respectively.
This study considers the effects of the transmission line
length by changing the number of the inductors and capacitors. Furthermore, we investigate the parameter effects ofthe transmission line model.
2. Two van der Pol Oscillators Coupled by Transmission
Line Model
Figure 1 shows the conceptual circuit model of this study.
Two van der Pol Oscillators are coupled by transmission line
as shown in Fig. 2. In this circuit model, the transmission
line is modeled by the ladder circuit composed of inductorsand capacitors.
TML model

iR1

i1
C0v1

iR2

L0van der Pol -1

i2
C0v2

L0van der Pol -2

Figure 1: Conceptual circuit model.
The number of inductors and capacitors is fixed with 11and 10, respectively. In this study, we change the numberof inductors and capacitors of the transmission line model.

- 455 -L1

i3iR1i5 L1

i6 L1i7 L1i8 L1

i9 L1

i10 L1

i11 L1i12 L1i13 L1

L1

i1C0v1

L0C1

C1v3v5v6C1C1

C1C1v7v8C1v9C1v10

i4 L1iR2

C1C1v11v12i2
v2

C0v4

van der Pol -1

L0

van der Pol -2

Figure 2: Two van der Pol Oscillators coupled by transmission line model (Ltml =10).
Three types of transmission line models are considered as fol- 3. Basic Synchronization Phenomena (Ltml =10)
lows.
Figures 3-5 show the attractors and the Lissajous when the• Inductors: 6, Capacitors: 5 (Ltml =5)
parameters α and β are changed.
In the case of α=β=1.0, coexistence of in-phase and anti• Inductors: 11, Capacitors: 10 (Ltml =10)
phase states of periodic solutions are observed as shown in• Inductors: 21, Capacitors: 20 (Ltml =20)
Fig. 3. Figure 4 shows the simulation results when α andNext, we develop the expression for the circuit equations β are set to 1.9. In this case, we also confirm coexistenceof the circuit model as shown in Fig. 2. The vk − iRk char- of in-phase and anti-phase state. We confirm that the torus atacteristics of the nonlinear resistor are approximated by the tractor can be obtained when two oscillators are synchronized
at in-phase, while the periodic attractor is generated when
following third order polynomial equation,
two oscillators are synchronized at anti-phase. In the case ofiRk = −g1 vk + g3 vk 3 (g1 , g3 > 0), (k = 1, 2). (1) α=β=5.0, we observe coexistence of in-phase and anti-phasestates as shown in Fig. 5.
The normalized circuit equations governing the circuit are
expressed as [van der Pol Oscillator]
dxk1
= ε 1 − xk 2 xk − yk − yn3 dτ(2)
dy k

= yk
dτ

(k = 1, 2).
[Transmission Line Model] dxk
= β(yk − yn ) dτdy k= α(xk − xn )

 dτ

t=L0 C0 τ , vk =ik =rr

g1
xk
3g3

X2

X2

X1

(a) In-phase.
y2, 4

y1, 3

X2

X2

X1

(b) Anti-phase.

(3)

Figure 3: Attractor (α = β = 1.0).

(k = 3...N).
y1, 3y2, 4

X1

X2X2

(k = 1...N + 2),

rC0ykL0

α=L0C0, β=,L1
C1

g1
3g3

X1X1

where√y2, 4y1, 3

X1(a) In-phase.

(k = 1...N + 3),
y1, 3

In this equations, ε denotes nonlinearity of the oscillators. αand β denote the ratio of inductor and capacitor of the oscillators and transmission line, respectively.
For the computer simulations, we calculate Eqs. (2), (3)using the fourth-order Runge-Kutta method with the step size
h = 0.005. The parameter ε is fixed with 0.1.

- 456 -y2, 4

X1X2X2X1(b) Anti-phase.

Figure 4: Attractor (α = β = 1.9).

y2, 4

X210101099888877776666555544443333

2

22

2

1

1

1

"period-in.dat"

9X1

X2

X1

y1, 3

y2, 4

beta

(a) In-phase.
X2

1X1

X2

X1

2

3

4

56
alpha

7

89

beta

y1, 3

10

10
"period-anti.dat"

1
1(a) Initial: in-phase.

23456
alpha

78

910(b) Initial: anti-phase.

1.8(b) Anti-phase.

910"period-inZ.dat"

410"period-antiZ.dat"

91.7

93.5
8

1.6

Figure 5: Attractor (α = β5.0).

87

376
1.4
5

beta

beta

1.5
6
2.5
5

1.3
4
1.2

4. Synchronization Phenomena Depending on Parameters

3
1.5

1.1

2

12

1
5.5

6

6.5
alpha

7

7.5

18

11(c) Initial: in-phase.

In this section, we investigate the effects of the transmission line length and the parameter effects of the transmission
line model.

1.5

22.5
alpha

3

3.5

4(d) Initial: anti-phase.

Figure 6: Solution (Ltml = 5).

101099888877776666555544443333222

11"period-in.dat"

beta91123456
alpha

789beta

104.1 Solution- 457 -

4351010"period-anti.dat"

92
11(a) Initial: in-phase.

23456alpha78

910(b) Initial: anti-phase.

410"period-inZ.dat"

410"period-antiZ.dat"

93.5

93.5
88762.5
523betabeta34762.5524331.5

1.5
21211

1.522.5
alpha

3

3.5

14

11(c) Initial: in-phase.

1.5

22.5
alpha

3

3.54(d) Initial: anti-phase.

Figure 7: Solution (Ltml = 10).

1010

99888877

77666655554444

333

3222

11

"period-in-all.dat"

9

1
12345

67

alpha(a) Initial: in-phase.

8

beta

10

beta

We investigate the solution type by changing the length of
the transmission line and the circuit parameters α and β. α
corresponds to the ratio of inductors between van der Pol andthe transmission line. β corresponds to the ratio of capacitors
between van der Pol and the transmission line. The obtainedsolutions are distinguished to one periodic solution and theothers. The simulation results of the solutions when the α andβ are changed from [1:10] are shown in Figs. 6-8. In thesefigures, the black and white region denote one periodic andquasi-periodic solutions, respectively. The horizontal axis denotes α and the vertical axis denotes β.
In the case of Ltml = 5, one periodic solution can beobserved almost every area in the both cases as shown inFig. 6(a) and (b). The extended graphs of quasi-periodic solution area are shown in Fig. 6(c) and (d). The structure of
the quasi-periodic solution is very simple. Figure 7 shows theresults of Ltml = 10. The area of one periodic solution becomes large when α and β show small value. The extended
graphs of quasi-periodic solution area are shown in Fig. 7(c)and (d). The area of the quasi-periodic solution seems likeshrimp structure. By increasing the length of the transmission
line (Ltml = 20), the area of one periodic solution becomeslarge for whole area. The form of the quasi-periodic solution
also seems like shrimp structure as shown in Fig. 8.
From these results, we can see that the quasi-periodic solution area becomes large when the length of transmission line
becomes long. Furthermore, we observe the shrimp structureof quasi-periodic solutions when the length of transmission
line is set to Ltml = 10 and 20.

210"period-anti-all.dat"

92
1
1

2

3

4

567

alpha(b) Initial: anti-phase.

Figure 8: Solution (Ltml = 20).

8

4.2 Phase Difference

5. Conclusions

The simulation results of the phase difference are shownin Figs. 9-11. In these figures, the black and white region
denotes in-phase and anti-phase states, respectively. The horizontal axis denotes α and the vertical axis denotes β.
In the case of Ltml = 5, the coexistence of in-phase andanti-phase states can be observed almost every area as shownin Fig. 9. In the case of Ltml = 10, the anti-phase area increases when the initial condition is set to in-phase mode asshown in Fig. 10(a). Figure 11 shows the phase difference ofLtml = 20. We confirm that anti-phase area can be obtainedlarge area in Fig. 11(a). We can see that the coexistence area
decreases by increasing the length of the transmission line.

We have investigated synchronization phenomena in two
van der Pol oscillators coupled by transmission line modelwith different length. By using the computer simulations, weobserve the shrimp structure of quasi-periodic solution in thecircuit parameter space when the length of the transmission
line is longer than Ltml = 10. Furthermore, several types ofsynchronization states such as coexistence in-phase and antiphase were confirmed by changing the circuit parameter.
In our future works, we would like to investigate mechanism of the obtained synchronization phenomena in detailand apply this model to more complex networks such as smart
grid network and social network.

101210"phase3d-in.dat"

9160

1088

6654

beta

beta

743

Acknowledgment

180"phase3d-anti.dat"

98140

7120

6100

580

460

340

220This work was partly supported by JSPS Grant-in-Aid forChallenging Exploratory Research 26540127.
References

2210123456alpha78911001(a) Initial: in-phase.

23456
alpha

78910(b) Initial: anti-phase.

Figure 9: Phase difference (Ltml = 5).

beta[2] A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno and C.
Zhou, “Phase Synchronization of Chaotic Oscillators”
Physics Reports, 469, pp. 93-153, 2008.

18010160

98140

8140

7120

7120

6100

6100

580

580

460

46034034020

201"phase3d-in.dat"

921123456alpha789beta

1010180"phase3d-anti.dat"

160

2001(a) Initial: in-phase.

23456alpha78910160

98140

8140

7120

7120

6100

6100

580

580

460

460

340

340

220

2200

1112345678beta

beta

180
"phase-in.dat"

180"phase-anti-all.dat"234

56[6] Y. Nakaaji and Y. Nishio, “Synchronization of Chaotic
Circuits Linked by Cross Talk” Proc. of NOLTA’06, pp.
43-46, 2008.

160

01

[4] J. Kawata, Y. Nishio and A. Ushida, “Chaos Synchronization by Transmission Line Coupling” Proc. ofNOLTA’96, pp. 109-112, 1996.
[5] Y. Nakaaji and Y. Nishio, “Synchronization of Chaotic
Circuits with Transmission Line” Proc. of NCSP’06, pp.
353-356, 2006.

Figure 10: Phase difference (Ltml = 10).

9[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.

10(b) Initial: anti-phase.

10[1] S. Boccaletti, J. Kurths, G. Osipov, D. Valladares andC. Zhou, “The Synchronization of Chaotic Systems”
Physics Reports, 366, pp. 1-101, 2002.

7alphaalpha(a) Initial: in-phase.

(b) Initial: anti-phase.

8[7] N. Kondo, Y. Uwate and Y. Nishio, “Synchronization of
Two Chaotic Circuits with Transmission Line Coupledby Resistors” Proc. of NCN’13, pp. 17-19, 2013.
[8] Y. Uwate, Y. Tanji and Y. Nishio, “Synchronization inTwo van der Pol Oscillators Coupled by Transmission
Line Model” Proc. of NOLTA’15, pp. 109-112, 2015.

Figure 11: Phase difference (Ltml = 20).

- 458 -