2014 International Symposium on Nonlinear Theory and its Applications
NOLTA2014, Luzern, Switzerland, September 14-18, 2014

Synchronization Phenomena in van der Pol Oscillatorswith Diﬀerent Amplitudes
Vu Minh Thuan† , Yoko Uwate† and Yoshifumi Nishio††Department of Electrical and Electronic Engineering, Tokushima University
2–1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {thuan, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—This article gives an explanation of the synchronization phenomena in a circuit which contains threeor more van der Pol oscillators when one of them has different amplitude from the others.

1. Introduction
We are living in the world where there are so many
example of synchronization: firefly luminescence, cry ofbirds and frogs, human applause, etc. Studies of synchronization phenomena have been reported in so many research of engineering filed: about synchronization in three
coupled van der Pol oscillators with diﬀerent couplingstrength [1], about frustrated synchronization in coupledoscillator chains with unbalanced parametric distribution[2], synchronization-inspired partitioning and hierarchicalclustering [3], about detection of synchronization phenomena in networks of hindmarsh-rose neuronal models [4],and also about synchronization in agents harvesting gamewith regional nature [5], etc. Furthermore, the applicationsof synchronization phenomena have also found in chemical, physical and biological fields: about chaos synchronization of chemical models [6], about synchronization andcoupling analysis of applied cardiovascular physics in sleep
medicine [7], or about multistate and multistage synchronization with excitatory chemical and electrical synapses[8], etc.
In this article, we propose a new type of coupled van derPol oscillator system, and observe its synchronization phenomena by theoretical analysis, computer simulation andcircuit experiment. In the first step, we concern about asystem which contains only 3 or 4 van der Pol oscillators,
then we hope to find a perfect solution of larger coupledsystems.

Figure 1: Circuit model.

dvk
dt
diLk
L
dt
C=−iLk − iRk=−vk − Rn∑

(1)
iLk

(2)

j=1

The v − i characteristics of the nonlinear resistor are expressed by:(iRk=−g1)( )3vkvk+ g3, k = 1, .., n.
βkβk(3)Figure 2 shows the relation between amplitude of the kth oscillator and βk . In this study, we first concern about
3 and 4 coupled oscillators system, assign the value of βk(k = 1, .., n − 1) as 1 by default, and change the value of βn(n = 3 or 4) to observe synchronization phenomena.
3. Theoretical Analysis for n=3
To normalize circuit equations, we use Eqs. (4)-(6) tochange the variables:

2. Circuit ModelIn this arcticle, n > 2 van der Pol oscillators are used
to build the circuit model (Fig. 1). One van der Pol oscillator contains one capacitor (C), one inductor (L), and one
nonlinear resistor.
The circuit equations are described as follows:- 365 -t=vk=

α =

√LCτ
√
√
g1
Cg1
xk , iLk =
yk
3g3
3Lg3
√√
CL, ε = g1
RLC(4)
(5)
(6)

α ∑
ρi sin ϕki
2ρk i=13θ˙kWith the new variables, Eqs. (1)-(3) are normalized as:dxkε 1 2 =xk 1 − 2 x k  − yk(7)dτβk3βk=xk − α3∑y j (k = 1, .., 3)

(8)

j=1

ρ sin ϕ12 − ρ3 sin ϕ31 = 0(21)sin ϕ31 + sin ϕ32 = 0(22)ϕ32 = ϕ12 + ϕ31(23)Actually:

then Eqs. (19)-(22) have two solution.
Solution 1:

where α is the coupling factor and ε is the strength of nonlinearity. We will simulate the above equations later, andnow we first use the averaging method to analyze xk , yk .
We assume that xk , yk can be considered as bellow:
xk (τ) =ρk (τ) cos(τ + θk (τ))(9)

yk (τ) =ρk (τ) sin(τ + θk (τ))(10)

ρ˙k=Mk cos φk − α sin φk

ρi sin φi(11)i=1

θ˙k=−3cos φk ∑Mksin φk − αρkρk

ρi sin φi(12)i=1whereMk=φk=ε1
xk 3f (xk ), f (xk ) = xk −βk3βk 2
τ + θk(13)(14)

By averaging Eqs. (11)-(12) over one period, as averaging method’s theory, there is a positive εr which smallenough that satisfied: for every ε inside (0, εr ), ρk (τ) andθk (τ) can be considered as constant and the values of ρ˙k , θ˙kcan be calculated as:∫ 2π∑1(Mk cos φk − αsin φk 3i=1 ρi sin φi )dφk (15)ρ˙k = 2π0∫ 2π −M sin φφk ∑31kk− α cosθ˙k =i=1 ρi sin φi )dφk (16)2π 0 (ρkρk
We obtain the following equations:()
3ερk 3α∑
ρ˙k =
ρk −−
ρi cos ϕki
2βk
2 i=14βk 2(17)

ρ =
ρ3 =
ϕ31 =

00mπ(24)(25)(26)

ϕ12lπ(27)=Solution 2:

Assign Eqs. (9)-(10) to Eqs. (7)-(8), and we obtain:
3∑(18)

where ϕi j = θi − θ j is the phase diﬀerence between i-thand j-th oscillator for i, j = 1, 2, 3. Because the role of thefirst and the second oscillator are equal in this 3 coupledoscillators system, we can assume that ρ1 = ρ2 = ρ. In
steady state, ρ˙k and θ˙k equal zero, then Eqs. (17)-(18) canbe rewritten as:()ρ2α1ερ 1 −− (ρ + ρcos ϕ12 + ρ3 cos ϕ31 ) = 0 (19)242()
ρ21 εαρ3 1 −− (ρ3 cos ϕ31 +ρ cos ϕ32 +ρ3 ) = 0 (20)2 β34β3 2 2Figure 2: Relation between amplitude and βk .

dykdτ

=

ρ =2(28)ρ3=2β3(29)

ϕ31=

ϕ12=

( β )
3± cos−1 −2( β )32kπ ∓ 2cos−1 −2(30)(31)

where k, m, l are integers. We use Jacobi matrix to concern
about stability of each solution. For the solution 1, the Jacobi matrix contains positive eigenvalues, and it is unstablesolution, but for the solution 2, the Jacobi matrix contains
only negative eigenvalues, consequently it is stable solution. In other words, solution 2 is the theoretical analysis
results of this 3 coupled oscillators system. The graph ofthis theoretical analysis results is shown in Fig. 3.
In this article, we assume that β3 is changed inside [0.2,2] only but actually, when β3 is lager than 2, Eq. (30) is notmathematically correct, and in this case, phase diﬀerence
ϕ31 and ϕ12 are both zero.
4. Synchronization for n = 3
For the case of n = 3, β1 = β2 = 1, Fig. 4 shows threeof computer simulation results obtained by using RungeKutta method when β3 equals 0.1, 1.5 and 2.0. In this- 366 -Figure 3: Theoretical analysis results when n = 3.

figure, there are xi − x j windows show the phase relation
between a pair of oscillators, while the x1 , x2 , x3 windows
show the time waveform of each single oscillator.
The phase diﬀerences are summarized in Fig. 5. By comparing Fig. 3 and Fig. 5, the theoretical analysis results
and computer simulation results agree well. Furthermore,
there is not α in the theoretical analysis results. To explain
this, we can calculate the current of resistor R by calculat∑
ing 3k=1 yk (t). By using Eqs. (28)-(29), we obtain:
yi = 2βi sin(τ + θi ), β1 = β2 = 1(a) β3 = 0.5.

(32)

then:
3
∑yk (τ − θ3 ) = 2 sin(τ − ϕ31 ) + 2 sin(τ + ϕ23 )(33)

k=1+2β3 sin(τ)(b) β3 = 1.5.

Substituting Eqs. (28)-(31) into Eq. (32), we obtain:3∑yk (τ) = 0(34)k=1

this corresponds i(t) = 0 at every value of t. In other words,
the electric current of resistor R is always zero wheneversynchronization phenomena occurs. Figure 6 shows one of
our circuit experimental results. In Fig. 6, we obtain (ϕ12 ,ϕ23 , ϕ31 ) equals (−150◦ , −105◦ , −105◦ ). By comparing
these values to Figs. 4-5 above, it is shown that Fig. 6 expresses for the circuit experimental results when (β1 , β2 , β3 )are about (1, 1, 0.5).
5. Synchronization for n = 4
For the case of n = 4, β1 = β2 = β3 = 1, Fig. 7 shows
the computer simulation results when β4 equals 3.0. In
Fig. 7, the first three oscillators which have the same amplitude reach the in-phase, while each one of them and thefourth oscillator reach the anti-phase synchronized oscillatory state.

- 367 -(c) β3 = 2.0.
Figure 4: Computer simulation results when n = 3.

Figure 5: Computer simulation results when n = 3.
Figure 7: Computer simulation results when n = 4, β4 = 3.
[4] N. Corson, S. Balev and M. Azizalaoui, “Detection ofsynchronization phenomena in networks of hindmarshrose neuronal models”, Proc. of ECCS’10, pp. 112118, Sep. 2010[5] J. Wu and Z. Yang, “Synchronization in agents harvesting game with regional nature”, Proc. of CCDC’10, pp.
3955-3960, May. 2010
Figure 6: Circuit experimental results when n = 3.

[6] N. Vasegh and F. Khellat, “Chaos synchronization ofchemical models”, International Journal of Applied
Science and Technology, vol. 1, no. 5, pp. 233-243,
Sep. 2011.

6. Conclusions
This study mainly concerned about the synchronization
phenomena, especially about phase diﬀerence between theoscillators when the amplitudes are not equal. However,there is slight diﬀerence between computer simulation results and theoretical analysis results when β3 is small, andthe reason of this diﬀerence has not been perfectly explained. In the next step, it is necessary to increase thenumber of ocsillators and complete the theoretical analysis
when n is larger than 4, and it is expected to bring us moreinteresting phenomena.

[7] N. Wessel, “Synchronization and coupling analysis:
Applied cardiovascular physics in sleep medicine”,
Proc. of EMBC’13, pp. 6567-6570, Jul. 2013.
[8] F. Jhou, “Multistate and multistage synchronization ofhindmarsh-rose neurons with excitatory chemical andelectrical synapses”, Proc. of IEEE’12, pp. 1335-1347,
Jan. 2012.

References[1] Y. Uwate, Y. Nishio and R. Stoop, “Synchronization inthree coupled van der Pol oscillators with diﬀerent coupling strength”, Proc. of NCSP’10, pp. 109-112, Mar.
2010.
[2] K. Matsumura, T. Nagai, H. Kumeno, Y. Uwate and Y.
Nishio, “Frustrated synchronization in coupled oscillator chains with unbalanced parametric distribution ”,Proc. of NCSP’12, pp. 679-682, Mar. 2012.
[3] J. Shao, X. He, C. Bohm, Q. Yang and C. Plant,“Synchronization-inspired partitioning and hierarchical clustering”, IEEE Transaction on Knowledge andData Engineering, vol. 25, no. 4, pp. 893-905, Apr.
2003- 368 -