Synchronization Phenomena inThree Oscillators Coupled by a Resonator
Chikayasu HIGASHI and Yoshifumi NISHIO
Department of Electrical & Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506 JAPAN
Email: { higashi, nishio }@ee.tokushima-u.ac.jp
Abstract
Synchronization is common phenomenon in the field of
natural science. It should be noted that mutual synchronization phenomenon of oscillators gives various phasestates and there have been many investigations on these
phenomenon. In this study, we investigate synchronization
phenomena observed from three oscillators with the same
natural frequencies coupled by a resonator.

ir (v1)ir (v2 )ir (v3 )

CNR

v1v2

NR

L

In our surroundings, there are a lot of synchronous phenomena. The synchronous luminescence of firefly group,
cell of heart producing pulses at equal intervals and revolution of the moon etc. are examples in which a synchronous
phenomenon is comprehensible. Similarly, synchronization is common phenomenon in the field of natural science.
There have been many investigations of the mutual synchronization of oscillators ([1]-[6] and therein). Moro andone of the authors have confirmed that N oscillators withsame natural frequencies mutually coupled by one resistor
give N-phase oscillations. Their system can take (N − 1)!
phase states, because of their system tends to minimize thecurrent through the coupling resister [7][8]. They thoughtthat these coupling structure and huge number of steadystates (for example, when their system take 479,001,600
steady states when N = 13.), would be structural elementof cellular neural network or may be used as an extremely
large memory.
In this study, we observe synchronization of three oscillators coupled by a resonator. Resonator is consisted ofparallel circuit of a capacitor and an inductor. Computer
simulations and circuit experiments are carried out to investigate the phenomenon in detail.

NRv3LLi2

i1

1. Introduction

CC

i3vCci Lc
Lc

NR : Nonlinear Resistor

Cc

Figure 1: Circuit model.

scribed as Eq. (1).
dvk


C

dt


dikLdt
dvCcCc
dtdi Lc Lcdt= −ik − ir (vk )= vk − vCc=(k = 1, 2, 3)∑3(1)j=1 i j − iLc= vCc

where ir (vk ) indicates the v − i characteristics of the nonlinear resistor, which is approximated by Eq. (2).
2. Circuit Model
The circuit model is shown in Fig. 1. Three oscillatorswith the same natural frequencies are mutually coupled bya resonator (LC CC circuit). The circuit equations are de-ir (vk ) = −g1 vk + g3 v3k .

(2)For circuit experiments, the nonlinear resister is realized asshown in Fig. 2. Note that when r is small, the nonlinearity- 35 -

1k Ωv1v2

3
12v3
v1

1k Ωv2

rv3
GND

v1

Figure 2: Nonlinear resister.

v2v3

is strong. By using the following variables and parameters,
√

√


gCg1
1



vk =
xk , ik =
yk ,



g
Lg3
3




√


√



g
Cg1
1



X, iLc =Y, vCc =g3
Lg3
(3)


√d


t = LC τ, “ · ” =,dτ√
LCL =g1 , β =, γ=,CCc
Lc
the normalized circuit equations are given as follows.
ẋk = −yk + (xk − xk3 )








ẏk = xk − X(k = 1, 2, 3)


(4)
(∑)


3

Ẋ
=βy−Y

j=1 j Ẏ = γX
3. Synchronization Phenomena for LC = L and CC = C
As shown in Fig. 3, we can observe three patterns of oscillations for the same parameter; in-phase oscillation andtwo types of three-phase oscillations. Which synchronous
pattern was seen depends on the initial states. It is interesting that three patterns can be observed for the same parameter and the oscillation frequency depends on the synchronization patterns. Also, as Fig. 4, the circuit experimentalresults show similar phenomena to the numerical results.
4. Synchronization Phenomena for LC , L or CC , C
To investigate the synchronization phenomenon when
the parameters of the coupling resonator are changed, oneof the parameters are fixed to 1.0, and the other parameter(β or γ) is changed.

Figure 3: Time waveform of in-phase oscillation andtwo types of three-phase oscillations (numerical results).
=β=γ=1.0.

4.1. Synchronization when γ is changedFirst, the parameter β is fixed to 1, and the parameter γ ischanged from 0.3 to 2.7. Three typical examples are shownin Figs. 5(a), (b) and (c). For any values of γ, the threepatterns of synchronization; in-phase oscillation and twotypes of three-phase oscillations, are able to be confirmed.
We should note that the oscillation frequency of the threephase oscillations is almost the same for diﬀerent γ. On
the other hand, the frequency of the in-phase oscillation
decreases as γ decreases.

4.2. Synchronization when β is changed
Secondly, the parameter γ is fixed to 1, and the parameter β is changed from 0.3 to 2.7. Two typical examples
are shown in Figs. 6(a) and (b). We can observe the threepatterns of synchronization as well as the previous case.
However, in this case, when β was changed, the oscillation
frequency of either in-phase oscillation or three-phase oscillation does not change.
Further, as shown in Figs. 7(a) and (b), we observedsome strange waveforms for some parameter values. ForFig. 7(a), the waveform of the three-phase oscillations becomes distorted, while for Fig. 7(b) the waveform of thein-phase oscillation becomes distorted.
It is very interesting that a variety of synchronization
phenomena can be seen for diﬀerent coupling parametervalues.

- 36 -5. Conclusion
In this study, we have investigated the synchronization
phenomena in three oscillators coupled by a resonator.
From the coupled oscillators, we observed the in-phase oscillation and two types of three-phase oscillations. Moreover, we investigated the dependences of the oscillation
frequencies on the coupling parameter values. Our future
work is to investigate these synchronization phenomena inmore detail.
Acknowledgments
This work was partly supported by Yazaki Memorial Foundation for Science and Technology.
References[1] L.L. Bonilla, C.J. Perez Vicente and R. Spigler,“Timeperiodic phases in populations of nonlinearly
coupled oscillators with bimodal frequency distributions,” Physica D: Nonlinear Phenomena, vol.113,no.1, pp.79-97, Feb. 1998.

(a)[2] J.A. Sherratt, “Invading wave fronts and their oscillatory wakes are linked by a modulated traveling phase
resetting wave,” Physica D: Nonlinear Phenomena,
vol.117, no.1-4, pp.145-166, June 1998.
[3] G. Abramson, V.M. Kenkre and A.R. Bishop, “Analytic solutions for nonlinear waves in coupled reacting systems,” Physica A: Statistical Mechanics andits Applications, vol.305, no.3-4, pp.427-436, Mar.
2002.

(b)

[4] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer,“Synchronization and graph topology,” Int. J. Bifurcation and Chaos, vol.15, no.11, pp.3423-3433, Nov.
2005.
[5] C.M. Gray, “Synchronous oscillations in neural systems: mechanisms and functions,” J. Computational
Neuroscience, vol.1, pp.11-38, 1994.
[6] J. Cosp, J. Madrenas, E. Alarcon, E. Vidal and G. Villar, “Synchronization of nonlinear electronic oscillators for neural computation,” IEEE Trans. Neural Networks, vol.15, no.5, pp.1315-1327, Sep. 2004.
[7] Y. Nishio and S. Mori, “Mutually Coupled Oscillators with an Extremely Large Number of SteadyStates,” Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS’92), vol. 2, pp.
819-822, May 1992.

(c)
Figure 4: Time waveform of in-phase oscillation andtwo types of three-phase oscillations (experimental results). L=Lc =10mH, C=Cc =68nF and r=250Ω. Horizontal
scale: 50µs/div. and Vertical scale: 1.0V/div.

[8] S. Moro, Y. Nishio and S. Mori, “Synchronization
Phenomena in Oscillators Coupled by One Resistor,”
IEICE Transactions on Fundamentals, vol. E78-A,
no. 2, pp. 244-253, Feb. 1995.

- 37 -v1v2v3v1v2v3v1v1v2v2v3v3(a)v1v1v2v2v3v3

β＝2.7, γ=1.0v1v1v2v2v3v3(a)v1v1v2v2v3

β＝1.0, γ=2.7v3(b)v1Figure 6: Numerical results for CC , C.
(a) β=2.7. (b) β=0.6.

v2

=γ=1.0.

v3v1v1v2v2v3v3(b)v1v1v2v2v3

β＝1.0, γ=0.9v3v1v1v2v2v3v3(a)v1v1v2v2v3v3

β＝2.1, γ=1.0(c)v1Figure 5: Numerical results for LC , L.
(a) γ=2.7. (b) γ=1.2. (c) γ=0.3.

=β=1.0.

v2v3v1v2v3(b)Figure 7: Distorted waveforms observed for CC , C.
=γ=1.0. (a) β=2.1. (b) β=0.3.

- 38 -