RISP

NCSP’16

International Workshop on
Nonlinear Circuits, Communications and Signal Processing

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

Synchronization of Coupled Oscillators Containing Ring and Star Structures
Kazuki Ueta, Seiya Kita, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minami-Josanjima, Tokushima, 770-8506, Japan
Phone: +81-88-656-7470, Fax: +81-88-656-7471
E-mail: {kazuki, kita, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract
Analysis of synchronization phenomena is expected for applying in various feilds. In this study, we propose a new system model using the van del pol oscillator. We investigate
synchronization phenomena observed in the system model
containing ring and star. In addition, we observe several
types of synchronization phenomena by increasing the coupling strength of ring.

2. System modelFig. 2 shows a system model constituted van del pol oscilators (VDP-A and VDP-B). We couple each VDP-B via
inductor L and ground by coupling resistor R0 . 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.

1. Introduction
i1i1g

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
study of 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 thephase propagation wave, clustering and complex patterns.
In this study, we focus on the synchronization phenomena
coupled by van del pol oscillators containing ring and starstructures. van del pol oscillator is shown in Fig. 1.

-gV DP Aingv1
LC

V DP B-gvnCR
VDP A

R2L

RR
R02L
VDP B

2L

R02L

2L

R0

R0

VDP B

VDP B

Figure 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:-gCLikg = −g1 vk + g3 v3k ,(1)(g1 , g3 > 0),(k = 1, 2, 3, ..., N),
Figure 1: van del pol oscillator.

The charactaristic of ring coupling has in-phase, anti-phaseand N-phase. The charactaristic of star coupling has in-phase- 463 -

and anti-phase.
First, the circuit equations of VDP-A are given as follows:NXdvk1C=−i−i+((N−1)v−vn ),n1gk
dtR
n=2(2)
di1
= vk , L
dt(n = 2, 3, ..., N),

3. Simulation Result
We calculate Eqs. (5) and (6) using the Runge-Kutta
method with the step size h = 0.02. We show the simulation result of the synchronization phenomena when N = 5 inFig. 3. In this figure, we show the attractor of each oscillator
and the horizontal axis is the voltage of each oscillator, andthe vertical 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 = 5 in Fig. 4.

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 as
3
1
follows:2
dvn
C= −ikg − inR − ina − inb ,dt3
2dina
1(3)=v−R(i+i),2L
n0na(n+1)bdtdinb 2L= vn − R0 (inb + i(n−1)a ).
dt
54
By using the parameters and variables as follows:
sss
g1C
g1C
g1C
4
yc , ina =yn , inb =zn ,
in =
5
3g3 L
3g3 L
3g3 L
r
r
g1
g1
vk =
xc , vn =xn ,3g3
3g3(4) Figure 3: Attractor between adjacent oscillators (horizontalr
axis:xk , vertical axis:yk ) (k = 1, 2, 3, 4, 5).
√Ld, α = g1,t = LCτ, “ · ” =dτCrr
1 LCβ=, γ = R0,R C
LyyyxxxyyxxR0
NXno
1 xn , x˙c = αxc 1 − xc2 −yc + β (N − 1)xc −3n=2 y˙ = x .
cc

VDP B

VDP B

R2LR0R

2LR0

VDP A

2L

(5)

R

R
2L

2L

VDP B

The normalized circuit equations of VDP-B are given as follows:
1
x˙n = αxn (1 − xn2 ) − yn − zn − β(xc − xn ),31y˙n = {xn − γ(yn + zn+1 )},


2

1
 z˙n = {xn − γ(yn + zn−1 )}.
22L

2L

where α is the nonlinearity, β is the coupling strength, γ indicates the resistive component and yn denotes the current ofneighbor oscillator on coupling resistor R0 . The normalized
circuit equations of VDP-A are given as follows:

2LVDP B

R0Figure 4: System model of N = 5.

(6)
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.

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It seems that two electric currents of VDP-B are piled up as
we underestand it from the figures like 3 phase.

x3x2

x1x1

xnx5x4x6x1

x1

x1

time
Figure 5: The Time waveforms of the each oscillator.
(a) β=0.001.

xxx

432xxxx2(b) β=0.0085.

x

4522xxx1x1xx

1x1x1xx

43xx11xxxx

5x

11x25x3x2xx3x5x4x6

3x1x1x1x1x1

Figure 6: Lissajous figures.

Finaly, the simulation results of the system model containing six circuits are shown in Fig. 7. The value of the parameters are fixed with β=0.001, 0.0085 and 0.05. In the case of
β=0.001, 5 phase synchronization appeared because the coupling strength of VDP-A is weak. The current of the VDPA and one of the current of the VDP-B are in phase at thattime. Therefore, we assume 5 phase synchronization. Whenthe value of β sets 0.0085, some time waveforms of VDP-B
come close on in-phase synchronization. When we increase
coupling strength, most of two electric currents of VDP-B
are piled up as we understand it from the Lissajous figures.
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.

(c) β=0.05.

Figure 7: Simulation Results for N=6 (α = 0.1 and γ = 0.02).timewaveform.Red and other colors denote x1 and xn respectively.(n =2,3,...,6)

4. Conclusion
In this study, we have proposed a system model using fiveand six circuits that are combined the ring and star structures. We have observed the synchronization phenomena
by increasing the coupling strength of ring. When the coupling strength is sufficiently small, system model becomes
like function of ring coupling therefore, 5 phase synchronization can be observed. By increasing the coupling strength,- 465 -

time wave forms of VDP-B have come close in-phase synchronization. When the coupling strength is sufficiently large,
time wave forms of VDP-A and VDP-B become anti-phasesynchronization. In the future, we investigate synchronization phenomena using other circuits.

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