Effect of Changing Coupling Strength to Synchronization Phenomena
in Coupled van der Pol Oscillators
Vu Minh Hien, Minh Hai Tran, 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. Furthermore,
we also investigate the effect of the initial condition. By using
computer simulations, we observe synchronization phenom- 2. Circuit Modelena of the system and apply theoretical analysis to confirm
computer simulation results.
Abstract

1. Introduction
Synchronization phenomena can be observed everywherein our world, such as in firefly luminescence, birds and frogs
crying, human aplause and so on. Synchronization phenomena in coupled van der Pol oscillators are good models todescribe various higher-dimensional nonlinear phenomena in
the field of natural science. Synchronization phenomena has
a long history of researches and they have been reported inmany researches of engineering fields [1] - [2]. Furthermore,
the applications of synchronization phenomena have been
also found in chemical, physical and biological fields [3] [4]. Therefore, investigation of synchronization phenomenaobserved in coupled oscillatory systems is an important issue. However, there are not many discussions about coupledcomplex oscillatory networks by using electrical oscillators.
The van der Pol oscillator is a simple circuit as shown in
Fig. 1. It consists of an inductor, a capacitor and a nonlinear
resistor.

Figure 2: Circuit model.

Figure 1: van der Pol oscillator.

- 148 -The proposed circuit is shown in Fig. 2. We use three van(k = 2, 3, 4).
der Pol oscillators coupled as star structure that connected toanother oscillator via resistors r. We investigate synchroniza- where parameters α, β, and the γ denote nonlinearity, thetion phenomena by changing coupling strength of the resis- resistors r, and the resistor R, respectively.
tors.
With vC1 vC2 , vC3 , and vC4 denote capacitor’s voltage and3. Simulation ResultsiL1 iL2 , iL3 , and iL4 denote inductor’s electric current.
The circuit equations of VDP-A1 are given as folows:
For the computer simulations, we calculates Eqs. (2)-(5) bydvC1
using
the Runge-Kutta method with the step size h = 0.05.
C= −ig1 − iL1 − iR1 − iR2 − iR3 ,dt
(1)
We
show
the simulation results when the parameters of thediL1
L= v1 .
circuitmodel
are fixed as α = 0.01, β = 0.01 and γ = 0.006.
dt
Figure3shows
the attractor of each oscillator with the horThe circuit equations of Circuit-A2, Circuit-A3, Circuitizontal
axis
isthevoltage and the vertical axis is the electric
A4 are given as follows:
current of each oscillator.
dvCk
First, we investigate the effect of different the initial conC= −igk − iLk + iRk ,dtdition.
As simulation results are shown in Fig. 4, we obtain4
(2)
∑diLk
that,
bychanging initial values, 2 oscillators of Circuit-A2,L= vk − RiLm .
dtCircuit-A3,Circuit-A4 can become in-phase.
m=2
where:iRk =v1 − vk,r(k = 2, 3, 4).
The characteristics of the nonlinear resistors are defined asfollows:(3)
igk = −g1 vk + g3 vk3 .
By changing the variables and parameters:√√
g1t = LCτ, vk =xk ,3g3√√
g1 CL
iLk =yk , α = g1,3g L
C
√ 3√
1 L
C
β=,γ = R,r CL(4)Figure 3: Attractor between adjacent oscillators (α = 0.01,β = 0.01, γ = 0.006).

(k = 1, 2, 3, 4).
the normalized circuit equations of VDP-A1 are given as follows:
()
()
4
∑

1 3dx1= α x1 − x1 − y1 + β 3x1 −xmdτ
3m=2 dy1 = x .
1dτ(5)
the normalized circuit equations of Circuit-A2, Circuit-A3,Circuit-A4 are given as follows:()dxk1 3 dτ = α xk − 3 xk − yk − β(x1 − xk )4(6)∑dyk
=x−γy,km
 dτ
m=2Figure 4: Phase difference (α = 0.01, β = 0.01, γ = 0.006).
Next, we decrease the values of parameter γ from 0.006 to0.005 and continue conducting computer simulation. In thiscase, all four oscillatos become synchronization with second,
third and fourth oscillator (Circuit-A2, Circuit-A3, CircuitA4) are in-phase and only first oscillator of VDP-A1 is antiphase. The result of simulation is shown in Fig. 5.

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Figure 5: Phase difference (α = 0.01, β = 0.01, γ = 0.005).
Figure 6, figure 7 and figure 8 show the computer simulation results of the phase difference when parameter γ ischanged from 0.001 to 0.009, α is changed from 0.001 to 0.01
and β is changed from 0.007 to 0.016 with step size 0.001. In
this case, we can obtain that, when value of parameter γ issmaller than 0.005, α is smaller than 0.006 and β is lager
than 0.011, second, third, fourth oscillator are in-phase andonly first oscillator VDP-A1 is anti-phase.

Figure 8: Phase difference (α = 0.01, β = 0.01).
changing initial values or the coupling strengths.
4. Theoretical Analysis
In this section, we apply theoretical analysis to comfirm
above computer simulation results by using averaging method
for Eqs. (5) and (6). We assume that x1,k , y1,k can be considerd as below:
x1,k (τ ) = ρ1,k (τ ) cos(τ + θ1,k (τ ))y1,k (τ ) = ρ1,k (τ ) sin(τ + θ1,k (τ )).

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

ρ1 = α(x1 − 31 x31 ) cos ϕ1 − y1 cos ϕ1+β(3x1 −4∑xn ) cos ϕ1 + x1 sin ϕ1 ≡ X1n=2.

θ1 =Figure 6: Phase difference (β = 0.01, γ = 0.006).

α(x1 − 13 x31 ) sin ϕ1y1 sin ϕ1x1 cos ϕ1−+ρ1ρ1ρ14∑
β(3x1 −xn ) sin ϕ1n=2−− 1 ≡ Y1 .
ρ1(8)Circuit-A2, Circuit-A3, Circuit-A4:.

ρk = α(xk − 13 x3k ) cos ϕk − yk cos ϕk−β(x1 − xk ) cos ϕk + xk cos ϕk−γ4∑yn sin ϕk ≡ Xk

n=2.
α(xk − 13 x3k ) sin ϕkxk cos ϕkyk sin ϕkθk =−+ρk
ρk
ρk4
∑

γ

Figure 7: Phase difference (α = 0.01, γ = 0.006).

−Therefore, we can control synchronization phenomena byyn cos ϕkn=2where- 150 -ρk− 1 ≡ Yk ,(9)ϕk = τ + θk(k = 2, 3, 4) .

These theoretical rersults correspond with the computer
simulation results. Table 1 summarizes the comparison between theoretical and simulation results when the parameters
By averaging Eqs. (8)-(9) over on period, as averaging α, β are set as α = 0.01, β = 0.01 and parameter γ ismethod’s theory, ρ1,k and θ1,k can be considered as constant changed from 0.001 to 0.003. By solving Eq. (14), we can. .
and the values of ρ1 ,θ1 can be calculated as:
see that they match very well from below table.
VDP-A1:∫T.
ρ1 = lim X1 dτTable 1: Comparison between theoretical and simulation reT →∞ 0(10) sults (α = 0.01, β = 0.01).
T.
∫θ1 = lim Y1 dτ .
ρk /ρ1T →∞ 0γ
TheorySimulation
Circuit-A2, Circuit-A3, Circuit-A4:
0.001 0.711499
0.711512
0.002 0.686578
0.686588∫T.
0.003
0.660892
0.660902ρk = lim Xk dτ.
θk =T →∞ 0
∫Tlim

T →∞ 0(11)Yk dτ .

By solving the above equations, Eqs. (12) and (13) are
obtained:
VDP-A1:131αρ1 − αρ1 3 + β ρ12824∑1
βρn cos(θn − θ1 )−2n=24.
1 ∑ ρnθ1 =sin(θn − θ1 ).
2 n=2 ρ1.

ρ1 =Circuit-A2, Circuit-A3, Circuit-A4:
1
11αρk − αρk 3 − βρ1 cos(θ1 − θk )2824
∑11+ βρk −γρn cos(θn − θk )22n=24.
1 ∑ ρnθk =sin(θk − θn ).
2 n=2 ρk.

ρk =5. Conclusions

In this study, we have investigated the synchronization phenomena in coupled oscillators containing star structure connected to another oscillator. By computer simulation, we haveobserved the synchronization phenomena by changing initialvalues and coupling strength. In the case of changing initial(12) values, in three oscillators of Circuit-A2, Circuit-A3, CircuitA4, two oscillators can become in-phase. By chaning coupling strength, all four oscillators can become synchronization with oscillators of Circuit-A2, Circuit-A3, Circuit-A4 arein-phase and only oscillator VDP-A1 is anti-phase. We also
apply theoretical analysis to confirm these computer simulation results.
In the future, we experiment the proposed circuit model.
References(13)[1] K. Ueta, Y. Uwate and Y. Nishio, “Synchronization Phenomena in Complex Networks of van der Pol Oscillators”, Proc. of NCSP’17, pp. 285-288, Mar. 2017.

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[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.

[3] F. Jhou, “Multistate and Multistage Synchronization ofHindmarsh-rose Neurons with Excitatory Chemical and
3 4311311
Electrical Synapses”, Proc. of IEEE, pp. 1335-1347,
βa +α + β a3 −α + β − γ a − β = 0.
2222222Jan. 2012.
(14)
For the phase difference:[4] W. Wessel, “Synchronization and Coupling Analysis:
Applied Cardiovascular Physics in Sleep Medicine”,
θ1 − θk = π
Proc. of EMBC’13, pp. 6567-6570, Jul. 2013.
()()

θ2 − θ3 = θ3 − θ4 = θ4 − θ2 = 0.

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