2017 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing (NCSP'17)
Guam, USA, February 28th to March 3rd, 2017

Synchronization Phenomena in Complex Networks of van der Pol Oscillators
Kazuki Ueta, 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, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract

2. System model

Synchronization phenomena of frustration network by coupled oscillators has been studied in a wide range of ﬁelds,such as medicine and engineering. It is investigated towards
various systems up to now. However, analysis of regarding
more complex systems are little. In our study, we developed
the system model so that a basic minimum unit even in more
complex systems. In addition, we observed synchronization
phenomena about the system.

Figure 2 shows a system model constituted van der Pol oscillators (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
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 andfrogs, applause of many people and so on. Synchronization
phenomena have a feature that the set of small power canproduce very big power by synchronizing at a time. Therefore study of synchronization phenomena have been widelyreported not only engineering but also the physical and biological ﬁelds[1]–[8]. Investigation of coupled oscillators attention from many researchers because coupled oscillatory
network produces interesting phase synchronization such as
the phase propagation wave, clustering and complex patterns.
In this study, we focus on the synchronization phenomena
coupled by van der Pol oscillators containing ring and starstructures. Then, we observe the synchronization phenomena
with computer simulation. van der Pol oscillator is shown inFig. 1.

  





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:

Figure 1: van del pol oscillator.



ikg = −g1 vk + g3 v3k ,





(1)

(g1 , g3 > 0),(k = 1, 2, 3, ..., N),
The characteristic of ring coupling has in-phase, anti-phaseand N-phase. The characteristic of star coupling has in-phaseand anti-phase.

- 285 -First, the circuit equations of VDP-A are given as follows:⎧N⎪1dvk⎪⎪⎪−i+−vn ),=−i((N−1)vC
⎪
n1gk⎪
⎪dtR
⎨
n=2(2)
⎪⎪
⎪
⎪
di1⎪⎪
⎪
= vk ,⎩ Ldt(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 = 4 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
follows:
⎧dvn⎪⎪⎪= −ikg − inR − ina − inb ,C⎪⎪⎪dt⎪⎪⎪⎪dina⎨(3)2L= vn − R0 (ina + i(n+1)b ),
⎪⎪
⎪
dt⎪
⎪⎪
⎪
⎪
dinb

⎪
⎪
⎩ 2L= vn − R0 (inb + i(n−1)a ).
dt
By using the parameters and variables as follows:



g1C
g1C
g1C
in =yc , ina =yn , inb =zn ,
3g3 L
3g3 L
3g3 L


g1
g1


xc , v n =xn ,vk =3g3
3g3(4) Figure 3: Attractor between adjacent oscillators (horizontal
√Ldaxis:xk , vertical axis:yk ) (k = 1, 2, 3, 4, 5).
t = LCτ, “ · ” =, α = g1,dτC
C1 Lβ=, γ = R0,R CL


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:
⎧N

⎪
⎪
1
⎪⎪
xn ,
⎨ x˙c = αxc 1 − xc2 −yc + β (N − 1)xc −3⎪⎪n=2⎪⎪⎩ y˙ = x .
cc  (5)   

The normalized circuit equations of VDP-B are given as follows:⎧1⎪⎪x˙n = αxn (1 − xn2 ) − yn − zn − β(xc − xn ),⎪⎪
⎪
3⎪⎪
⎪
1
⎨
{x=−γ(yy˙⎪
nnn + zn+1 )},
⎪⎪
2
⎪
⎪
⎪
1⎪
⎪
⎩ z˙n = {xn − γ(yn + zn−1 )}.
2 
Figure 4: System model of N = 4.

(6)
Next, the time waveforms of the voltage of each capacitor
C after suﬃcient time has elapsed are shown in Fig. 5. Andthe phase diﬀerences between the adjacent oscillator of thiscase is equal to the result as shown in Fig. 6. It seems that- 286 -two electric currents of VDP-B are piled up as we understandit from the ﬁgures like 3 phase.





















Figure 5: The Time waveforms of the each oscillator for N =4.
(a) β=0.001.












(b) β=0.0085.





Figure 6: Lissajous ﬁgures of N = 4.

Second, the simulation results of the system model containing six circuits are shown in Fig. 7. The value of theparameters are ﬁxed with β=0.001, 0.0085 and 0.05. In the(c) β=0.05.
case of β=0.001, 5 phase synchronization appeared because
the coupling strength of VDP-A is weak. The current of theVDP-A and one of the current of the VDP-B are in phase Figure 7: Simulation Results for N=6 (α = 0.1 and γ = 0.02). timeat that time. Therefore, we assume 5 phase synchronization. waveform. Red and other colors denote x1 and xn respectively.(n =When the value of β sets 0.0085, some time waveforms of 2,3,...,6)
VDP-B come close on in-phase synchronization. When we
increase coupling strength, most of two electric currents ofVDP-B are piled up as we understand it from the Lissajous
Finaly, we summarize the simulation results in Fig. 8. Inﬁgures.
the ﬁgure, we show the results when we increase the circuit
In the case of β=0.05, 5 phase synchronization become in- numbers N = 3, 4, ..., 7. The phase diﬀerence is based on one
phase synchronization. And then VDP-A becomes anti-phase voltage waveform of VDP-B. The broken line in the ﬁgure
synchronization with VDP-B by increasing the value of β. represent asynchronous. The solid line in the ﬁgure repreWe understand that VDP-B synchronizes even if we read ei- sent synchronous. From this result, It turns out that an eventher ﬁgure.
number circuits become in-phase as increasing the couplingstrength. Similarly, It turns out that an even number circuits- 287 -become in-phase as increasing the coupling strength. However, we could’t conﬁrm the synchronization state between
β=0 and β=0.016 in an odd number circuits.

[5] L. M. Pecora and T. L. Carrol, “Synchronization in
Chaotic Systems”, Physical Review Letters, vol. 64,
pp. 821-824, 1990.
[6] K. Kaneko, “Spatiotemporal Intermittency in Coupled
Map Lattices”, Progress of Theoretical Physics, vol. 74,
No. 5, pp. 1033-1044, 1985.
[7] K. Kaneko, “Clustering, Coding, Switching, Hierarchical Ordering, and Control in a Network of Chaotic Elements”, Physica D, vol. 41, pp. 137-172, 1990.
[8] M. Shinriki, M. Yamamoto and S. Mori, “MultimodeOscillations in a Modiﬁed van der Pol Oscillator Containing a Positive Nonlinear Conductance”, Proc. IEEE,vol. 69, pp. 394-395, 1981.

Figure 8: Relationship between coupling strength and phase diﬀerence.

4. Conclusion
In this study, we have proposed a system model using six
circuits that is combined the ring and star structures. We
have observed the synchronization phenomena by increasing
the coupling strength of ring. When the coupling strengthis suﬃciently small, system model becomes like function ofring coupling therefore, 5 phase synchronization can be observed. By increasing the coupling strength, time wave forms
of VDP-B have come close in-phase synchronization. Whenthe coupling strength is suﬃciently large, time wave forms of
VDP-A and VDP-B become anti-phase synchronization. In
the future, we investigate synchronization phenomena using
other circuits.

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