2014 RISP International Workshop on Nonlinear Circuits,
Communications and Signal Processing
NCSP'14, Honolulu, Hawaii, USA. February 28 - March 3, 2014

Propagation of Frustration in Three Coupled Oscillators as a Ring Topology
Yoko Uwate† , Yosifhimi Nishio† and Ruedi Stoop‡†Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470
Email: {uwate, nishio}@ee.tokushima-u.ac.jp

Abstract
This paper considers frustration mechanism of three coupled
van der Pol oscillators. By using a parallel circuit composedof a resistor and a capacitor as a coupling element, the dynamical behavior of frustration can be observed. We confirm
that the propagation speed of frustration diﬀers by the initialconditions.

‡Institute of Neuroinfomatics, UZH/ETH Zurich,
Winterthurerstrasse 190,
CH-8057 Zurich, Switzerland,
Email: ruedi@ini.phys.ethz.ch

switch is closed. When the switch is oﬀ, the adjacent oscillators tend to synchronize with anti-phase state. In this case,the coupled oscillators have no frustration because the energyof the coupling term is zero. While, in the case of that theswitch is on, 3-phase synchronization can be observed.
In order to make clear the frustration mechanism, we investigate how to change the synchronization from anti-phaseto 3-phase state by measuring the capacitance voltage of thecoupling term.

1. Introduction

1st oscillator

i R1

2L

rm

ia1
2L

C
v1

SW
rm

C

2L

C

vC

vA
2L

2L
rm

rm

ib3

2L

R

R

ia3

rm

Coupled oscillatory systems are good models to express
essential role of high-dimensional nonlinear phenomena occurring in the field of natural sciences. Therefore, studies ofsynchronization phenomena have been extensively reported
in physical, biological and electrical systems. Moreover, coupled oscillatory systems can also produce interesting phasepatterns, including wave propagation, clustering and complex
patterns [1], [2].
In our previous study, our research group have reported interesting synchronization phenomena observed from several
types of the oscillatory systems with frustration [3]-[5]. Here,
we call frustration which can produce diﬀerent synchronization state from original system. Frustration can be occurredby network structure, parameter mismatch and so on.
In this study, we consider that the dynamic behavior offrustration can be observed by using a inductor or a capacitor
in the coupling element. As a first step, we set the parallel circuit composed of a capacitor and a resistor up as the couplingelement. By using a parallel circuit composed of a resistor
and a capacitor as the coupling elements, we expect that the
dynamical behavior of frustration in the coupling term can beobserved.

ib1

ib2

ia2
rm

i R3

iR2
v3
R

C

3rd oscillator

C

vB

C

v2

2nd oscillator

Figure 1: Three coupled van der Pol oscillators as a ring
topology.
In the computer simulations, we assume that the vk − iRkcharacteristics of the nonlinear resistor in each oscillator isgiven by the following third order polynomial equation
iRk = −g1 vk + g3 vk 3 (g1 , g3 > 0),

2. Circuit Model(1)(k = 1, 2, 3).

The normalized circuit equations (SW:ON) governing the
The circuit model of three coupled van der Pol oscillator as
ring topology is shown in Fig. 1. In this circuit model, the net- circuit in Fig. 1 are expressed aswork structure is changed by the switch. The adjacent oscilla- [First oscillator]

(
dx1
1 2)
tors are coupled by the parallel circuit composed of a resistor



=
ε
1
−
x1 x1 − (ya1 + yb1 )
and a capacitor. By using this coupling method, the adjadτ
3



{
}


cent oscillators tend to synchronize with anti-phase state. The dya1 1
(2)
=
x
−
ηy
−
x

1
a1
A

conceptual circuit model and switching operation are shown

dτ
2



}

in Fig. 2. At the beginning, the switch is opened. When
dyb1 1 {=
x1 − ηyb1 − xC
the count of Runge-Kutta simulation reaches to 500, 000, thedτ2- 545 -

3.1. Initial Condition: Anti-Phase State

1
SW

2

3

(a) Conceptual circuit model with switch.
SW: ON

SW: OFF1

1
anti-phase3

anti-phase

23-phase3-phase?

33-phase2(b) Synchronization and switching operation.

Figure 2: Conceptual circuit model.

[Second oscillator]

(
dx2
1 2)



=
ε
1
−
x2 x2 − (ya2 + yb2 )



dτ
3



{
}


 dya2 1(3)=x2 − ηya2 − xB



dτ
2



{
}



dyb2 1=x2 − ηyb2 − xAdτ2[Third oscillator]

(
dx3
1 2)



=
ε
1
−
x3 x3 − (ya3 + yb3 )



dτ
3



{
}


 dya3 1
(4)
=
x
−
ηy
−
x

3
a3
C


dτ
2




}


dyb3 1 {=
x3 − ηyb3 − xBdτ2[Coupling terms]

dxA



= β(ya1 + yb2 ) − αβxA



dτ





 dyB
(5)
= β(ya2 + yb3 ) − αβxB



dτ






dyC



= β(ya3 + yb1 ) − αβxC
dτ
where
√
√
√
g1
C
t = LCτ, vk =
xk ,
3g3 L
√
√
√
√
g1
C
g1
C
iak =
yak , ibk =
ybk ,
3g3 L
3g3 L
√
√
1
C
L
C
ε = g1, η = rm
, α= , β=
,
C
L
R
C0
(k = 1, 2, 3).
In this equations, α and β correspond the coupling strengthand ε denotes the nonlinearity of the oscillators.

First, we investigate the capacitance voltages of the coupling term when the initial condition of the coupled oscillators is fixed with anti-phase state.
The simulation result is shown in Fig. 3. xA , xB and xC
are corresponding on the capacitance voltages of the couplingterm. From this result, before the switch is on, the value ofxA , xB and xC show zero. This means that the coupled oscillators have no frustration, because the adjacent oscillators aresynchronized with anti-phase state. After the switch is on, allcapacitance voltages start to oscillate. The capacitance voltage of xC shows larger voltage than the others. After certain
time, three voltages converge to same amplitude. In this case,three voltages are synchronized with 3-phase state as shownin Fig. 4(a). We also confirm same 3-phase synchronizationby using circuit experiment (see. Fig. 4(b)). The parametersof the circuit experiment are set to L = 50[mH], C = 68[nF],C0 = 68[nF] and R = 50[Ω].
0.1

0.05

XA

0

-0.05
-0.1
0.1
0.05

XB

0

-0.05
-0.1
0.1
0.05

XC

0

-0.05

-0.1
4000006000008000001e+06

1.2e+06

1.4e+06

1.6e+06

1.8e+06

2e+06

iteration

SW:ON

Figure 3: Capacitance voltages of coupling element.

0.1

0.05

XA XC XB

0

-0.05

-0.1
1.995e+06

1.996e+06

1.997e+06

1.998e+06(a)1.999e+06

2e+06
iteration(b)

Figure 4: 3-phase synchronization. (a) Computer simulation.
(b) Circuit experiment.

3. Synchronization Phenomena (α = 20.0 and β = 1.0)
For the computer simulations, we calculate Eqs. (2)-(5) 3.2. Initial Condition: In-Phase State
using a fourth-order Runge-Kutta method with the step size
h = 0.005. The parameters of this circuit model are setting
Next, we investigate the capacitance voltages of the coufor ε = 0.1, η = 0.001, α = 20.0 and β = 1.0.
pling term when the initial condition of the coupled oscilla-- 546 -

SW:ON

tors is fixed with in-phase state.
We observe similar result of the capacitance voltages withabove section. However, we find the diﬀerence of the convergence time to 3-phase state between in-phase and anti-phase
initial conditions. Figure 5 shows the simulation result of xA
obtained from diﬀerent initial conditions. We confirm that theconvergence time of in-phase initial condition is shorter thananti-phase initial condition.

200
180
160

phase difference

140
120
100

Initial condition: In-phase
Initial condition: Anti-phase

80
60
40

0.1

Initial condition: In-phase (red)

20

0.05

0

XA

0

0

2000004000006000008000001e+06

1.2e+06 1.4e+06 1.6e+06 1.8e+06

2e+06

iteration-0.05

Initial condition: Anti-phase (green)-0.1
4000006000008000001e+06

1.2e+06

1.4e+06

1.6e+06

1.8e+06

Figure 7: Phase diﬀerences (α=20).

2e+06

iteration

Figure 5: Comparison of capacitance voltages depending on
4. Parameter Dependency (α and β)
initial conditions (xA ).
Finally, the parameter dependency of the circuit model isinvestigated.
In this study, we change the parameters of the
We count the convergence time from iteration=500,000
coupling
strength(α and β). α and β correspond to the resis(SW:ON) to 3-phase synchronization (see. Fig. 6). Table 1
tance
and
capacitanceof the parallel circuit of the couplingsummarizes the convergence times of in-phase and anti-phaseelement.
initial conditions. Two convergence times are diﬀerence andT c of in-phase initial condition is shorter than the anti-phase
initial condition.
4.1. Dependence on α
From this result, we consider that the convergence time
Figure 8 shows the convergence time with α when the inicould be depending on the past synchronization state. Figtial
condition is set to in-phase and anti-phase states. In theure 7 shows the phase diﬀerences obtained from in-phase andbothcases, the convergence time has peak around α=10. By
anti-phase initial conditions. Before the switch is closed, twoincreasing
α, the convergence time of in-phase initial conphase diﬀerences show the diﬀerent synchronization states.
ditiondecrease
gradually. In the case of anti-phase initial
In the case of in-phase initial condition, the phase diﬀerence
condition,
theconvergencetime also decrease after the peak.
changes from 0 to 180 degrees. While, in the case of antiWhile,
when
α
is
largerthan
20, the convergence time of antiphase initial condition, the phase diﬀerence shows 180 dephase
initialcondition
increases
linearly.
grees from the beginning.
Figure
9
shows
the
phase
diﬀerences
obtained from inWe assume that the propagation speed of the frustration
phase
and
anti-phase
initialconditions.
Inthe both cases, theis depending on the time of the stable synchronization state.
phase
diﬀerenceofin-phase
initialcondition
converges faster
Namely, by using the convergence time, we can expect thethananti-phase
initialcondition.
past synchronization state.
SW:ON

1.4e+063-phase

1.3e+06

0.1

1.2e+06

0.05

1.1e+06
1e+06

0

Tc

XA

convergence time
(Tc)

Initial condition: Anti-phase

900000-0.05

800000-0.1
4000006000008000001e+06

1.2e+06

1.4e+06

1.6e+06

1.8e+062e+06

700000

iteration

600000Figure 6: Definition of convergence time (T c ).

400000

Table 1: Convergence time (T c ).
Initial condition Convergence time (T c )In-phase
720,453
Anti-phase
787,824

Initial condition: In-phase

500000

0

10

20

30

40

50

α

60

70

80

90

100

Figure 8: Convergence time depending on α.
In our past study, we have used the amplitude as the frustration evaluation. Figure 10 shows the amplitude value obtained from in-phase and anti-phase initial conditions when α
is changed. From this figure, we can see that two lines of the- 547 -

200

180

180

160

160

140

140

100

Initial condition: In-phase
Initial condition: Anti-phase

80

100

Initial condition: In-phase
Initial condition: Anti-phase

80
60

40

40

1.88

1.875

20

0

Initial condition: In-phase
Initial condition: Anti-phase

1.885

120

60

20

1.89

Amp

120

phase difference

phase difference

200

0
0

2000004000006000008000001e+06

1.2e+06 1.4e+06 1.6e+06 1.8e+06

2e+06

0

2000004000006000008000001e+06

1.2e+06 1.4e+06 1.6e+06 1.8e+06

2e+06

iteration

iteration(a) α=40.

(b) α=60.

1.87

1.865

Figure 9: Phase diﬀerences.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

β

Figure 12: Amplitude depending on β.

2

1.9

Initial condition: In-phase
Initial condition: Anti-phase

5. Conclusions

1.8

Amp

0

1.7

In this study, we have investigated synchronization phenomena of ring oscillators coupled by a parallel circuit composed of a resistor and a capacitor. By using such coupling
1.5
elements, we observed interesting frustration mechanism de1.4
0
10
20
30
40
50
60
70
80
90
100
pending on the initial conditions.
αThe convergence times of in-phase initial condition isshorterthan the anti-phase initial condition. Namely, the conFigure 10: Amplitude depending on α.
vergence time could be depending on the past synchronization state. We assume that the propagation speed of the frustration is depending on the time of the stable synchronization
amplitudes are completely same. Namely, we consider that state. By using the convergence time, we can expect the past
the convergence speed of frustration will be used for evalua- synchronization state.
In the future works, we would like to investigate frustration of frustration in the coupled oscillatory systems.
tion mechanism of more complex oscillatory systems by using passive device in the coupling elements.
4.2. Dependence on β
1.6

Acknowledgment
Figure 11 shows the convergence time with α when theinitial condition is set to in-phase and anti-phase states. In the This work was partly supported by JSPS Grant-in-Aid forboth cases, the convergence time increases when β is smaller Young Scientists 23700269.
than 0.5. By increasing β, the convergence time of the both
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Tc

720000Initial condition: In-phase
700000680000660000640000620000600000
0

0.2

0.4

0.6

0.8

1

β

1.2

1.4

1.6

1.8

Figure 11: Convergence time depending on β.

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