IEEE Workshop on Nonlinear Circuit Networks
December 12-13, 2014

On Five Coupled van der Pol Oscillatorswith Different Oscillation Amplitudes
Vu Minh Thuan Yoko Uwate Yoshifumi Nishio
Dept. Electrical and Electronic Eng., Tokushima University
Email: {thuan,uwate,nishio}@ee.tokushima-u.ac.jp
Abstract—This article gives an explanation of the synchronization phenomena in a circuit which contains five van der Pol
oscillators when some of them have different amplitudes fromthe others.

I. I NTRODUCTION
We are living in the world where there are so many
example of synchronization: firefly luminescence, cry of birds
and frogs, human applause, etc. Studies of synchronization
phenomena have been reported in so many research of engineering filed: about synchronization in three coupled vander Pol oscillators with different coupling strength [1], about
frustrated synchronization in coupled oscillator chains with unbalanced parametric distribution [2], synchronization-inspired
partitioning and hierarchical clustering [3], about detectionof synchronization phenomena in networks of hindmarsh-rose
neuronal models [4], and also about synchronization in agents
harvesting game with regional nature [5], etc. Furthermore, theapplications of synchronization phenomena have also found inchemical, 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 the past research, we have proposed a new type of
coupled van der Pol oscillators system and discovered thatin 3 coupled oscillators system, if we changed the amplitudeof one oscillator to some appropriate values, we can observe
the synchronization phenomena by computer simulation, andproved these results by both averaging method and circuit
experimental results [9]. For 4 coupled oscillators system, wehave changed the amplitude of not only 1 but also 2 and 3oscillators and introduced the computer simulation results ofsynchronization phenomena in [10].
In this article, in order to expand the number of oscillators,
we give some computer simulation results about synchronization phenomena of 5 coupled van der Pol oscillators systemand compare them to obtained results in [9] and [10].
II. C IRCUIT M ODEL [9][10]In our study, 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 non-linear resistor.
The circuit equations are described as follows:

Fig. 1. Circuit model.

Fig. 2. Relation between amplitude and βk .

dvk
dt
diLkL
dt
C

= −iLk − iRk= −vk − Rn∑

(1)
iLj

(2)

j=1

The v − i characteristics of the nonlinear resistor areexpressed by:( )( )3vkvkiRk = −g1+ g3, k = 1, .., n. (3)βkβkFigure 2 shows the relation between amplitude of the k-thoscillator and βk . Generally, βk = 1 mean that the amplitudeof k-th oscillator is kept in the normal situation, while in theother cases, the amplitude of k-th oscillator is changed.
To normalize circuit equations, we use Eqs. (4)-(6) tochange the variables:√t =LCτ
(4)
√√g1
Cg1
vk =xk , iLk =yk
(5)
3g3
3Lg3

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Fig. 3. Phase difference result when n = 3.

√CL, ε = g1
α = RLC
With the new variables, Eqs. (1)-(3) are normalized as:)(dxkε1 2=xk 1 − 2 x k − ykdτβk3βkn∑dyk= xk − αyj (k = 1, .., n)
dτ
j=1
√

(6)

(7)

(a) (β1 , β2 , β3 , β4 , β5 )=(1.00,1.00,.0.33,0.33,0.33)(8)where α is the coupling factor and ε is the strength ofnonlinearity.
III. S YNCHRONIZATION FOR n=3 [9]As the results of our past article [9], in 3 coupled oscillators
system, if we change only 3rd oscillator’s amplitude (in otherwords, we change only β3 while keeping β1 = β2 = 1), we
can obtain the result about phase difference as shown in Fig. 3.
This result was obtained by both computer simulation method,
theoretical analysis method and was proved by some circuit
experimentals. We will use this to compare with 5 coupledoscillators system.

(b) (β1 , β2 , β3 , β4 , β5 )=(1.00,1.00,.0.66,0.66,0.66)

IV. S YNCHRONIZATION FOR n = 5In 5 oscillators system, we first divide them to 2 groupsof amplitude: β1 = β2 = 1 and β3 = β4 = β5 are changed
and obtain computer simulation results as shown in Fig. 4.
In Fig. 4(a), when the oscillator 3rd, 4th, and 5th have same
phase and their sum can be concerned as one oscillator with 3times bigger of amplitude. By this way, we can concerned thesystem as 3 oscillators system which have 3 same amplitudes,
and these obtained results are similar to Fig. 3 when β3 = 1.
Similarly, the phase differences shown in Fig. 4(b) are samewith the phase difference shown in Fig. 3 when β3 = 2: 1stand 2nd oscillator are in-phase synchronization, and each oneof them is anti-phase synchronization with 3rd, 4th, or 5thoscillator.
When we divide the system to 3 groups of amplitude:
β1 = 1, β2 = β3 and β4 = β5 , and change both of β3
and β5 separately, we obtain the computer simulation results
as shown in Figs. 5. The phase relation in Fig. 5(a), Fig.
5(b) can be compared with the phase relation in Fig. 3 whenβ3 equals 1, 2, respectively. For more details, Fig. 6 shows
when synchronization phenomena occurs. In Fig. 6, the stepof changing β3 , β5 is 0.25. Currently, mathematical difficulty- 103 -(c) (β1 , β2 , β3 , β4 , β5 )=(1.00,1.00,.1.50,1.50,1.50)
Fig. 4. Simulation results when dividing oscillators into 2 group.

(a) (β1 , β2 , β3 , β4 , β5 )=(1.0,0.5,.0.5,0.5,0.5)
Fig. 6. Distribution map of synchronization between phase 1-2 when n = 5.

V. C ONCLUSIONSAs development and extending of our past results reported
in [9] and [10], in this article we have concerned about thesynchronization phenomena in 5 coupled van der Pol oscillators system, when a part of them have different amplitudes. In
the next step, it is necessary to complete theoretical analysis
and circuit experiment to perfectly verify obtained results in
5 coupled oscillators system.
R EFERENCES[1] Y. Uwate, Y. Nishio and R. Stoop, “Synchronization in three coupled vander 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, “Frustratedsynchronization 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-inspiredpartitioning and hierarchical clustering”, IEEE Transaction on Knowledge
and Data Engineering, vol. 25, no. 4, pp. 893-905, Apr. 2003.
[4] N. Corson, S. Balev and M. Azizalaoui, “Detection of synchronization
phenomena in networks of hindmarsh-rose neuronal models”, Proc. ofECCS’10, pp. 112-118, Sep. 2010.
[5] J. Wu and Z. Yang, “Synchronization in agents harvesting game withregional nature”, Proc. of CCDC’10, pp. 3955-3960, May. 2010.
[6] N. Vasegh and F. Khellat, “Chaos synchronization of chemical models”,
International Journal of Applied Science and Technology, vol. 1, no. 5,
pp. 233-243, Sep. 2011.
[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 of hindmarsh-rose
neurons with excitatory chemical and electrical synapses”, Proc. ofIEEE’12, pp. 1335-1347, Jan. 2012.
[9] Vu Minh Thuan, Y. Uwate and Y. Nishio, “Synchronization phenomena
in van der Pol oscillators with different amplitudes”, Proc. of NOLTA’14,
pp. 365-368, Sep. 2014.
[10] Vu Minh Thuan, Y. Uwate and Y. Nishio, “Synchronization of Coupled
van der Pol Oscillators by Changing their Oscillation Amplitudes”,
ISCAS’15 (being submitted).

(b) (β1 , β2 , β3 , β4 , β5 )=(1.0,0.5,0.5,1.0,1.0)(c) (β1 , β2 , β3 , β4 , β5 )=(1.0,1.0,1.0,1.5,1.5)
Fig. 5. Simulation results when dividing oscillators into 3 group.

does not allow us to prove theoretically our results. However,
the obtained results in this article have some similarities withthe proved results in [9], so we can consider that they arecredible.
As an idea, in larger coupled system, by dividing them to2 or 3 groups we can estimate phase difference between eachphase of oscillators.

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