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

Synchronization Phenomena in Diﬀerent Sizes of Rings of Coupled Oscillators
Honami KUME, Yoko UWATE and Yoshifumi NISHIO
Department of Electrical and Electronic Engineering, Tokushima University
2–1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: honami@ee.tokushima-u.ac.jp, uwate@ee.tokushima-u.ac.jp, nishio@ee.tokushima-u.ac.jp
Abstract—In this study, two rings of 5 and 7 van der Pol
oscillators coupled by resistors are researched. These rings
show 5-phase and 7-phase synchronizations when they are
not coupled. Hence, the whole system becomes a frustrated network after coupling. We investigate synchronization phenomena by computer simulations.

1. Introduction
Figure 2: Anti-phase of two oscillators.
System of coupling oscillators is useful as a simple
model that represents the natural phenomena, because
it has a variety of phenomena depending on coupling
method[1][2]. We focus on the “Frustration Phenomena”
occurred by coupling two diﬀerent characteristics.
In this study we prepare two diﬀerent sizes of rings ofcoupled van der Pol oscillators, and we jointed two rings
like n1 and n2 in Fig. 1. In our past study, coupling vander Pol oscillators as a ring show unique phenomenon [3].
Neighboring two oscillators have anti-phase as Fig. 2. The
phase state depends on the number of oscillators in a ring.
Even number of oscillators reach only anti-phase, but odd
number of oscillators reach some states with phase diﬀerences. For example, 5 oscillators coupled as a ring show 5phase synchronization, and 7 oscillators case reach 7-phasesynchronization. If we couple two diﬀerent sizes of rings,
5 oscillators and 7 oscillators, they may aﬀect each otherand frustration occurs. We research synchronization phenomena in the 5-7 rings shown in Fig. 3.

2. Circuit Model
The 5-7 rings have three kinds of coupling types asshown in Fig. 4. We name Type A, B and C. The two oscillators in the lower ring (among oscillators) with no directconnections to the upper ring are called “Free Oscillator”
in this article. In Type A, Free Oscillators are neighboring.
In Type B, there is an oscillator connected to the upper ring
between two Free Oscillators. In Type C, there are twooscillators connected to the upper ring between two FreeOscillators.

Figure 3: Circuit model of 5-7 rings.

Figure 1: Two rings of coupled oscillators.

- 369 -where√

vk =

√

iak =
Figure 4: Three types of coupling methods. Black oscillators are Free Oscillators.

t=

3. Circuit Equations

η=rWe show the circuit equations of Type A. Type B andType C have similar equations to Type A. The circuit equations for Type A are shown below.
−irk − (iak + ibk ) − R1 (vk − v j ): (k = U1 ∼ U5): ( j = L1 ∼ L5)



dvk 
 −irk − (iak + ibk ) + R1 (v j − vk )C=
: (k = L1 ∼ L5)
dt




: ( j = U1 ∼ U5)−i−(i+i)rkakbk: (k = L6, L7)
Ldiak= vk − r(iak + ib(k+1) )dtL

dibk= vk − r(ia(k−1) + ibk )dt

g13g3 xk ,√g1
3g3LCτ,√CL,√CL yak ,ibk =ε = g1
γ = R1

√√

√

g13g3

√CL ybk ,(4)
LC,LCThe parameter γ is a coupling strength between the tworings. We research synchronization phenomena when wechange γ from 0.001 to 1.0. The parameter η shows coupling strength between oscillators in the same ring. The
strength of the nonlinearity is ε. We fix η = 0.1 and ε = 1.0.
4. Simulation Results(1)

Figures 5, 6 and 7 show attractors of Type A, B and C,
respectively, and the horizontal axis is x that means voltage and the vertical axis is y that means current. In all
the figures, the coupling strength between the rings γ arechanged as 0.001, 0.01, 0.1, and 1.0 from the top. For
each case, the horizontally-aligned 5 attractors in the figure correspond to the oscillators in the upper ring, while
the horizontally-aligned 7 attractors correspond to the oscillators in the lower ring. The position of the 5 attractors: (k = U1 ∼ U5, L1 ∼ L5)
We assume that the characteristics of the nonlinear resistors
are described by the following equation.
irk = −g1 vk + g3 v3k(2)We normalize the above equations and they are shown below.

ε(xk − 13 xk3 ) − (yak + ybk ) − γ(xk − x j ): (k = U1 ∼ U5)





: ( j = L1 ∼ L5)




dxk 
 ε(xk − 31 xk3 ) − (yak + ybk ) + γ(x j − xk )=: (k = L1 ∼ L5)
dτ




:( j = U1 ∼ U5)


1 3
ε(x−x)−(ykak + ybk )3 k: (k = L6, L7)

(3)

dyak
= xk − η(yak + yb(k+1) )dτdybk= xk − η(ya(k−1) + ybk )dτFigure 5: Attractors of Type A.

: (k = U1 ∼ U5, L1 ∼ L5)- 370 -

Figure 7: Attractors of Type C.

Figure 6: Attractors of Type A.

corresponds to the connections between the rings.
For γ = 0.001, the 5 attractors are the same size for theall types of the connections as shown in Figs. 5, 6 and 7.
This is because the upper 5 oscillators are not concerned
from lower 7 oscillators.
For γ = 0.01, the amplitudes of attractors keep changing. Because one ring is concerned by the other ring, andfrustration between 5-phase and 7-phase occurs.
For γ = 0.1, all oscillators synchronize except the oscillator between the Free Oscillators in Type B. Note that theamplitudes of the two oscillators between the Free Oscillators in Type C become small. These are the eﬀect of thefrustration.
For γ = 1.0, the amplitudes of the oscillators become
smaller by the frustrations and we can see that the eﬀect
depends on the position of the oscillators.
Figure 8 shows the synchronization between the tworings. From Fig. 8(a), we can understand that xU1 and xL1
become to synchronize as γ increases. From Fig. 8(b), wecan see that the oscillators with strong frustrations become
to stop oscillating.
Figure 9 shows characteristics of average phase diﬀerences from xU1 . Upper 5 oscillators of all types exhibit 5phase synchronization for γ = 0.001. In Type A, the phasebecome a change from 5-phase but not reach anti-phase.
As γ increases, Type B and Type C becomes anti-phasesynchronization, although Type A does not show any synchronization.

5. Conclusions
In this study, we have researched synchronization phenomena observed from two rings with diﬀerent sizes ofcoupled van der Pol oscillators. We observed attractors
and phase relationship by computer simulations. We could
observe frustration phenomena between the rings and obtained diﬀerent results from three coupling method. In future research, we carry out circuit experiments and collectdata of many cases from computer simulations.
References[1] S. Yamane, T. Imoto and M. Yamauchi, “Investigation and Application of the Phenomena in Coupled
Oscillators as a Lattice”, Technical Report of IEICE,NLP107(400), pp.51-56, Jan. 2008.
[2] K. Ochiai and K. Shimizu, “Experimental Study ofa Switching Phenomenon in Two Coupled Hard-Type
Oscillators”, Technical Report of IEICE, NLP113(69),pp.7-11, May. 2013.
[3] Y. Setou, Y. Nishio and A. Ushida, “Synchronization
Phenomena in Oscillators Coupled by Resistors as aRing”, Technical Report of IEICE, NLP93-89, pp.3137, Mar. 1994.

- 371 -

(a)(b)Figure 8: Synchronization between the two rings for diﬀerent γ. (a) xU1 − xL1 . (b) xU5 − xL5 for Type A and xU5 − xL6 forTypes B and C.

Figure 9: Average phase for diﬀerent γ.

- 372 -