IEEE Workshop on Nonlinear Circuit Networks
December 7-8, 2018

Synchronization Phenomena in Coupled van der Pol Oscillators
Containing Three Oscillators with Star Structure Connected to Another Oscillator
Vu Minh Hien, Yoko Uwate and Yoshifumi Nishio†Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minamijosanjima, Tokushima, 770-8506 Japan
Email: {hien, minhhai, uwate, nishio}@ee.tokushima-u.ac.jp

In this study, we propose a new type of coupled van der Pol
oscillators: Star structure connected to another oscillator. By
In this study, synchronization phenomena in coupled oscilla- carring out computer simulations and theoretical analysis, thetors containing star structure connected to another oscillator relationship of the model between synchronization phenomis investigated. We focus on the phase difference between two ena and coupling strength is investigated.
oscillators when coupling strength is changed. By using computer simulations, we observe synchronization phenomena of
the system and use theoretical analysis and circuit experiment
2. Circuit Modelto confirm computer simulation results.
Abstract

The proposed circuit is shown in Fig. 2. We use three vander Pol oscillators coupled as star structure that connected toanother oscillator via resistors r. We investigate synchronizaWe are living in the world where there are so many ex- tion phenomena by changing coupling strength of the resisample of synchronization: firefly luminescence, cry of birds tors.
and frogs, human applause, etc. Synchronization phenomena have a long history of researches and they have beenreported in many researches of engineering fields [1] - [2].
Furthermore, the applications of synchronization phenomena have been also found in chemical, physical and biological fields [3] - [4]. Synchronization phenomena in coupled
van der Pol oscillators are good models to describe various
higher-dimensional nonlinear phenomena in the field of natural science. However, in each one of them, van der Pol oscilators is either coupled by different method or has different
feature. Therefore, investigation of synchronization phenomena observed in coupled oscillatory systems is an important
issue.
The van der Pol oscillator is a simple circuit as shown inFig. 1. It consists of an inductor, a capacitor and a nonlinearresistor.
1. Introduction

Figure 2: Circuit model.

Figure 1: van der Pol oscillator.

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With vC1 vC2 , vC3 , and vC4 denote capacitor’s voltage andiL1 iL2 , iL3 , and iL4 denote inductor’s electric current.
The circuit equations of VDP-A1 are given as folows:
dvC1= −ig1 − iL1 − iR1 − iR2 − iR3 ,dt
(1)
diL1
L
= v1 .
dt
The circuit equations of Circuit-A2, Circuit-A3, CircuitA4 are given as follows:
CdvCk= −igk − iLk + iRk ,dt
4∑
diLkL= vk − RiLm .
dt
m=2C(2)where:
iRk =v1 − vk,r

3. Simulation Results
For the computer simulations, we calculates Eqs. (2)-(3)by using Runge-Kutta method with the step size h = 0.05.
When the parameters are fixed as α = 0.1,γ = 0.006, we
control synchronization phenomena of this circuit model bychanging the coupling strengths β2 , β3 , β4 .
First, in the case of parameters β2 , β3 , β4 are set to 0.015,Fig. 3 shows the attractor of each oscillator. Next, we sightly
increase the parameters β2 , β3 , β4 to the same value as 0.017,
all fours oscillators become in-phase as Fig. 4. When weonly change β2 to 0.005, the oscillator of Circuit A2 becomesanti-phase with other oscillators, when we change β2 , β3 to0.0005, oscillator of VDP-A1 becomes in-phase with oscillator of Circuit A2 and the oscillators of Circuit A2, A3, A4
become 3-phase synchronization. These results are shown inFigs. 5-6.

(k = 2, 3, 4).
The characteristics of the nonlinear resistors are defined asfollows:igk = −g1 vk + g3 vk3 .
(3)
By changing the variables and parameters:√√
g1
t = LCτ, vk =xk ,3g3√√
g1 CLiLk =yk , α = g1,
3g3 LC√√
1 LC
β=,γ = R,r C
L(4)Figure 3: Phase differences (β2 = β3 = β4 = 0.015).

the normalized circuit equations of VDP-A1 are given as follows:
()
()
4∑

 dx1 = α x − 1 x3 − y − β 3x −xm
1
11dτ3 1m=2
 dy1= x1 .
dτ(5)the normalized circuit equations of Circuit-A2, Circuit-A3,Circuit-A4 are given as follows:()
1 3dxk=αx−x− yk + β(x1 − xk )k dτ
3 k4(6)∑ dyk

=x−
γy,
km
 dτ
m=2

Figure 4: Phase differences (β2 = β3 = β4 = 0.017).

(k = 1, 2, 3, 4).

(k = 2, 3, 4).
where parameters α, β, and the γ denote nonlinearity, theresistors r, and the resistor R, respectively.

Figure 5: Phase differences (β2 = 0.005, β3 = β4 = 0.015).

Figure 6: Phase differences (β2 = β3 = 0.0005, β4 = 0.015).
Figures 7, 8 and 9 show the computer simulation results ineach case when parameters β2 , β3 , β4 are changed in rangesof values. From these result, we can control synchronization
phenomena of this system by changing coupling strengths.

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ρ1 = α(x1 − 31 x31 ) cos ϕ1 − y1 cos ϕ1−β(3x1 −4∑xn ) cos ϕ1 + x1 sin ϕ1 ≡ X1

n=2.

θ1 =α(x1 − 13 x31 ) sin ϕ1x1 cos ϕ1y1 sin ϕ1−+
ρ1
ρ1
ρ14
∑
β(3x1 −xn ) sin ϕ1n=2+− 1 ≡ Y1 .
ρ1(8)Circuit-A2, Circuit-A3, Circuit-A4:Figure 7: Phase differences in the case of changing β2 .

.

ρk = α(xk − 31 x3k ) cos ϕk − yk cos ϕk+β(x1 − xk ) cos ϕk + xk cos ϕk−γ

4∑yn sin ϕk ≡ Xkn=2α(xk − 13 x3k ) sin ϕkyk sin ϕkxk cos ϕk−+ρkρkρk4∑γyn cos ϕk− n=2− 1 ≡ Yk ,ρk.

(9)

θk =Figure 8: Phase differences in the case of changing β2 , β3 .

whereϕk = τ + θk(k = 2, 3, 4) .
By averaging Eqs. (8)-(9) over on period, as averaging
method’s theory, ρ1,k and θ1,k can be considered as constant. .
and the values of ρ1 ,θ1 can be calculated as:
VDP-A1:∫T.
ρ1 = lim X1 dτ
T →∞ 0(10).
∫Tθ1 = lim Y1 dτ .
T →∞ 0Figure 9: Phase differences in the case of changing β2 , β3 , β4 .

Circuit-A2, Circuit-A3, Circuit-A4:
Therefore, we can control synchronization phenomena bychanging the coupling strengths.

.

∫T.

T →∞ 0∫Tρk = lim
θk = lim

T →∞ 04. Theoretical Analysis

Xk dτ(11)Yk dτ .

By solving the above equations, Eqs. (10) and (11) are obIn this section, we apply theoretical analysis to comfirm tained:
above computer simulation results by using averaging method
VDP-A1:for Eqs. (5) and (6). We assume that x1,k , y1,k can be consid1
31.
erd as below:
ρ1 = αρ1 − αρ1 3 + β ρ1282
4x1,k (τ ) = ρ1,k (τ ) cos(τ + θ1,k (τ ))∑1(7)+βρn cos(θn − θ1 )y1,k (τ ) = ρ1,k (τ ) sin(τ + θ1,k (τ )).
(12)
2n=24.
1 ∑ ρnθ1 =sin(θn − θ1 ).
2 n=2 ρ1

Assign Eqs. (5)-(6) to Eq. (7), we obtain:
VDP-A1:- 78 -

Circuit-A2, Circuit-A3, Circuit-A4:
1
11αρk − αρk 3 + βρ1 cos(θ1 − θk )2824∑
11+ βρk −γρn cos(θn − θk )22n=24.
1 ∑ ρnsin(θk − θn ).
θk =2 n=2 ρk.

ρk =(13)
Figure 10: Circuit experiment for r2 = 47[kΩ], r3 = r4 = 2[kΩ].

In the steady state,.

.

ρ1,k = 0 and θ1,k = 0 must be satisfield. By solving
Eqs. (12)-(13) we obtain:
ρkas solution of below equation:
ρ1

Figure 11: Circuit experiment for r2 = r3 = 82[kΩ], r4 = 2[kΩ].

)()131131α − β a3 −α − β − γ a − β = 0.
222222(14)For the phase difference:

3 4βa +2(Figure 12: Circuit experiment for r2 = r3 = r4 = 2[kΩ].

θ2 − θ3 = θ3 − θ4 = θ4 − θ1 = 0.
These theoretical rersults correspond with the computer
simulation results. Table 1 summarizes the comparison between theoretical and simulation results when the parametersα, γ are set as α = 0.05, γ = 0.006 and parameter β1,2,3 arechanged together to 0.018, 0.02, 0.05. By solving Eq. (14),
we can see that they match very well from below table.

6. Conclusions

In this study, we have investigated the synchronization phenomena in coupled oscillators containing star structure connected to another oscillator, and observe its synchronization
phenomena by theoretical analysis, computer simulation andcircuit experiment. In the next step, it is necessary to increase
the number of oscillators and complete the theoretical analysis, and it is expected to bring us more interesting phenomena
Table 1: Comparison between theoretical and simulation reand we can find a good solutions for larger coupled systems.
sults (α = 0.05, γ = 0.006).
ρk /ρ1References
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[2] Y. Uwate, Y. Nishio and R. Stoop, “Synchronization in
Three Coupled van der Pol Oscillators with Different
Coupling Strength”, Proc. of NCSP’10, pp. 109-112,
Mar. 2010.

5. Circuit Experiments
In this section, we build a real circuit to comfirm these result above. We set L = 20[mH], C = 33[nF] for each oscillator
and change value of r. We also obtain the same synchronization from the circuit experiments. Figure 10 shows the resultof oscillator 1, 2, 3 (Oscillator 2 is anti-phase). Figure 11
shows the result of oscilator 2, 3, 4. Figure 12 shows theresult of oscillator 1, 2, 3.

[3] F. Jhou, “Multistate and Multistage Synchronization ofHindmarsh-rose Neurons with Excitatory Chemical andElectrical Synapses”, Proc. of IEEE, pp. 1335-1347,
Jan. 2012.
[4] W. Wessel, “Synchronization and Coupling Analysis:
Applied Cardiovascular Physics in Sleep Medicine”,
Proc. of EMBC’13, pp. 6567-6570, Jul. 2013.

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