IEEE Workshop on Nonlinear Circuit Networks
December 15-16, 2017

Analysis of Frustration by Coupled Ring and Starof BVP-VDP Oscillator
Kazuki Ueta, Yoko Uwate and Yoshifumi nishio
Dept. Electrical and Electronic Eng.,
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 offields, such as BVP oscillator is the simplest mathematicalmodel describing a neural activity. However, even if two BVPoscillators are merely completed by a linear element, the wholesystem exhibits complicated behavior. In this study, we focus
on the synchronization phenomena coupled by Bonhöffer-vander Pol oscillator and van der Pol oscillators (BVP and VDP)
containing ring and star structures. At that time, we observe thesynchronization phenomena with computer simulation.

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. Synchronazation 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]–[6]. 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 addition, it has theadvantage of being able to manufacture for circuit on theboard.
In this study, we focus on the synchronization phenomena
coupled by Bonhöffer-van der Pol oscillator and van del pol
oscillators (BVP and VDP) containing ring and star structures.
Here, we used BVP as frustration. Then, we obserb thesynchronization phenomena with computer simulation.

L-gII. P REVIOUS CIRCUIT MODEL [7]
A. Circuit Model
Figure 2 shows a system model constituted van del pol
oscilators (VDP-A and VDP-B) in our previous study[7]. We
couple each VDP-B via inductor L and ground by coupling
resistor R. In addition, we couple VDP-A via resistor r. VDPA is the only one central circuit which is connected to all
VDP-B in this system by resistor r.

iAiAg

ikgvA
L

VDP B

VDP ArC

VDP ArCCCrr

2L

2Lr

R
2L
VDP B

2L
R
VDP B

R

2L

2L
R

VDP B

Fig. 2. System model.

In the computer simulations, we assume that the voltage
and current characteristics of the nonlinear resistor in eachoscillator are given by the follows:ig = −g1 v + g3 v 3 ,Cvk(1)L(g1 , g3 > 0).

Fig. 1. Bonhöffer-van der Pol oscillator and van del pol oscillator.

When we coupled ring of each VDP, the feature of ringcoupling has in-phase and N-phase. On the other hand, thefeature of star coupling has in-phase and anti-phase.

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First, the circuit equations of VDP-A are given as follows: y1
dvA1
= −iA − iAg + (N vA − v1 − v2 − ... − vN ), C
dtr(2)diA
= vA ,
 L
dt
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:dvk1
= −ika − ikb − ikg − (vk − vA ),C


dtr

dika
= vk − R(ika + ik+1,b ),
2L(3)
dt


dikb= vk − R(ikb + ik−1,a ),
2L
dt

yy

2xy

4x

1

5x

2x

3x

45Fig. 3. Attractor between adjacent oscillators for N = 4 (horizontal axis:xk ,vertical axis:yk ) (k = 1, 2, 3, 4, 5).

R2L

2L
VDP B

VDP Brr

2L
R2L
R

VDP Ar(k = 1, 2, ..., N ).
By using the following parameters and variables:
√√√
g1 C
g1 C
g1 C
yA , ika =yka , ikb =ykb ,iA =
3g L
3g3 L
3g3 L
√ 3
√
g1
g1
vA =
x A , vk =xk ,
3g3
3g3
√
√
d
L
, α = g1,
t = LCτ, “ · ” =dτC√√
1 LCβ=, γ=R,r CLy3r

2L

2L 2L

2L

VDP B

VDP BR(4)where α is the nonlinearity, β is the coupling strength between
VDP-A and VDP-B. γ indicates the coupling strength between
VDP-B. The normalized circuit equations of VDP-A are given
as follows:(1 ) x˙A = αxA 1 − x2A −yA
3(5)+β(N xA − x1 − x2 − ... − xN ),y˙A = xA .

Fig. 4. System model of N = 4.

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 twoelectric currents of VDP-B are piled up as we underestand it
from the figures like 3 phase.

xntimeThe normalized circuit equations of VDP-B are given asfollows:1 x˙k = αxk (1 − x2k ) − yka − ykb − β(xA − xk ),3(6)y ˙ = 0.5{xk − γ(yka + yk+1,b )}, kaykb
˙ = 0.5{xk − γ(yka + yk−1,b )}.

Fig. 5. The Time waveforms of the each oscillator for N = 4.

xxxx

4325x

1xxx

4xx

1x

11B. Simulation Results
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 = 4 in Fig. 3.
In this figure, we show the attractor of each oscillator andthe 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.

x3x2x2x545x2x3x3Fig. 6. Lissajous figures of N = 4.

Second, the simulation results of the system model containing six circuits are shown in Fig. 7. The value of the- 73 -parameters 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 we
increase 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 either
figure.

x4x3x2x1x1x5x1numbers 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 number
circuits become in-phase as increasing the coupling strength.
Similarly, It turns out that an even number circuits become
in-phase as increasing the coupling strength. However, wecould not confirm the synchronization state between β=0 and
β=0.016 in an odd number circuits.

x6x1

x1
Fig. 8. Relationship between coupling strength and phase difference.

III. P ROPOSED SYSTEM MODEL(a) β=0.001.

xxxxx

1x432x1x56x1x

11Next, we propose the system model which replaces centralcircuit with chaotic circuit[8]. Figure 9 shows a system model
constituted Bonhöffer-van der Pol oscillator and van der Poloscillators (BVP and VDP). A VDP is coupled adjacent VDP
via inductor L and ground by coupling resistor R0 . In addition,a VDP is coupled BVP via resistor R. BVP is the only one
central circuit which is connected to all VDP in this system
by resistor R.
L(b) β=0.0085.

x4x3x2

BVP

x5-gr

C-gVDP

CCx6
Rx1x1

x1x1x1RR0VDP
BVPA

2L
2L
VDP

2L

2L

RR
R0VDP

R02L

2L
R0

VDP

Fig. 9. System model.

(c) β=0.05.
Fig. 7. Simulation Results for N=5 (α = 0.1 and γ = 0.02). time-waveform. Red and other colors denote xc and xN respectively. (N= 1,2,...,6)

In Fig. 8, we show the results when we increase the cuircuit

When we coupled ring of each VDP, the feature of ringcoupling has in-phase and N-phase. On the other hand, thefeature of star coupling has in-phase and anti-phase. Where Ndenotes the number of VDP. VDP is connected to the adjacent
VDP and BVP.

- 74 -Where α is the nonlinearity, β is the coupling strength ofstar, σ is internal parameter of BVP, γ indicates the resistive
component and yn denotes the current of neighbor oscillator
on coupling resistor R0 . The normalized circuit equations ofBVP is given as follows:N∑
({}1 )
xn , x˙c = αxc 1 − x2c −zc − β N xc −3n=1(7)y˙c = zc − σyc ,z˙c = xc − yc .
The normalized circuit equations of VDP are given as follows:1 x˙n = αxn (1 − x2n ) − yn − zn + β(xc − xn ),3(8)y˙ = 0.5{xn − γ(yn + zn+1 )}, nz˙n = 0.5{xn − γ(yn + zn−1 )}.
IV. S IMULATION R ESULTWe show the simulation result of the synchronization phenomena when N = 3 in Figs. 10 and 11. In Fig. 10, weshow the attractor of each oscillator. The horizontal axis is thevoltage of each oscillator, and the vertical axis is the electric
current of each oscillator. We set the parameters α = 1.46, σ= 1.25, γ = 0.1 and β = 0. The BVP changes from a chaotic
solution to a periodic solution according to the initial value.
Therefore, Fig. 11 shows the phase difference compared in twopatterns (chaotic and periodic) when we increase the couplingstrength β.

xc

x1x2
y1

yc
x1

xc

x3
y2

y3x2x3y1

ycy2y3Fig. 10. Attractors of chaotic and periodic solution when β = 0.
180
VDP2 Chaos MIN
VDP2 Chaos MAXIn Fig. 11, we show the phase difference among the surrounding circuits. These broken lines are chaotic solution andsolid lines are periodic solution. This figure shows a comparison between chaotic solution and periodic solution. We
measured the phase difference with reference to one Poincar
section. As a result, as increasing β, the chaotic solution isdrawn into the periodic solution. And also, surrounding VDP
synchronised in-phase according to increasing β. Since A
and B are in agreement, it is considered that there is somecorrelation between both. Finally, as BVP is synchronized, it
can be said that frustration bacame weak.
V. C ONCLUSIONSIn this study, we have proposed the system model using
Bonhöffer-van der Pol that is combined the ring and star
structures. We have observed the synchronization phenomena
by increasing the coupling strength of star. When the couplingstrength is sufficiently small, system model becomes like function of ring coupling therefore, N phase synchronization canbe observed. By increasing the coupling strength, waveformsof VDP have come close in-phase synchronization. In thefuture, we investigate synchronization phenomena using othercircuits.
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VDP3 Chaos MIN

120

VDP3 Chaos MAX
VDP2 Periodic MIN
VDP2 Periodic MAX

Phase Difference of each VDP

VDP3 Periodic MIN60
VDP3 Periodic MAX

0-60-120-18000.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Coupling Strength βFig. 11. The phase difference among the surrounding circuits.

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