2017 IEEE Asia Pacific Conference on Postgraduate Research in Microelectronics and Electronics (PrimeAsia)

Frustrated Coupled Oscillators with Anomalous
Coupling Method
Kazuki Ueta, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {kazuki, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract—Synchronization phenomena of frustration networkby coupled oscillators has been studied in a wide range of ﬁelds,such as medicine and engineering. It is investigated towards
various systems up to now. However, analysis of regarding morecomplex systems are little. In our study, we developed the systemmodel so that a basic minimum unit even in more complexsystems. In addition, we observed synchronization phenomena
about the system. In this paper, we show the result of circuit
experiment with oscilloscope. Finally, we state the conclusion
accompanying the experiment results.

II. S YSTEM MODEL
Figure 2 shows a system model constituted van der Pol
oscillators (VDP-A and VDP-B). We couple each VDP-B via
inductor L and ground by coupling resistor R. In addition, We
couple VDP-A via resistor r. VDP-A is the only one central
circuit which is connected to all VDP-B in this system byresistor r.

I. I NTRODUCTION

There are a lot of synchronization phenomena in this world.
This is one of the nonlinear phenomena that we can often
observe by natural animate beings which do collective actions.
For example, ﬁreﬂy luminescence, cry of birds and frogs,
applause of many people and so on. Synchronization phenomena have a feature that the set of small power can produce
very big power by synchronizing at a time. Therefore studyof synchronization phenomena have been widely reported not
only engineering but also the physical and biological ﬁelds[1]–[8]. Investigation of coupled oscillators attention from many
researchers because coupled oscillatory network produces interesting phase synchronization such as the phase propagation
wave, clustering and complex patterns.
In this study, we focus on the synchronization phenomena
coupled by van der Pol oscillators containing ring and star
structures. Van der Pol oscillator is structured by inductor,capacitor and nonlinearity resistor. Van der Pol oscillator has
the feature of complicated oscillation pattern changing byfrequency. Therefore, it can be approximated to any oscillation
pattern in the natural world. We observe the synchronization
phenomena with computer simulation. van der Pol oscillator
is shown in Fig. 1.

 

 























Fig. 2. System model.

In the computer simulations, We assume that the voltage vk
and ampere ikg characteristics of the nonlinear resistor in eachoscillator are given by the follows:

ikg = −g1 vk + g3 vk3 ,(1)(g1 , g3 > 0),(k = 1, 2, 3, ..., N ),The characteristic of ring coupling has in-phase, anti-phaseand N-phase. The characteristic of star coupling has in-phaseand anti-phase.

Fig. 1. van der Pol oscillator.

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2017 IEEE Asia Pacific Conference on Postgraduate Research in Microelectronics and Electronics (PrimeAsia)

the horizontal axis is the voltage of each oscillator, and thevertical axis is the electric current of each oscillator. We set
the parameters α = 0.1, β = 0.0075 and γ = 0.02. In addition,
We show the system model of N = 4 in Fig. 4.

First, the circuit equations of VDP-A are given as follows:
⎧N
⎪dv1
⎪
⎪ C k = −in − i1g + ((N − 1)vk −vn ),⎨dt
R
n=2(2)⎪⎪
di1⎪⎩ L= vk ,dt(n = 2, 3, ..., N ),where N denotes the number of VDP-B.
On the other hand, VDP-B is connected to the adjacent VDPB and VDP-A. The circuit equations of VDP-B are given asfollows:⎧dvn⎪⎪C= −ikg − inR − ina − inb ,⎪⎪dt⎪⎨dina(3)= vn − R0 (ina + i(n+1)b ),2L⎪dt⎪⎪⎪⎪⎩ 2L dinb = vn − R0 (inb + i(n−1)a ).
dt
By using the parameters and variables as follows:


g1 C
g1 C
g1 C
in =yc , ina =yn , inb =zn ,
3g3 L
3g3 L
3g3 L

g1
g1
xc , v n =xn ,vk =3g3
3g3(4)
√Ld, α = g1,t = LCτ, “ · ” =dτC
1 LCβ=, γ = R0,R CL



Fig. 3. Attractor between adjacent oscillators (horizontal axis:xk , verticalaxis:yk ) (k = 1, 2, 3, 4, 5).

  where α is the nonlinearity, β is the coupling strength, γ
indicates the resistive component and yn denotes the currentof neighbor oscillator on coupling resistor R0 . The normalized
circuit equations of VDP-A are given as follows:   ⎧⎪⎨ N

1x˙c = αxc 1 − x2c −yc + β (N − 1)xc −xn ,3n=2⎪⎩y˙c = xc .

(5)

Fig. 4. System model of N = 4.

The normalized circuit equations of VDP-B are given asfollows:⎧1⎪⎪x˙n = αxn (1 − x2n ) − yn − zn − β(xc − xn ),⎪⎪3⎨
1y˙n = {xn − γ(yn + zn+1 )},⎪2⎪⎪⎪⎩ z˙n = 1 {xn − γ(yn + zn−1 )}.
2III. S IMULATION R ESULTNext, the time waveforms of the voltage of each capacitor
C after sufﬁcient time has elapsed are shown in Fig. 5. And
the phase differences between the adjacent oscillator of thiscase is equal to the result as shown in Fig. 6. It seems that
two electric currents of VDP-B are piled up as we understandit from the ﬁgures like 3 phase.

(6)A. Computer Simulation
We calculate Eqs. (5) and (6) using the Runge-Kutta methodwith the step size h = 0.02. We show the simulation resultof the synchronization phenomena when N = 5 in Fig. 3.
In this ﬁgure, We show the attractor of each oscillator andFig. 5. The Time waveforms of the each oscillator for N = 4.

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2017 IEEE Asia Pacific Conference on Postgraduate Research in Microelectronics and Electronics (PrimeAsia)

(b) β=0.0085.
































Fig. 6. Lissajous ﬁgures of N = 4.

(c) β=0.05.

Second, the simulation results of the system model containing six circuits are shown in Fig. 7. The value of theparameters are ﬁxed with β=0.001, 0.0085 and 0.05. In thecase of β=0.001, 5 phase synchronization appeared because
the coupling strength of VDP-A is weak. The current of theVDP-A and one of the current of the VDP-B are in-phaseat that time. Therefore, we assume 5 phase synchronization.
When the value of β sets 0.0085, some time waveforms ofVDP-B come close on in-phase synchronization. When weincrease coupling strength, most of two electric currents ofVDP-B are piled up as we understand it from the Lissajousﬁgures.
In the case of β=0.05, 5 phase synchronization become inphase synchronization. And then VDP-A becomes anti-phasesynchronization with VDP-B by increasing the value of β. We
understand that VDP-B synchronizes even if we read eitherﬁgure.











Fig. 7. Simulation Results for N=6 (α = 0.1 and γ = 0.02). time-waveform. Red and other colors denote x1 and xn respectively.(n =2,3,...,6)

B. Relationship between coupling strength and phase difference
Third, we summarize the simulation results in Fig. 8. In
the ﬁgure, we show the results when we increase the circuitnumbers N = 3, 4, ..., 7. The phase difference is based on one
voltage waveform of VDP-B. The broken line in the ﬁgure
represent asynchronous. The solid line in the ﬁgure represent
synchronous. From this result, It turns out that an even numbercircuits become in-phase as increasing the coupling strength.
Similarly, It turns out that an even number circuits become inphase as increasing the coupling strength. However, we could’t
conﬁrm the synchronization state between β=0 and β=0.016
in an odd number circuits.





(a) β=0.001.
Fig. 8. Relationship between coupling strength and phase difference.

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2017 IEEE Asia Pacific Conference on Postgraduate Research in Microelectronics and Electronics (PrimeAsia)

C. Comparison with circuit experiment
Finally, we show the results of circuit experiment. In thisexperiment, we observed the waveform when N = 3 withoscilloscope. From this results, we can observe that the 3phase synchronized in-phase and anti-phase. Each VDP-B
synchronised in-phase, VDP-A is anti-phase.

Fig. 9. Results of circuit experiment.

IV. C ONCLUSIONSIn this study, we have proposed a system model using ﬁve
circuits that is combined the ring and star structures. We
have observed the synchronization phenomena by increasing
the coupling strength of ring. When the coupling strengthis sufﬁciently small, system model becomes like function ofring coupling therefore, N phase or in-phase and anti-phasesynchronization can be observed. By increasing the couplingstrength, time wave forms of VDP-B have come close in-phasesynchronization. When the coupling strength is sufﬁciently
large, time wave forms of VDP-A and VDP-B become antiphase synchronization. In the future, we investigate synchronization phenomena using other circuits.
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[5] L. M. Pecora and T. L. Carrol, “Synchronization in Chaotic Systems”,
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