2009 International Workshop on Nonlinear Circuits and Signal Processing
NCSP'09, Waikiki, Hawaii, March 1-3, 2009

Synchronous Oscillation of Resonator-Coupling Oscillators
Chikayasu Higashi† and Yoshifumi Nishio‡†Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone/FAX:+81-88-656-7470
Email: higashi@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 phase states andthere have been many investigations on these phenomenon.
In previous study, we confirmed the basic synchronization of
three oscillators with the same natural frequencies coupled
by a resonator. In this study, we investigate oscillation frequency of resonator-coupling three oscillators to carried out
the sychronization phenomenon in detail.
1. Introduction‡Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone/FAX:+81-88-656-7470
Email: nishio@ee.tokushima-u.ac.jp(for example, when their system take 479,001,600 steadystates when N = 13.), would be structural element of cellular neural network or may be used as an extremely large
memory.
In our previous stady, we observed synchronization of threeoscillators coupled by a resonator. Resonator was consistedof parallel circuit of a capacitor and an inductor. We observed in-phase oscillation and two types of three-phase oscillations from the coupled oscillators by using the Runge-Kutta
method [9].
In this study, we investigate frequency characteristics of thesynchronized oscillations and distorted waveforms by using
SPICE in detail.

In our surroundings, there are a lot of synchronous phe2. Circuit Modelnomena. It is not unusual at all. For example, synchronus luminescence of firefly group is widely known for synchronous
phenomena. When you observe the firefly group, the intervalof firefly luminescence is gradually becomes the equal interval. However, this mechanism has not been clarified in detail.
In a familiar point, the activity of our brain is a synchronousir (v1)ir (v2 )ir (v3 )phenomenon. We feel, think about something, at the moment,
the hundreds million of neuron of the brain synchronizes andCCC
exchanges pulses. In the others, cell of heart producing pulses
at equal intervals and revolution of the moon etc. are good exNR
NRNRv2v3v1
amples in which a synchronous phenomenon is comprehensible. Similarly, synchronization is common phenomenon inLLL
the field of natural science. However, it is deﬃcult to clarii
ii
2
3
1
fied the mechanism of synchronous phenomena in natural science. We believe that investigating simpler synchronous phei Lc vCc NR : Nonlinear Resistor
nomena (ex.synchronous oscillators) is the key that arrives atsynchronous of firefly group and more complex synchronousLcCcphenomenon.
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
Figure 1: Circuit model.
give N-phase oscillations. Their system can take (N − 1)!
phase states, because of their system tends to minimize thecurrent through the coupling resistor [7][8]. They thought
The circuit model is shown in Fig. 1. Three oscillatorsthat these coupling structure and huge number of steady states- 297 -with the same natural frequencies are mutually coupled by aresonator (LC CC circuit). The circuit equations are describedas Eq. (1).
dvk
C

dt



dik



L


dt



dvCc

Cc
dt
di Lc Lcdt====

the normalized circuit equations are given as follows.
ẋkẏkẊ Ẏ

−ik − ir (vk )vk − vCc(k = 1, 2, 3)(1)∑3

j=1 i j − iLc=−yk + ε(xk − xk3 )=xk − X(k = 1, 2, 3)

=β(∑)

=

γX

3
j=1 y j − Y(4)3. Basic Synchronization Phenomena

vCc

where ir (vk ) indicates the v − i characteristics of the nonlinear

1k Ω

Basic synchronization phenomena are shown in Fig. 3.
We can observe three patterns of oscillations for the same
parameter; in-phase synchronization and two types of threephase synchronizations. The observed synchronous patterns
depend on the initial states. Also the circuit experimental results show similar phenomena to the numerical results.

31v12v2v31k Ωv1v2rv3v1

GNDv2v3

Figure 2: Nonlinear resistor.

(a)
(b)

resistor, which is approximated by Eq. (2).
ir (vk ) = −g1 vk + g3 v3k .

(2)For circuit experiments, the nonlinear resistor is realized asshown in Fig. 2. Note that when r is small, the nonlinearity
is strong. By using the following variables and parameters,√√gCg11vk =x k , ik =
yk ,
gLg33√√
gCg11
X, iLc =Y,
 vCc =g3
Lg3


√d


t = LC τ, “ · ” =,
dτ




√



LCL

 ε=
g1 , β =, γ=,
C
Cc
Lc

(3)

Figure 3: (a)Time waveform of in-phase oscillation andtwo types of three-phase oscillations (numerical results).
ε=β=γ=1.0. (b)Time waveform of in-phase oscillationand two 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.

To investigate the synchronization phenomenon when theparameters of the coupling resonator are changed, one of theparameters are fixed to 1.0, and the other parameter (β orγ) is changed. First, the parameter β is fixed to 1, and theparameter γ is changed from 0.3 to 2.7. For any values ofγ, the three patterns of synchronization; in-phase oscillationand two types of three-phase oscillations, are able to be confirmed. We should note that the oscillation frequency of thethree-phase oscillations is almost the same for diﬀerent γ. On
the other hand, the frequency of the in-phase oscillation increases as γ increases.

- 298 -

Secondly, the parameter γ is fixed to 1, and the parameter β
is changed from 0.3 to 2.7. We can observe the three patternsof synchronization as well as the previous case. However, in
this case, when β was changed, the oscillation frequency ofeither in-phase oscillation or three-phase oscillation does notchange.

Frequency[rad/sec]

100806040

βbeta

0γgammav1

v1v2v2200v3v3

180v1v1v2v2v3v3v1v1v2v2v3v3

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3(a)Frequency[rad/sec]

5. Analysis of distorted waveforms
Also, we observed distorted waveforms of synchronizations for some parameter values in SPICE. The distorted
oscillations obtained by using the Runge-Kutta method areshown in Fig. 5 for the reference. The distorted oscillations
obtained by using SPICE are shown in Figs. 6 and 7. The frequency characteristics of the distorted oscillations obtainedby using FFT of SPICE are shown in Figs. 8 and 9. We confirmed that the distortion comes from the third-order harmonics of the oscillations.

120

20

fluenced to the oscillation frequencies than the value of Cc .
We think the reason of this result is that L of the oscillator isconnected to the resonator.

160
140120

βbata

100

γ

gamma

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3(b)(a)

Figure 4: The change of the frequency for changing theparameters. (a)The frequency of the in-phase oscillation.
(b)The frequency of the three-phase oscillation. Horizontalaxis is the change β or γ.

(b)

Figure 5: Distorted oscillation by Runge-Kutta method for
ε=γ=1.0. (a) β=0.3. (b) β=2.1.

6. Conclusions
4. Frequency Analysis by SPICE

In this study, we have investigated frequency characteristics of the synchronized oscillations in three oscillators coupled by a resonator by using SPICE. Further, detailed analysisof the distorted waveforms was carried out.
Our future work is to investigate synchronous oscillatorswith various circuits in more detail.

Because of the detailed oscillation frequency is not obtained by the Runge-Kutta method, we analyzed the oscillation frequency by using SPICE. We investigated the change
in the oscillation frequency as changing β or γ. The change
in oscillation frequency of the in-phase oscillation and thethree-phase oscillation are shown in Fig. 4. Horizontal axis
Acknowledgments
is the changing β or γ. When β is changed, γ is fixed to 1.0.
Also, When γ is expanded, β is fixed to 1.0. The oscillation
This work was partly supported by Yazaki Memorial Foundafrequency of the in-phase oscillation increases as γ increases.
tion for Science and Technology.
On the other hands, there are no remarkable change in otheroscillation frequencies. The value of Lc is more strongly in-- 299 -Figure 6: The distorted in-phase oscillation by SPICE for
ε=γ=1.0, β=2.1.

Figure 8: The frequency of the distorted in-phase oscillation
by SPICE for ε=γ=1.0, β=2.1.

Figure 7: The distorted three-phase oscillation by SPICE for
ε=γ=1.0, β=0.3.

Figure 9: The frequency of the distorted three-phase oscillation by SPICE for ε=γ=1.0, β=0.3.

tems: mechanisms and functions,” J. Computational
Neuroscience, vol.1, pp.11-38, 1994.

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